Sugawara Operators for Classical Lie Algebras

Mathematical Surveys and Monographs Volume 229

Alexander Molev Sugawara Operators for Classical Lie Algebras

Mathematical Surveys and Monographs Volume 229

Sugawara Operators for Classical Lie Algebras

Alexander Molev EDITORIAL COMMITTEE Robert Guralnick Benjamin Sudakov Michael A. Singer, Chair Constantin Teleman MichaelI.Weinstein

2010 Mathematics Subject Classiﬁcation. Primary 17B35, 17B63, 17B67, 17B69, 16S30.

For additional information and updates on this book, visit www.ams.org/bookpages/surv-229

Library of Congress Cataloging-in-Publication Data Names: Molev, Alexander, 1961- author. Title: Sugawara operators for classical Lie algebras / Alexander Molev. Description: Providence, Rhode Island : American Mathematical Society, [2018] | Series: Mathe- matical surveys and monographs ; volume 229 | Includes bibliographical references and index. Identifiers: LCCN 2017041529 | ISBN 9781470436599 (alk. paper) Subjects: LCSH: Lie algebras. | Affine algebraic groups. | Kac-Moody algebras. | AMS: Nonas- sociative rings and algebras – Lie algebras and Lie superalgebras – Universal enveloping (super)algebras. msc | Nonassociative rings and algebras – Lie algebras and Lie superalgebras – Poisson algebras. msc | Nonassociative rings and algebras – Lie algebras and Lie superalgebras – Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras. msc | Nonas- sociative rings and algebras – Lie algebras and Lie superalgebras – Vertex operators; vertex operator algebras and related structures. msc | Associative rings and algebras – Rings and algebras arising under various constructions – Universal enveloping algebras of Lie algebras. msc Classification: LCC QA252.3 .M6495 2018 | DDC 512/.482–dc23 LC record available at https://lccn.loc.gov/2017041529

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Contents

Preface xi Chapter 1. Idempotents and traces 1 1.1. Primitive idempotents for the symmetric group 1 1.2. Primitive idempotents for the Brauer algebra 6 1.3. Traces on the Brauer algebra 14 1.4. Tensor notation 17 1.5. Action of the symmetric group and the Brauer algebra 19 1.6. Bibliographical notes 21 Chapter 2. Invariants of symmetric algebras 23 2.1. Invariants in type A 23 2.2. Invariants in types B,C and D 28 2.3. Symmetrizer and extremal projector 39 2.4. Bibliographical notes 41 Chapter 3. Manin matrices 43 3.1. Deﬁnition and basic properties 43 3.2. Identities and invertibility 45 3.3. Bibliographical notes 51

Chapter 4. Casimir elements for glN 53 4.1. Matrix presentations of simple Lie algebras 53 4.2. Harish-Chandra isomorphism 55 4.3. Factorial Schur polynomials 58 4.4. Schur–Weyl duality 60 4.5. A general construction of central elements 61 4.6. Capelli determinant 63 4.7. Permanent-type elements 65 4.8. Gelfand invariants 66 4.9. Quantum immanants 67 4.10. Bibliographical notes 69

Chapter 5. Casimir elements for oN and spN 71 5.1. Harish-Chandra isomorphism 71 5.2. Brauer–Schur–Weyl duality 74 5.3. A general construction of central elements 76 5.4. Symmetrizer and anti-symmetrizer for oN 78 5.5. Symmetrizer and anti-symmetrizer for spN 83 5.6. Manin matrices in types B, C and D 89 5.7. Bibliographical notes 90

vii viii CONTENTS

Chapter 6. Feigin–Frenkel center 91 6.1. Center of a vertex algebra 91 6.2. Aﬃne vertex algebras 93 6.3. Feigin–Frenkel theorem 96 6.4. Aﬃne symmetric functions 101 6.5. From Segal–Sugawara vectors to Casimir elements 103 6.6. Center of the completed universal enveloping algebra 104 6.7. Bibliographical notes 106

Chapter 7. Generators in type A 107 7.1. Segal–Sugawara vectors 107 7.2. Sugawara operators in type A 114 7.3. Bibliographical notes 117

Chapter 8. Generators in types B, C and D 119 8.1. Segal–Sugawara vectors in types B and D 119 8.2. Low degree invariants in trace form 128 8.3. Segal–Sugawara vectors in type C 134 8.4. Low degree invariants in trace form 142 8.5. Sugawara operators in types B, C and D 145 8.6. Bibliographical notes 147

Chapter 9. Commutative subalgebras of U(g) 149 9.1. Mishchenko–Fomenko subalgebras 149 9.2. Vinberg’s quantization problem 155 9.3. Generators of commutative subalgebras of U(glN ) 157 9.4. Generators of commutative subalgebras of U(oN )andU(spN ) 165 9.5. Bibliographical notes 167

Chapter 10. Yangian characters in type A 169 10.1. Yangian for glN 169 10.2. Dual Yangian for glN 177 10.3. Double Yangian for glN 180 10.4. Invariants of the vacuum module over the double Yangian 183 10.5. From Yangian invariants to Segal–Sugawara vectors 185 10.6. Screening operators 186 10.7. Bibliographical notes 190

Chapter 11. Yangian characters in types B, C and D 191 11.1. Yangian for gN 191 11.2. Dual Yangian for gN 202 11.3. Screening operators 206 11.4. Bibliographical notes 211

Chapter 12. Classical W-algebras 213 12.1. Poisson vertex algebras 213 12.2. Generators of W(g) 216 12.3. Chevalley projection 226 12.4. Screening operators 228 12.5. Bibliographical notes 241 CONTENTS ix

Chapter 13. Aﬃne Harish-Chandra isomorphism 243 13.1. Feigin–Frenkel centers and classical W-algebras 243 13.2. Yangian characters and classical W-algebras 255 13.3. Harish-Chandra images of Sugawara operators 259 13.4. Harish-Chandra images of Casimir elements 263 13.5. Bibliographical notes 268

Chapter 14. Higher Hamiltonians in the Gaudin model 269 14.1. Bethe ansatz equations 269 14.2. Gaudin Hamiltonians and eigenvalues 271 14.3. Bibliographical notes 275 Chapter 15. Wakimoto modules 277 15.1. Free field realization of glN 277 15.2. Free field realization of oN 280 15.3. Free field realization of sp2n 284 15.4. Wakimoto modules in type A 287 15.5. Wakimoto modules in types B and D 289 15.6. Wakimoto modules in type C 292 15.7. Bibliographical notes 294 Bibliography 295 Index 303

Preface

In representation theory of Lie algebras, Casimir operators are commonly understood as certain expressions constructed from generators of a Lie algebra which commute with its action. Their spectra are useful for understanding the representation. In particular, ﬁnite-dimensional irreducible representations of a simple Lie algebra g over the ﬁeld of complex numbers are characterized by the eigenvalues of the Casimir operators. This fact is based on a theorem of Harish-Chandra describing the center Z(g) of the associated universal enveloping algebra U(g). The center is isomorphic to an algebra of polynomials via the Harish-Chandra isomorphism ∼ (0.1) Z(g) = C L1,...,Ln .

Here n is the rank of g and L1,...,Ln are polynomial functions in the highest weights of the representations, each Li is invariant under a certain action of the Weyl group of g. The isomorphism (0.1) relies on a theorem of Chevalley which can also be recovered as a ‘classical limit’ of (0.1). Namely, the symmetric algebra S(g) is isomorphic to the graded algebra gr U(g), and the subalgebra of g-invariants in S(g) is isomorphic to gr Z(g). Taking the symbols Mi of the polynomials Li,we get the Chevalley isomorphism g ∼ (0.2) S(g) = C M1,...,Mn .

The respective degrees d1,...,dn of the Weyl group invariants M1,...,Mn coincide with the exponents of g increased by 1. A vast amount of literature both in mathematical physics and representation theory has been devoted to understanding the correspondence in (0.1) in terms of concrete generators on both sides, especially for the Lie algebras g of classical types A, B, C and D. Various families of generators of the center Z(g) were discovered together with their Harish-Chandra images. The simple Lie algebras g can be regarded as a part of the family of Kac–Moody algebras parameterized by generalized Cartan matrices. Of particular importance is the class of affine Kac–Moody algebras which admits a simple presentation. The (untwisted) affine Kac–Moody algebra g is the central extension g[t, t−1] ⊕ CK of the Lie algebra of Laurent polynomials with coefficients in g. Basic results of representation theory of these Lie algebras together with applications to conformal field theory, modular forms and soliton equations can be found in the book by V. Kac [86]. Motivated by the significance of the Lie algebras g,onecomesto wonder what the center of U(g) looks like. However, this straight question turns out to be too naive to have a meaningful answer. First of all, the enveloping algebra is ‘too small’ to contain central elements beyond polynomials in K.The canonical quadratic Casimir element is already a formal series of elements of the algebra U(g), so it is necessary to consider its completion. As a natural choice,

xi xii PREFACE one requires that the action of elements for such a completion is well-defined on certain smooth modules over g. Secondly, the central element K must be given a unique constant value known as the critical level. With a standard choice of the invariant bilinear form on g, this value is the negative of the dual Coxeter ∨ number, K = −h . The suitably completed universal enveloping algebra U−h∨ (g) at the critical level does contain a large center Z(g), and the qualified question has a remarkably comprehensive answer which is explained in detail in the book by E. Frenkel [46]. Namely, similar to (0.1), the center Z(g)isacompletionofthe algebra of polynomials C S1[r],...,Sn [r] | r ∈ Z in infinitely many variables. Moreover, the elements Si [r] whichareknownas Sugawara operators, can be produced from a family of generators S1,...,Sn of a commutative differential algebra z(g) by employing instruments of the vertex algebra theory: the vacuum module at the critical level over g is equipped with a vertex algebra structure, and z(g)isthecenter of this vertex algebra. Thus the key to understanding the center Z(g) lies within the smaller object z(g). Its structure was described by a theorem of B. Feigin and E. Frenkel [39]and hence is known as the Feigin–Frenkel center. The theorem states that z(g)isan algebra of polynomials r r z(g)=C T S1,...,T Sn | r =0, 1,... , where T is a derivation defined as the translation operator of the vertex algebra. For type A this theorem can be derived from a previous work of R. Goodman and N. Wallach [58], and for types A, B, C from an independent work of T. Hayashi [65]. Both papers were concerned with a derivation of the character formula for the irreducible quotient L(λ) of the Verma module M(λ) at the critical level over g.The Sugawara operators form a commuting family of g-endomorphisms of M(λ)which leads to a computation of the character and thus proves the Kac–Kazhdan conjecture [89]. Our choice for the title of the book was motivated by the terminology used in both pioneering papers [58]and[65], although the term Segal–Sugawara operators is also common in the literature. The origins of the terminology go back to the paper by H. Sugawara [144] and an unpublished work of Graeme Segal; see e.g. I. Frenkel [52]. We chose to reserve the longer name, Segal–Sugawara vectors, for elements of z(g) to make a clearer distinction between the vectors and operators. More recently, new families of Segal–Sugawara vectors were constructed by A. Chervov and D. Talalaev [24]fortypeA, by the author [110]intypesB, C and D, and in joint work with E. Ragoucy and N. Rozhkovskaya [116]intype G2. Furthermore, these constructions lead to a direct proof of the Feigin–Frenkel theorem in those cases relying on an affine analogue of the Chevalley isomorphism (0.2). This analogue provides an isomorphism −1 −1 g[t] ∼ r r S t g[t ] = C T M1,...,T Mn | r =0, 1,... , for the ‘classical limit’ of z(g), and is due to M. Ra¨ıs and P. Tauvel [134]andto A. Beilinson and V. Drinfeld; see [46, Theorem 3.4.2]. Our goal in the book is to review these constructions of Segal–Sugawara vectors and to give an introduction to the subject. We hope that together with the general results explained in the book [46], they would bring more content to make the beau- tiful theory more accessible via concrete examples. The explicit Segal–Sugawara PREFACE xiii vectors will also be used in the applications of the theory as envisaged by the seminal work of B. Feigin, E. Frenkel and N. Reshetikhin [40]. Elements S ∈ z(g) give rise to Hamiltonians of the Gaudin model describing quantum spin chain. Their eigenvalues on the Bethe vectors can be calculated by using an affine version of the Harish-Chandra isomorphism for the algebra z(g).Theroleoftheinvariant polynomials occurring in (0.1) will now be played by elements of the classical W- algebra W(Lg) associated with the Langlands dual Lie algebra Lg. In parallel to the finite-dimensional theory, the affine Harish-Chandra isomorphism can be understood via the action of elements of the center Z(g)intheWakimoto modules over g: central elements act by scalar multiplication with the scalars interpreted as the Harish-Chandra images. As another application of the constructions of Segal–Sugawara vectors, an explicit solution of E. Vinberg’s quantization problem [149] will be given. It is based on the general results of L. Rybnikov [139] and B. Feigin, E. Frenkel and V. Toledano Laredo [42] which provide algebraically independent families of generators of commutative subalgebras of U(g) from generators of the algebra z(g). All constructions of the Segal–Sugawara vectors which we discuss in the book can be explained in a uniform way with the use of the fusion procedure allowing one to represent primitive idempotents for the centralizer algebras associated with representations of g, as products of rational R-matrices. This approach is therefore applicable, in principle, to all simple Lie algebras, depending on the availability of such a procedure. Its development for the exceptional types would give a uniform description of the Feigin–Frenkel center. An R-matrix is a solution of the Yang–Baxter equation.Givensuchaso- lution, one can define the corresponding algebra by an RT T -relation,wherethe generators of the algebra are combined into a matrix. This general approach originated in the work of L. Faddeev and the St. Petersburg (Leningrad) school on the quantum inverse scattering method in the early 1980s. Motivated by this work, V. Drinfeld [30]andM.Jimbo[81] came to the discovery of quantum groups.De- formations of universal enveloping algebras in the class of Hopf algebras form one of the most important families of quantum groups. The presentations of these Hopf algebras involving R-matrices give rise to special algebraic methods often referred to as the R-matrix techniques, to investigate their structure and representations; see e.g. [32], [96], [136] and references therein for more details on the origins of the methods. Moreover, these techniques can also be used to study the underly- ing Lie algebras themselves to bring new insights into their properties. It is these techniques which will underpin our approach. As a starting point, we will consider their applications to the simple Lie algebras g of classical types. Then we apply the R-matrix techniques to the corresponding affine Kac–Moody algebras g and a class of quantum groups Y(g)knownasYangians. In both cases, the defining relations of the algebras will be written in terms of certain generator matrices which can be understood as ‘operators’ on the space of tensors (CN )⊗ m with coefficients in the respective algebras. Therefore, as essential role will be played by the Schur–Weyl duality involving natural actions of the classical Lie algebras on the space (CN )⊗ m and the commuting actions of the symmetric group in type A or the Brauer algebra in types B, C and D. We will begin by reviewing constructions of primitive idempotents for the symmetric group and the Brauer algebra based on the respective fusion procedures xiv PREFACE which provide multiplicative R-matrix formulas for these idempotents (Chapter 1). We apply them to construct invariants in symmetric algebras S(g) in Chapter 2. Then we use the R-matrix techniques to derive some basic algebraic properties of Manin matrices (Chapter 3). They will be applied for constructions of Casimir elements for the general linear Lie algebras (Chapter 4). Similar constructions based on symmetrizers and anti-symmetrizers for the Brauer algebra will be used for the orthogonal and symplectic Lie algebras (Chapter 5). In Chapter 6 we introduce the center of the affine vertex algebra at the critical level associated with the affine Kac–Moody algebra g. We will produce explicit generators of the center in the classical types in Chapters 7 and 8 and show how this leads to a proof of the Feigin–Frenkel theorem. In Chapter 9 the generators are used to construct commutative subalgebras of the classical universal enveloping algebras which ‘quantize’ the shift of argument subalgebras of the symmetric algebras. Our calculation of the Harish-Chandra images of the Segal–Sugawara vectors will be based on explicit formulas for the characters of some finite-dimensional representations of the Yangian Y(g). The R-matrix techniques will play a key role in the derivation of the character formulas which we review in Chapters 10 and 11. In Chapter 12 we discuss the classical W-algebras and construct their generators. The images of the Segal–Sugawara vectors with respect to an affine version of the Harish-Chandra isomorphism will be calculated in Chapter 13. This will produce special families of generators of the classical W-algebra W(Lg) associated with the Langlands dual Lie algebra Lg. Applications to the Gaudin model will be discussed in Chapter 14. In the final Chapter 15 we will give a construction of the Wakimoto modules over g for all classical types and calculate the eigenvalues of the Sugawara operators in these modules. Bibliographical notes at the end of each chapter contain some comments on the origins of the results and references. An initial version of the exposition was based on the lecture courses delivered by the author at the Second Sino–US Summer School on Representation Theory at the South China University of Technology in 2011, organized by Loek Helminck and Naihuan Jing, and the International Workshop on Tropical and Quantum Ge- ometries at the Research Institute for Mathematical Sciences, Kyoto, in 2012, organized by Anatol Kirillov and Shigefumi Mori. I am grateful to the organizers of both events for the invitation to speak. My warm thanks extend to Alexander Chervov, Vyacheslav Futorny, Alexey Isaev, Evgeny Mukhin and Eric Ragoucy for collaboration on the projects which have formed the backbone of the book.

Alexander Molev Sydney, July 2017 CHAPTER 1

Idempotents and traces

We begin by reviewing some basic facts on representations of the symmetric group and the Brauer algebra. In addition to standard material which can be found in the books by Goodman and Wallach [59], James and Kerber [79] and Sagan [141], we discuss fusion procedures providing multiplicative formulas for primitive idempotents in both cases. The idempotents associated with one- dimensional representations will play a key role in our constructions of generators of the Feigin–Frenkel center in Chapters 7 and 8. We also introduce trace maps on the Brauer algebra and connect them with the traces of linear operators via natural actions of the symmetric group and the Brauer algebra in tensors.

1.1. Primitive idempotents for the symmetric group

We let Sm denote the symmetric group whose elements are permutations of the set {1,...,m}. We will identify the group Sm−1 with the subgroup of Sm which consists of the permutations s such that s(m)=m.For1 a

Aboxofλ is called removable if its removal leaves a diagram. Similarly, a box outside λ is addable to λ if the union of λ and the box is a diagram. In the example above, the boxes (1, 5), (3, 4) and (4, 2) are removable, while the boxes (1, 6), (2, 5), (4, 3) and (5, 1) are addable. A tableau U of shape λ m (or a λ-tableau U) is obtained by ﬁlling in the boxes of the diagram with the numbers in a given set {1,...,N}. The tableau is called semistandard if the entries weakly increase along each row from left to right and strictly increase in each column from top to bottom. We write sh(U)=λ if the shape of U is λ.

1 2 1. IDEMPOTENTS AND TRACES

A tableau U with entries in {1,...,m} which are ﬁlled in the boxes bijectively is called standard if its entries strictly increase along the rows and down the columns. The following is a standard tableau of shape (4, 4, 1): 1 3 4 5 2 6 7 9 8

The irreducible representations of Sm over C are parameterized by partitions of m.Givenλ m denote the corresponding irreducible representation of Sm by Vλ. The vector space Vλ is equipped with an Sm-invariant inner product ( , ). The orthonormal Young basis {vU } of Vλ is parameterized by the set of standard λ-tableaux U. The action of the generators sa = saa+1 of Sm in the Young basis is described as follows. We denote by cb = cb(U) the content of the box occupied by the number b in a standard λ-tableau U. Then for any a ∈{1,...,m− 1} we have · − 2 − −1 (1.1) sa vU = dvU + 1 d vsa U ,d=(ca+1 ca) , where the tableau sa U is obtained from U by swapping the entries a and a +1,and we assume v = 0 if the tableau s U is not standard. sa U a The group algebra C[Sm] is isomorphic to the direct sum of matrix algebras C ∼ C (1.2) [Sm] = Matfλ ( ), λm where fλ =dimVλ is the number of standard tableaux of shape λ. This number can be found by the hook length formula m! (1.3) f = ,h(λ)= (λ + λ − i − j +1), λ h(λ) i j (i,j)∈λ where λj denotes the number of boxes in the column j of the diagram λ. ∈ C The matrix units eUU Matfλ ( ) are parameterized by pairs of standard λ- tableaux (U, U ). We will identify C[Sm] with the direct sum of matrix algebras by the formulas

fλ (1.4) e = φ , UU m! UU where φUU is the matrix element corresponding to the basis vectors vU and vU of the representation Vλ, −1 (1.5) φUU = (s · vU ,vU ) s ∈ C[Sm].

s∈Sm

We will write φU = φUU for brevity. The diagonal matrix units eU = eUU are primitive idempotents of C[Sm]. They are pairwise orthogonal,

eU eV =0 if U = V, 2 with eU = eU and yield a decomposition of the identity element in C[Sm], 1= eU . λm sh(U)=λ The following simple properties of the matrix units will be needed later on. 1.1. PRIMITIVE IDEMPOTENTS FOR THE SYMMETRIC GROUP 3

Lemma 1.1.1. Let U be a standard tableau of shape λ and let a ∈{1,...,m−1}. Then − (1.6) eU (sa d)=eU sa esaU and − 2 (1.7) esaU sa eU sa esaU =(1 d ) esaU , where d is deﬁned in (1.1) and we suppose that e =0if the tableau s U is not saU a standard. Proof. We have −1 −1 −1 φU sa = (s · vU ,vU )(sas) = (sat · vU ,vU ) t = (t · vU ,sa · vU ) t .

s∈Sm t∈Sm t∈Sm Hence, applying (1.1) we ﬁnd − 2 φU sa = dφU + 1 d φU saU and so, by (1.4), − − 2 (1.8) eU (sa d)= 1 d eU saU . This element remains unchanged when multiplied by the matrix unit e from the saU right. So (1.6) follows since e e = 0. Furthermore, assuming that e =0and U saU saU replacing U by s U in (1.8) we get a − 2 esaU (sa + d)= 1 d esaUU. Together with (1.8) this yields − − 2 esaU sa eU sa esaU = esaU (sa + d) eU (sa d) esaU =(1 d ) esaU proving (1.7).

We will regard the character of the representation Vλ as an element of the group algebra −1 χλ = χλ(s) s ∈ C[Sm]

s∈Sm so that χλ = φU = h(λ) eU . sh(U)=λ sh(U)=λ The following identity will also be useful: for any standard λ-tableau U we have −1 (1.9) χλ = seU s .

s∈Sm

The primitive idempotents eU can be expressed explicitly in terms of the Jucys– Murphy elements x1,...,xm of the group algebra C[Sm] deﬁned by

(1.10) x1 =0 and xa = s1 a + ···+ sa−1 a for a =2,...,m.

Note that xm commutes with any element of the subgroup Sm−1. Therefore the Jucys–Murphy elements generate a commutative subalgebra of C[Sm]. Further- more, the vectors of the Young basis are eigenvectors for the action of xa on Vλ; for any standard λ-tableau U we have

xa · vU = ca(U) vU ,a=1,...,m. 4 1. IDEMPOTENTS AND TRACES

These relations imply

(1.11) xa eU = eU xa = ca(U) eU ,a=1,...,m. In particular, we have the identity in C[S ], m (1.12) xm = cm(U) eU , λm sh(U)=λ so that under the identiﬁcation (1.2), xm can be viewed as a diagonal matrix. Now let m 2 and let λ be a partition of m. Fix a standard λ-tableau U and denote by V the standard tableau obtained from U by removing the box α occupied by m. Then the shape of V is a diagram which we denote by μ.Weletc denote the content of the box α.Letu be a complex variable. Due to (1.12), the expression u − c (1.13) eV u − xm is a rational function in u with values in C[S ]. Since e = 1 for the (1)-tableau U m U0 0 with the entry 1, the following recurrence relations allow one to write any primitive idempotent eU in terms of the elements xa.

Proposition 1.1.2. We have the relation in C[Sm],

(xm − a1) ...(xm − al) (1.14) eU = eV , (c − a1) ...(c − al) where a1,...,al are the contents of all addable boxes of μ except for α,whilec is the content of the latter. Moreover, the rational function (1.13) is regular at u = c, and we have u − c (1.15) eU = eV . u − xm u=c Proof. The branching properties of the Young basis imply that (1.16) eV = eU , U summed over the standard tableaux U obtained from V by adding one box with the entry m. The right hand side of (1.14) now reduces to eU since by (1.11) the product (xm − a1) ...(xm − al) eU (c − a1) ...(c − al) is zero for all U = U and it equals 1 for U = U. Similarly, by (1.11) and (1.16) we have u − c u − c u − c eV = eU = eU + eU . u − xm u − cm(U ) u − cm(U ) U U = U Since cm(U ) = c for all standard tableaux U distinct from U, the value of this rational function at u = c is eU .

Example 1.1.3. The trivial one-dimensional representation of Sm corresponds to the partition (m); its diagram is a row with m boxes. By (1.4) and (1.5), the idempotent eU associated with the unique standard tableau U of shape (m) coincides with the symmetrizer 1 (1.17) h(m) = s. m! s∈Sm 1.1. PRIMITIVE IDEMPOTENTS FOR THE SYMMETRIC GROUP 5

Proposition 1.1.2 yields the product formula (1 + x )(1 + x ) ...(1 + x ) (1.18) h(m) = 2 3 m . m!

Example 1.1.4. The sign representation of Sm corresponds to the partition (1m); its diagram is a column with m boxes. By (1.4) and (1.5), the idempotent m eU associated with the unique standard tableau U of shape (1 ) coincides with the anti-symmetrizer 1 (1.19) a(m) = sgn s · s. m! s∈Sm Proposition 1.1.2 yields the product formula (1 − x )(1 − x ) ...(1 − x ) (1.20) a(m) = 2 3 m . m!

Example 1.1.5. For m = 2, the two primitive idempotents are the symmetrizer h(2) and anti-symmetrizer a(2).Form = 3, in addition to the symmetrizer h(3) and anti-symmetrizer a(3) there are two more primitive idempotents associated with the standard tableaux U = 1 2 and V = 1 3 3 2 given by 1 1 e = 1+(12) 2 − (1 3) − (2 3) and e = 1 − (1 2) 2+(13)+(23) . U 6 V 6

We record a property of the Jucys–Murphy elements to be used for calculation of traces in Section 1.3. Lemma 1.1.6. We have the identity for rational functions in u with values in the group algebra C[Sm], 1 1 1 1 1 (1.21) = sm−1 sm−1 + sm−1 + . u − xm u − xm−1 u − xm−1 u − xm u − xm−1 Proof. We have the relation sm−1xm = xm−1sm−1 + 1 in the group algebra. It implies 1 1 (1.22) sm−1 + =(u − xm−1) sm−1 . u − xm u − xm

Hence, multiplying from the left by the inverse of (u − xm−1) sm−1 we get 1 1 1 = sm−1 sm−1 + u − xm u − xm−1 u − xm 1 1 = sm−1 sm−1 + sm−1 . u − xm−1 (u − xm−1)(u − xm) Therefore, permuting the two factors in the denominator of the last fraction and applying (1.22) once again, we come to (1.21). 6 1. IDEMPOTENTS AND TRACES

An alternative way to express the primitive idempotents eU is provided by the fusion procedure originated in the work of Jucys [84], and its various versions have since been developed by many authors. Take m complex variables u1,...,um and consider the rational function with values in C[Sm] deﬁned by sab (1.23) φ(u1,...,um)= 1 − , ua − ub 1a

Proof. The group algebra C[Sm] can be regarded as the quotient of the Brauer algebra Bm(ω) by the ideal generated by all elements ab; see Section 1.2. Therefore, the statement is a consequence of Proposition 1.2.3 below. Alternatively, a direct argument is obtained by some obvious simpliﬁcations of the proof of that proposition. Example 1.1.8. The contents of the unique one-row standard tableau with m boxes are found by ca = a − 1fora =1,...,m. Hence the symmetrizer (1.17) is given by the multiplicative formula 1 s h(m) = 1+ ab m! b − a 1a

Example 1.1.10. By Proposition 1.1.7, the primitive idempotents eU and eV of Example 1.1.5 are also given by the formulas 1 (2 3) e = 1+(12) 1 − (1 3) 1 − , U 3 2 1 (2 3) e = 1 − (1 2) 1+(13) 1+ . V 3 2 1.2. Primitive idempotents for the Brauer algebra Let ω be an indeterminate. An m-diagram d is a collection of 2m dots arranged into two rows with m dots in each row connected by m edges such that any dot belongs to only one edge. The product dd of two diagrams d and d is determined by placing d under d and identifying the vertices of the bottom row of d with the 1.2. PRIMITIVE IDEMPOTENTS FOR THE BRAUER ALGEBRA 7 corresponding vertices in the top row of d.Lets be the number of closed loops obtained in this placement. The product dd is given by ω s times the resulting diagram without loops. For example, the product of the 8-diagrams rrrrrrrr rrrrrrrr ¦ ¥ ¦ ¥ H ¦ ¥ H @ ¨ @ H ¨H© §rrrrrrrr@ ¤ @ and §rrrrrrrr HH ¤ HH is found by

rrrrrrrr H ¦ ¥ H H ¨H© §rrrrrrrr HH ¤ HH rrrrrrrr P ¦ ¥ PP @ ¨ © = ω §rrrrrrrr PPP ¤@ rrrrrrrr ¦ ¥ ¦ ¥ @ ¨ @ §rrrrrrrr@ ¤ @

The Brauer algebra Bm(ω) is deﬁned as the C(ω)-linear span of the m-diagrams with the multiplication deﬁned above. The dimension of the algebra equals the number of the m-diagrams which is 1 · 3 ···(2m − 1). For 1 a

rrrrrr··· P ··· ··· rrrrrr··· ··· ··· PP ¦ ¥ rrrrrr··· ···PPP ··· and rrrrrr··· § ··· ¤ ···

1 abm 1 abm

The subalgebra of Bm(ω) generated over C by the elements sab is isomorphic to the group algebra of the symmetric group C[Sm] so that the m-diagram sab is identiﬁed with the transposition (ab). Sometimes we also use this notation with swapped subscripts, sba = sab and ba = ab. The Brauer algebra Bm−1(ω) will be regarded as a natural subalgebra of Bm(ω). This subalgebra is spanned by all m-diagrams in which the m-th dots in the top and bottom rows are connected by an edge. The algebra Bm(ω) is generated by the elements

(1.25) sa = saa+1,a = aa+1,a=1,...,m− 1, subject only to the following set of relations: 2 2 − sa =1,a = ωa,saa = asa = a,a=1,...,m 1, | − | sasb = sbsa,ab = ba,sab = bsa, a b > 1,

sasa+1sa = sa+1sasa+1,aa+1a = a,a+1aa+1 = a+1, − saa+1a = sa+1a,a+1asa+1 = a+1sa,a=1,...,m 2.

The Jucys–Murphy elements y1,...,ym for the Brauer algebra Bm(ω)aregiven by the formulas

− ω − 1 b 1 (1.26) y = + (s − ),b=1,...,m. b 2 ab ab a=1 8 1. IDEMPOTENTS AND TRACES

It is easily veriﬁed that ym commutes with all elements of the subalgebra Bm−1(ω). This implies that the elements y1,...,ym of Bm(ω) pairwise commute. By Wenzl’s theorem [152, Theorem 3.2], the algebra Bm(ω)overC(ω)is semisimple. It is isomorphic to a direct sum of matrix algebras with the sizes of matrices given by the dimensions of irreducible representations. Such representations are parameterized by the set of partitions λ of the numbers m − 2f with f =0, 1,...,m/2 .Anupdown λ-tableau is a sequence U =(Λ1,...,Λm)ofm diagrams such that for each r =1,...,m the diagram Λr is obtained from Λr−1 by adding or removing one box, where Λ0 = ∅ is the empty diagram and Λm = λ. The irreducible representation of Bm(ω) corresponding to λ admits a Young-type basis labeled by all updown λ-tableaux. To each updown tableau U we associate the sequence of contents (c1,...,cm), cr = cr(U), where ω − 1 ω − 1 c = + j − i or c = − + j − i , r 2 r 2 if Λr is obtained by adding the box (i, j)toΛr−1 or by removing this box from Λr−1, respectively. λ As with the isomorphism (1.2), the primitive idempotents eU = eU ∈Bm(ω) associated with updown tableaux U correspond to diagonal matrix units (we omit the superscripts indicating the diagrams since they are determined by the updown tableaux). When λ runs over all partitions of m, m − 2,... and U runs over all updown λ-tableaux, the elements eU yield a complete set of pairwise orthogonal primitive idempotents for Bm(ω). The following analogues of relations (1.11) hold:

(1.27) yr eU = eU yr = cr(U) eU ,r=1,...,m. The primitive idempotents can be found by the following recurrence formulas analogous to (1.14). Given an updown λ-tableau U =(Λ1,...,Λm), set μ =Λm−1 and consider the updown μ-tableau V =(Λ1,...,Λm−1). Let α be the box which is added to or removed from μ to get λ, and let c be the content of α.

Proposition 1.2.1. We have the relation in Bm(ω),

(ym − a1) ...(ym − ak) (1.28) eU = eV , (c − a1) ...(c − ak) where a1,...,ak are the contents of all boxes excluding α, which can be removed from or added to μ to get a diagram. Moreover, the following rational function is regular at u = c, and for its value we have u − c (1.29) eU = eV . u − ym u=c Proof. Both parts follow by the same arguments as for the proof of Proposi- tion 1.1.2 with the use of the branching properties of the irreducible representations of the Brauer algebra. Namely, given an updown tableau V =(Λ1,...,Λm−1), we have the relation (1.30) eV = eU , U U summed over all updown tableaux of the form =(Λ1,...,Λm−1, Λm). The proof is completed by applying the properties (1.27). 1.2. PRIMITIVE IDEMPOTENTS FOR THE BRAUER ALGEBRA 9

Example 1.2.2. The Brauer algebra B2(ω) has three one-dimensional representations. The corresponding updown tableaux associated with the diagrams (2), (12)and∅ are U 1 = , , U 2 = , , U 3 = , ∅ .

The respective contents (c1,c2)aregivenby ω − 1 ω +1 ω − 1 ω − 3 ω − 1 ω − 1 , , , , , − . 2 2 2 2 2 2 The corresponding primitive idempotents are

1+s1 1 1 − s1 1 eU = − ,eU = ,eU = . 1 2 ω 2 2 3 ω

The trivial one-dimensional representation of the Brauer algebra Bm(ω)isas- sociated with the partition (m). There is a unique updown (m)-tableau which can also be regarded as the standard tableau obtained by writing the numbers 1,...,m into the boxes of the row-diagram with m boxes from left to right. The corre- (m) sponding primitive idempotent is the symmetrizer s ∈Bm(ω) which is a unique idempotent determined by the properties

(m) (m) (m) (m) (m) (1.31) sab s = s sab = s and ab s = s ab =0 for all 1 a

Proposition 1.2.3. The following consecutive evaluations of the rational function ψ(u1,...,um) are well-deﬁned and we have (1.34) ψ(u1,...,um) ... = h(λ) eU , u1=c1 u2=c2 um=cm where h(λ) is the product of hooks of λ deﬁned in (1.3). 10 1. IDEMPOTENTS AND TRACES

Proof. As a ﬁrst step, we show that the following inductive formula for the function ψ(u1,...,um)holds: m−1 m 1 m (1.35) ψ(u1,...,um)=ψ(u1,...,um−1) 1 − ... 1 − um−1 + um u1 + um s s − × 1 − 1 m ... 1 − m 1 m . u1 − um um−1 − um This follows by using the easily veriﬁed identities for the rational functions in u and v with values in Bm(ω): if a

If indices a, b, c, d are distinct, then the elements ab and cd of Bm(ω)commute. Therefore, we can represent the ﬁrst product occurring in (1.33) as 1 − ab = 1 − ab ua + ub − ua + ub 1 a

Note that the Jucys–Murphy element ym commutes with eV ,andtheinversesof the expressions occurring in the product are found by −1 − srm − 1 srm 1 1 2 =1+ cr − u (u − cr) cr − u and −1 1 − rm =1+ rm , cr + u cr + u − ω 1.2. PRIMITIVE IDEMPOTENTS FOR THE BRAUER ALGEBRA 11

2 2 wherewehaveusedtherelationssrm =1andrm = ωrm. Hence, relation (1.37) is equivalent to sm−1 m s1 m (1.38) eV 1+ ... 1+ c − − u c − u m 1 1 1 m m−1 m u − ym × 1+ ... 1+ = eV . c1 + u − ω cm−1 + u − ω u − c1

Fix an integer n such that n m and regard Bm(ω) as a subalgebra of Bn(ω). We will verify by induction on m the following more general identity sm−1 n s1 n (1.39) eV 1+ ... 1+ cm−1 − u c1 − u (n) 1 n m−1 n u − ym × 1+ ... 1+ = eV , c1 + u − ω cm−1 + u − ω u − c1 where − ω − 1 m1 y(n) = + (s − ). m 2 an an a=1 It holds trivially for m = 1 so assume that 2 m n. By (1.16) we have eV = eV eW ,whereW is the standard tableau obtained from V by removing the box occupied by m − 1. Hence, using the induction hypothesis we can write the left hand side of (1.39) as − (n) sm−1 n u ym−1 m−1 n eV 1+ eW 1+ cm−1 − u u − c1 cm−1 + u − ω which equals

(n) s − (u − y ) 1 − (n) m 1 n m−1 eV u ym−1 + u − c1 cm−1 − u (n) (n) (u − y − )m−1 n sm−1 n (u − y − )m−1 n + m 1 + m 1 . cm−1 + u − ω (cm−1 − u)(cm−1 + u − ω)

Now use the following relations in Bn(ω) which hold for 1 r

sm−1 n srn = srm−1 sm−1 n,sm−1 n rn = rm−1 sm−1 n and

srnm−1 n = rm−1 m−1 n,rnm−1 n = srm−1 m−1 n. They imply that (n) sm−1 n ym−1 = ym−1 sm−1 n and (n) − − ym−1 m−1 n = ω 1 ym−1 m−1 n.

Together with the relation eV ym−1 = cm−1 eV implied by (1.27), this allows us to bring the left hand side of (1.39) to the form − (n) 1 − (n) − u ym eV u ym−1 sm−1 n + m−1 n = eV , u − c1 u − c1 as required. This proves (1.39) and hence (1.37). 12 1. IDEMPOTENTS AND TRACES

Finally, the product

m−1 u − c 1 1 1 − u − c (u − c )2 m r=1 r depends only on the shape μ of V so we may choose a particular tableau V for its evaluation. We take V to be the row tableau obtained by ﬁlling in the boxes of μ with the numbers 1,...,m by consecutive rows from left to right in each row. A short calculation shows that this product is regular at u = cm and the value equals the ratio h(λ)/h(μ). Thus, the proof is completed by applying (1.29).

Example 1.2.4. The expression for the symmetrizer (1.32) provided by (1.34) takes the form (1.40) s(m) = 1 − ab h(m), ω + a + b − 3 1a

(m) where h is the symmetrizer for Sm; see Example 1.1.8. Proposition 1.2.5. The expression (1.40) canalsobewrittenas

m/2 − (−1)r ω/2+m − 2 1 (1.41) s(m) = h(m) ... 2 r r! r a1b1 a2b2 ar br r=0 ai

Proof. Note that for each r the second sum commutes with any element of Sm and hence commutes with h(m). Now we apply (1.40) and write h(m) = h(m−1) h(m). (m−1) (m−1) (m−1) Since ambmh = amsabh = amh for distinct a and b,weget − 1m − m−1 m (m−1) 1 − ... 1 − h ω + m 2 ω +2m 4 + ···+ − = 1 − 1m m 1 m h(m−1) ω +2m − 4 which is a version of (1.38) and is veriﬁed by a similar calculation. Using induction on m we come to verifying that if (1.41) holds for m replaced by m − 1, then + ···+ − s(m−1) 1 − 1m m 1 m h(m) ω +2m − 4 coincides with the right hand side of (1.41). If none of the pairs (ai,bi) contains an ∈{ − } index c 1,...,m 1 , then the product a1b1 a2b2 ...ar br cm of r +1factors will contribute to the respective sum on the right hand side of (1.41). Otherwise, if c = ai or c = bi for some i,then

(m) (m) a1b1 a2b2 ...ar br cmh = a1b1 a2b2 ...ar br h by the relations

(m) (m) (m) (1.42) cbi cmh = cbi sbi m h = cbi h and similar in the c = bi case. The result follows by combining similar terms. 1.2. PRIMITIVE IDEMPOTENTS FOR THE BRAUER ALGEBRA 13

Corollary 1.2.6. We have the expression for s(m) in terms of the Brauer diagrams, m/2 − 1 ω/2+m − 2 1 (1.43) s(m) = (−1)r d, m! r r=0 d∈D(r) (r) where D ⊂Bm(ω) denotes the set of diagrams which have exactly r horizontal edges in the top (and hence in the bottom) row. Proof. The formula is essentially equivalent to (1.41). Indeed, the equivalence is implied by the relation r s a1b1 a2b2 ...ar br =2 r! d

s∈Sm ai

m/2 (−1)r (1.44) s(m) = h(m) r!(ω +2m − 4)(ω +2m − 6) ...(ω +2m − 2r − 2) r=0 × ab ab − (ω +2m − 4) ... ab − (r − 1)(ω +2m − 2r) . a

(m) Thus, counting the resulting coeﬃcient of the product h a1b1 a2b2 ...ar−1br−1 on the left hand side of (1.45) we ﬁnd (r − 1)ω +2(r − 1)(m − 2r +2)+2(r − 1)(r − 2) − (r − 1)(ω +2m − 2r)=0, as desired. We point out one more multiplicative formula for the symmetrizer. Proposition 1.2.8. We have 1 s (1.46) s(m) = 1+ ab − ab , m! b − a ω/2+b − a − 1 1a

It is well-known by [155] and easily veriﬁed that the rational functions ρab(u)satisfy the Yang–Baxter equation

(1.47) ρab(u) ρac(u + v) ρbc(v)=ρbc(v) ρac(u + v) ρab(u), with any distinct indices a, b, c,wherewesetρba(u)=ρab(u)fora

c ρcc+1(−1) = ρcc+1(−1) c =0, we derive that the product on the right hand side of (1.46) is an idempotent in (m) Bm(ω) which satisﬁes (1.31) and so must coincide with s . Example 1.2.9. If λ =(1m) is the single column-diagram with m boxes then the unique updown λ-tableau can be regarded as the standard tableau obtained by writing the numbers 1,...,m into the boxes of λ from top to bottom. The (m) corresponding primitive idempotent is the anti-symmetrizer a ∈Bm(ω)which coincides with the anti-symmetrizer in the group algebra for the symmetric group Sm in (1.19) and (1.24) and has the properties (m) (m) (m) (m) (m) (1.48) sab a = a sab = −a and ab a = a ab =0 for all 1 a

1.3. Traces on the Brauer algebra { } B Given a finite set of positive integers c1,...,cm denote by {c1,...,cm}(ω)the C(ω)-linear span of the Brauer diagrams, where the top and bottom dots are labeled by c1,...,cm. We define the product of two such diagrams in the usual way as in B B Section 1.2 so that the algebra {c1,...,cm}(ω)isisomorphicto m(ω). ∈{ } For any a 1,...,m define the partial trace tra (or the a-th closure map)as a linear map B →B (1.49) tra : m(ω) {1,...,a,...,m}(ω), 1.3. TRACES ON THE BRAUER ALGEBRA 15 where a indicates that the index a should be skipped. Take a diagram d ∈Bm(ω) and add the edge connecting the top dot a with the bottom dot a.Lets be the number of closed loops obtained in this operation (so that s =0ors =1).Then s tra(d)equalsω times the resulting diagram without loops and without the dots a in the top and bottom rows. More generally, the partial trace tra on the algebra B ∈{ } {c1,...,cm}(ω) with a c1,...,cm is defined in the same way. Given any subset {a1,...,ak} of {1,...,m}, define the corresponding partial trace tr on B (ω) as the composition tr ◦···◦tr . It is straightforward a1,...,ak m a1 ak ◦ ◦ B to verify that tra trb =trb tra for any a

Example 1.3.1. The trace of the following diagram d ∈B8(ω) rrrrrrrr H ¦ ¥ H H ¨H© §rrrrrrrr HH ¤ HH equals ω2. (m) Lemma 1.3.2. For the partial trace of the symmetrizer s ∈Bm(ω) we have (ω + m − 3)(ω +2m − 2) (1.51) tr s(m) = s(m−1). m m (ω +2m − 4) Hence, the full trace is found by ω +2m − 2 ω + m − 2 (1.52) tr s(m) = . 1,...,m ω + m − 2 m Proof. Expression (1.32) for the symmetrizer gives the recurrence relation − 1 m1 (1.53) s(m) = 1+ (s − ) m(ω +2m − 4) am am a=1 m−1 (m−1) × ω + m − 3+ (sam − am) s a=1 B in the Brauer algebra m(ω). We have trmsam =trmam = 1, while

trm samsbm =trm ambm = sab and trm sambm =trm amsbm = ab for a = b. Together with (1.31) this implies (1.51).

(m) Lemma 1.3.3. For the partial trace of the anti-symmetrizer a ∈Bm(ω) we have ω − m +1 (1.54) tr a(m) = a(m−1). m m 16 1. IDEMPOTENTS AND TRACES

Hence, the full trace is found by ω tr a(m) = . 1,...,m m Proof. As we pointed out in Example 1.2.9, a(m) coincides with the anti- symmetrizer in the group algebra for the symmetric group and is given by (1.20). Therefore, we have the recurrence relation − 1 m1 a(m) = 1 − s a(m−1). m am a=1 Since trmsam = 1, relation (1.54) follows.

(m) Lemma 1.3.4. For the partial trace of the element h ∈Bm(ω) we have ω + m − 1 (1.55) tr h(m) = h(m−1). m m Hence, the full trace is found by ω + m − 1 tr h(m) = . 1,...,m m Proof. By (1.18) we have the recurrence relation − 1 m1 h(m) = 1+ s h(m−1) m am a=1 which implies (1.55).

Now we will generalize the last two lemmas and calculate the traces of primitive idempotents associated with standard tableaux. They will imply hook dimension formulas for representations of GLN via the Schur–Weyl duality; see Section 4.4. Although the same approach can be used to prove such formulas for the orthogonal and symplectic groups, this leads to longer calculations which we omit and only state the dimension formulas in Section 5.2. Primitive idempotents for the symmetric group can be regarded as elements of the Brauer algebra due to the embedding C[Sm] ⊂Bm(ω). Recall that h(λ) denotes the product of hooks of λ as deﬁned in (1.3). Proposition 1.3.5. Suppose that λ is a diagram with m boxes and U is a standard tableau of shape λ. The trace of the corresponding primitive idempotent eU ∈ C[Sm] ⊂Bm(ω) is found by 1 (1.56) tr e = ω + j − i . 1,...,m U h(λ) (i,j)∈λ Proof. Since for partial traces we have 1 1 trm sm−1 sm−1 =trm−1 , u − xm−1 u − xm−1 we derive from Lemma 1.1.6 a recurrence relation for the rational functions 1 Am(u)=trm u − xm 1.4. TENSOR NOTATION 17 in the form 1 − Am(u)=Am 1(u)+ 2 1+Am(u) . (u − xm−1) That is, 2 (u − xm−1) 1 − Am(u)= 2 Am 1(u)+ 2 . (u − xm−1) − 1 (u − xm−1) − 1

Since for m =1wehaveA1(u)=ω/u, solving the recurrence relation we find − u + ω m1 (u − x )2 A (u)= a − 1. m u (u − x )2 − 1 a=1 a This calculation and the recurrence formula (1.15) allow us to find the partial trace trm eU . Indeed, with the notation used in (1.15), the properties (1.11) of the Jucys– Murphy elements imply − − m1 − 2 − u c (u ca) trm eU = eV (u c) Am(u) =(ω + c) eV , u=c u (u − c )2 − 1 u=c a=1 a where ca = j −i denotes the content of the box (i, j)ofU occupied by a and c = cm. The evaluation of the rational function in u is well-defined and it depends only on the shape μ of the standard tableau V but does not depend on V.Theresultof the evaluation is easily calculated by taking, say, the row tableau; cf. the proof of Proposition 1.2.3. It gives h(μ) tr e =(ω + c) e m U h(λ) V and so (1.56) follows by an obvious induction.

1.4. Tensor notation The endomorphisms of the vector space CN (linear maps CN → CN )form N an associative algebra which we denote by End C .Welete1,...,eN denote the vectors of the canonical basis of CN . Writing the endomorphisms as matrices with respect to this basis allows us to identify End CN with the algebra of N ×N matrices N over C. The matrix units eij with i, j ∈{1,...,N} form a basis of End C .The corresponding endomorphisms act on the basis vectors by

eij : ek → δjk ei, where δjk is the Kronecker delta, δjk =1ifj = k and δjk = 0 otherwise. For the product of the matrix units we have eij ekl = δjk eil. The vector space of tensors CN ⊗m CN ⊗ CN ⊗ ⊗ CN (1.57) ( ) = ... m ⊗ ⊗ is equipped with the basis formed by the tensor products ei1 ... eim ,wherethe indices i1,...,im run over the set {1,...,N}. The endomorphism algebra of the vector space (1.57) will be identiﬁed with the tensor product of the endomorphism algebras CN ⊗ ⊗ CN (1.58) End ... End m 18 1. IDEMPOTENTS AND TRACES via a natural isomorphism so that an element A1 ⊗ ...⊗ Am of the algebra (1.58) is understood as the endomorphism of (CN )⊗m deﬁned by N (A1 ⊗ ...⊗ Am)(v1 ⊗ ...⊗ vm)=A1(v1) ⊗ ...⊗ Am(vm),vi ∈ C .

We will often consider matrices⎡ ⎤ X11 ... X1N ⎢ . . ⎥ X = ⎣ . . ⎦ XN1 ... XNN whose entries Xij belong to a certain unital associative algebra A. The algebra of N × N matrices with entries in A is isomorphic to the tensor product algebra End CN ⊗Aso that the matrix X will be identiﬁed with the element N N (1.59) X = eij ⊗ Xij ∈ End C ⊗A. i,j=1 Given a scalar or variable u with values in A, the expression u + X will be understood as the sum u 1+X,whereu 1 is the scalar matrix of the same size as X. We will work with tensor product algebras of the form CN ⊗m ⊗A CN ⊗ ⊗ CN ⊗A (1.60) End ( ) =End ... End . m

For any a ∈{1,...,m} we will denote by Xa the element (1.59) associated with the a-th copy of End CN so that N ⊗(a−1) ⊗(m−a) N ⊗m (1.61) Xa = 1 ⊗ eij ⊗ 1 ⊗ Xij ∈ End (C ) ⊗A, i,j=1 where N N 1= eii ∈ End C i=1 is the identity endomorphism. Given any element N ⊗ ∈ CN ⊗ CN C = cijkl eij ekl End End , i,j,k,l=1 for any two indices a, b ∈{1,...,m} such that a

We will keep the same notation Cab for the element Cab ⊗ 1 of the algebra (1.60). By the trace we will mean the linear map N tr : End C → C,eij → δij.

Furthermore, for any a ∈{1,...,m} the partial trace tra will be understood as the linear map N ⊗m N ⊗(m−1) (1.63) tra :End(C ) → End (C ) 1.5. ACTION OF THE SYMMETRIC GROUP AND THE BRAUER ALGEBRA 19 which acts as the trace map on the a-th copy of End CN and is the identity map ◦···◦ on all the remaining copies. The full trace tr1,...,m is the composition tr1 trm. The following lemma will be often referred to as the cyclic property of trace; cf. (1.50). Lemma 1.4.1. Suppose that two elements X = e ⊗ ...⊗ e ⊗ X i1... im and i1j1 imjm j1... jm Y = e ⊗ ...⊗ e ⊗ Y i1... im i1j1 imjm j1... jm of the algebra (1.60) satisfy the property X i1... im Y k1... km = Y k1... km X i1... im j1... jm l1... lm l1... lm j1... jm for all values of the indices. Then

tr1,...,m XY =tr1,...,m YX. Proof. We have tr XY = X i1... im Y j1... jm = Y j1... jm X i1... im =tr YX, 1,...,m j1... jm i1... im i1... im j1... jm 1,...,m as required.

1.5. Action of the symmetric group and the Brauer algebra

The symmetric group Sm acts on the space (1.57) by permuting the tensor factors. More precisely, introduce the element N N N (1.64) P = eij ⊗ eji ∈ End C ⊗ End C i,j=1 which acts on the vector space CN ⊗ CN by swapping the tensor factors, P (v ⊗ w)=w ⊗ v, v, w ∈ CN . Then using the notation (1.62), for the images of the transpositions under the action of Sm on the space (1.57) we can write

(1.65) sab → Pab, 1 a

The image of an arbitrary element σ ∈ Sm under the map (1.65) will often be denoted by Pσ. Note the relations

(1.67) PabXa = Xb Pab, 1 a

(1.68) trb Pab =tra Pab =1.

We will need the action of the Brauer algebra Bm(ω) on the tensor space (1.57) with two particular specializations of the parameter ω.Thevaluesω = N and ω = −N (the latter with even N =2n) occur in the context of the Schur–Weyl duality, where the action of the Brauer algebra centralizers the respective actions 20 1. IDEMPOTENTS AND TRACES of the orthogonal and symplectic groups on the space (1.57). In the orthogonal case, the generators of Bm(N)actbytherule

(1.69) sab → Pab,ab → Qab, 1 a

tra B (N) −−−−→B{ }(N) m⏐ 1,...,a,...,m⏐ ⏐ ⏐ (1.72) 

tr End (CN )⊗m −−−−a→ End (CN )⊗m−1, where the vertical arrows denote the homomorphisms deﬁned in (1.69). In the symplectic case, the action of Bm(−N) with N =2n in the space (1.57) is deﬁned by

(1.73) sab →−Pab,ab →−Qab, 1 a

(1.76) trb Qab =tra Qab =1. However, the symplectic analogue of the commutative diagram (1.72) takes a different form, − tra B (−N) −−−−→B{ }(−N) m ⏐ 1,...,a,...,m⏐ ⏐ ⏐ (1.77)

tr End (CN )⊗m −−−−a→ End (CN )⊗m−1, where the vertical arrows denote the homomorphisms deﬁned in (1.73) and the top horizontal arrow is the composition of the map (1.49) with multiplication by −1. 1.6. BIBLIOGRAPHICAL NOTES 21

1.6. Bibliographical notes Proposition 1.1.2 is due to Jucys [85]andMurphy[120]. The version of the fusion procedure given in Proposition 1.1.7 is contained in [109, Section 6.4], where more detailed bibliographical notes can also be found. Originated from Jucys [84], the procedure was re-discovered by Cherednik [19] with detailed proofs given by Nazarov [124]. The deﬁnition of the algebra Bm(ω) is due to Brauer [14]. For the presentation in terms of the generators (1.25) see [10, Section 5]. The Jucys– Murphy elements (1.26) for the Brauer algebra Bm(ω) were introduced in [123]. They are also recovered from the corresponding elements of the Birman–Murakami– Wenzl algebras introduced in [101]atq = 1. The properties (1.27) were established in those papers. Two versions of the fusion procedure for the Brauer algebra were given in [74]and[75], respectively. The expression for the symmetrizer in terms of the Brauer diagrams given in (1.43) is provided by [69, Theorem 4.3]. The product formula (1.46) is contained in [74, Remark 3.8]. Its derivation from a fusion procedure was given in [75, Example 2.3].

CHAPTER 2

Invariants of symmetric algebras

Our goal in this chapter is to review some constructions of g-invariants of the symmetric algebra S(g), where g is a simple Lie algebra of classical type. These invariants will be produced with the use of the primitive idempotents for the symmetric group and Brauer algebra, discussed in Chapter 1. First recall some well- known facts on the invariants; see, for instance, Dixmier [29, Section 7.3]. The adjoint action of a simple Lie algebra g on itself extends to the symmetric algebra S(g)by k Y · X1 ...Xk = X1 ...[Y,Xi] ...Xk,Y,Xi ∈ g. i=1 Since any element Y ∈ g acts as a derivation, the subspace of g-invariants (2.1) S(g)g = {P ∈ S(g) | Y · P =0 forall Y ∈ g} is a subalgebra of S(g). Let n denote the rank of g.Then g (2.2) S(g) = C[P1,...,Pn] for certain algebraically independent invariants P1,...,Pn whose degrees d1,...,dn are the exponents of g increased by 1. Fix a Cartan subalgebra h of g and a triangular decomposition g = n− ⊕h⊕n+. Consider the projection S(g) → S(h) deﬁned by the decomposition S(g)=S(h) ⊕ J, where J is the ideal of S(g) generated by n− ∪ n+. The restriction of the projection to the subalgebra of invariants yields the Chevalley isomorphism (2.3) ς :S(g)g → S(h)W , where W is the Weyl group of the root system of g and S(h)W denotes the subalgebra of W -invariant elements in S(h).

2.1. Invariants in type A

The vector space MatN (C)ofN × N matrices over C is equipped with a Lie algebra structure deﬁned by the matrix commutator [X, Y ]=XY − YX.

This deﬁnes the general linear Lie algebra glN which has standard basis elements Eij with the commutation relations

(2.4) [Eij,Ekl]=δkj Eil − δil Ekl, i,j,k,l ∈{1,...,N}.

We thus make a notational distinction between the Eij and the introduced above N basis elements eij of the endomorphism algebra End C .Thespecial linear Lie algebra slN (the simple Lie algebra of type AN−1) is deﬁned as the subalgebra of glN spanned by the matrices whose trace is zero.

23 24 2. INVARIANTS OF SYMMETRIC ALGEBRAS

We will regard S(glN ) as the commutative associative algebra with the free generators Eij so that its elements are polynomials in the Eij. Introduce the matrix ⎡ ⎤ E11 ... E1N ⎢ . . ⎥ (2.5) E = ⎣ . . ⎦ EN1 ... ENN with entries in S(glN ) and write N N−1 (2.6) det(u + E)=u + C1 u + ···+ CN for a variable u. Here and below expressions like u + E should be understood as u1+E, where 1 is the identity matrix of the same size as E. By the well-known property of the characteristic polynomial, all coeﬃcients are invariant under the adjoint action of the general linear group GLN on S(glN ). More precisely, GLN acts on glN by the rule → −1 ∈ ∈ (2.7) Ad g : X gXg ,XglN , g GLN .

This extends to an action of GLN on the symmetric algebra S(glN )sothatelements glN of GLN act on S(glN ) as automorphisms. The subalgebra S(glN ) coincides with the subalgebra of GLN -invariants in S(glN ). Calculating the image of the basis element Eij under Ad g we ﬁnd that the image of the matrix E is given by (2.8) Ad g : E → gt E (gt)−1, t where g is the transpose of g. Hence the polynomial (2.6) is GLN -invariant. Moreover, we also have

glN C (2.9) S(glN ) = [C1,...,CN ] so that the coeﬃcients of the characteristic polynomial are algebraically independent generators of the algebra of invariants. This can be seen by calculating the Chevalley images of the Cm. Fix a standard triangular decomposition for glN so that the Cartan subalgebra h is spanned by the elements E11,...,ENN, while the subalgebras n− and n+ are spanned by the elements Eij with i>jand i

ς :det(u + E) → (u + λ1) ...(u + λN ) and so the Chevalley image ς(Cm) coincides with the elementary symmetric polynomial

(2.10) em(λ1,...,λN )= λp1 ...λpm .

1p1<···

The polynomials em(λ1,...,λN ) with m =1,...,N are algebraically independent generators of the algebra of symmetric polynomials which implies (2.9). By a similar argument, the traces of all powers of the matrix E belong to the algebra of invariants,

m ∈ glN (2.11) Tm =trE S(glN ) 2.1. INVARIANTS IN TYPE A 25 for all m 0. Moreover, we get an alternative family of generators,

glN C (2.12) S(glN ) = [T1,...,TN ] such that → m ··· m ς : Tm λ1 + + λN . To get a more general family of invariants, recall the action of the symmetric N ⊗m group Sm on the tensor product space (C ) as deﬁned in Section 1.5. The general linear group GLN acts on this space diagonally, N (2.13) h : v1 ⊗ ...⊗ vm → hv1 ⊗ ...⊗ hvm,vi ∈ C , h ∈ GLN , which can be written as h → h1 ...hm,where ⊗(a−1) ⊗(m−a) ha =1 ⊗ h ⊗ 1 . N ⊗m The action of any element of Sm on the vector space (C ) commutes with the action of any element h ∈ GLN . It suﬃces to check this property for generators of Sm which follows from (1.67):

(2.14) Paa+1 h1 ...hm = h1 ...ha+1 ha ...hmPaa+1 = h1 ...hm Paa+1 since ha+1 ha = ha ha+1. We will discuss the commuting actions of Sm and GLN in more detail in Section 4.4 within the context of the classical Schur–Weyl duality. Let S denote the image of s ∈ C[Sm] under the map (1.65). We regard S as an element of the algebra CN ⊗ ⊗ CN ⊗ (2.15) End ... End S(glN ) m by identifying it with S ⊗ 1. Also, as in (1.59), we regard the matrix E as the element N ⊗ ∈ CN ⊗ E = eij Eij End S(glN ). i,j=1

Proposition 2.1.1. For any s ∈ C[Sm] the element

(2.16) tr1,...,m SE1 ...Em

glN belongs to the algebra of invariants S(glN ) . Proof. As with the characteristic polynomial, we will be proving that the element (2.16) is invariant with respect to the adjoint action of the general linear group GLN . Due to (2.8), it suﬃces to verify that this element remains unchanged −1 if E is replaced by hE h for any h ∈ GLN .Wehave

−1 −1 −1 −1 tr1,...,m S h1E1h1 ...hmEmhm =tr1,...,m S h1 ...hm E1 ...Emh1 ...hm

−1 −1 =tr1,...,m h1 ...hm S h1 ...hm E1 ...Em =tr1,...,m SE1 ...Em, where we used the cyclic property of trace (Lemma 1.4.1) and the fact that the actions of S and h commute.

Proposition 2.1.1 is also a consequence of Theorem 4.5.1 below which will be proved by a slightly diﬀerent argument. 26 2. INVARIANTS OF SYMMETRIC ALGEBRAS

Remark 2.1.2. A seemingly more general family of invariants is obtained by taking the trace

k1 km (2.17) tr1,...,m SE1 ...Em with some nonnegative powers ka. This is veriﬁed by the same calculation as in the proof of Proposition 2.1.1. However, these elements can be written in the form (2.16) by applying a ‘linearization’ procedure so that (2.17) is written as

k1 km−1 tr1,...,m,m+1 E1 ...Em Em+1 Pmm+1 S in the ﬁrst step, due to (1.67) and (1.68). In particular, m tr E =tr1,...,m E1 ...Em Pm−1 m ...P12 =tr1,...,m E1 ...Em Pσ, for the cycle σ =(m, m − 1,...,1); see also Section 4.8 below. As in Section 1.1, suppose that U is a standard tableau of shape μ m and let eU ∈ C[Sm] be the associated primitive idempotent. We let EU denote the image of eU under the action of the symmetric group Sm given by (1.65). Consider the algebra (2.15) and employ the tensor notation of Section 1.4. Set E (2.18) I μ =tr1,...,m U E1 ...Em.

glN This element belongs to S(glN ) by Proposition 2.1.1 and is independent of the choice of the standard tableau U of shape μ. The latter property follows from the identity (1.9) in the group algebra of Sm, since by the cyclic property of trace, 1 tr E E ...E = tr P E E ...E P −1 1,...,m U 1 m m! 1,...,m s U 1 m s s∈Sm 1 1 = tr P E P −1 E ...E = tr X E ...E , m! 1,...,m s U s 1 m m! 1,...,m μ 1 m s∈Sm where Xμ denotes the image of the irreducible character χμ under the homomorphism (1.65). For a matrix X with entries in C,itsμ-immanant is deﬁned by immμ(X)= χμ(s) Xs(1) 1 ...Xs(m) m.

s∈Sm

The invariant I μ deﬁned in (2.18) can be written as a linear combination of the immanants of the principal m-submatrices of E with repeated rows and columns. Moreover, the image of I μ under the Chevalley isomorphism coincides with the Schur polynomial sμ,

(2.19) ς :Iμ → sμ(λ1,...,λN ), where sμ(λ1,...,λN )= λT (α), sh(T )=μ α∈μ summed over semistandard tableaux T of shape μ with entries in {1,...,N} as defined in Section 1.1, where T (α) denotes the entry of T in the box α.The property (2.19) can be verified directly (by a simplified version of the argument used in the proof of Theorem 10.1.2 below), but it is naturally explained within the context of Schur–Weyl duality which we will discuss in Section 4.4. Since the Schur polynomials form a basis of the algebra of symmetric polynomials in λ1,...,λN we come to the following result. 2.1. INVARIANTS IN TYPE A 27

Proposition 2.1.3. The elements I μ with μ running over all diagrams with at glN most N rows form a basis of S(glN ) .

Remark 2.1.4. From another perspective, the assignment χμ → ς(I μ)givesrise to the classical characteristic map providing an isomorphism between the algebra generated by the irreducible characters of the symmetric groups and the algebra of symmetric functions; see Macdonald [104, Section I.7]. We point out two particular cases of (2.19) corresponding to the row and column diagrams μ =(m)andμ =(1m) for which direct proofs are straightforward. Denote by H(m) and A(m) the elements of the algebra (2.15) (with the identity (m) components in S(glN )) which are the respective images of the symmetrizer h and anti-symmetrizer a(m) deﬁned in (1.17) and (1.19) under the map (1.65). Proposition 2.1.5. Under the Chevalley isomorphism we have (m) → (2.20) ς :tr1,...,m A E1 ...Em em(λ1,...,λN ), (m) → (2.21) ς :tr1,...,m H E1 ...Em hm(λ1,...,λN ), where the elementary symmetric polynomial e (λ ,...,λ ) is deﬁned in (2.10) and m 1 N

(2.22) hm(λ1,...,λN )= λp1 ...λpm

1p1···pmN is the m-th complete symmetric polynomial.

Proof. We will regard E as the diagonal matrix E =diag[λ1,...,λN ]and calculate the traces. Applying the operator occurring in (2.20) to a basis vector we ﬁnd 1 A(m)E ...E (e ⊗ ...⊗ e )= λ ...λ sgn σ · e ⊗ ...⊗ e . 1 m p1 pm m! p1 pm pσ(1) pσ(m) σ∈Sm This shows that the corresponding diagonal matrix element of the operator is nonzero only if all indices p1,...,pm are distinct. Moreover, all m! permutations of such m-tuple (p1,...,pm) give the same contribution to the trace and so it is found by (2.10). Similarly, for the operator in (2.21) we get 1 H(m)E ...E (e ⊗ ...⊗ e )= λ ...λ e ⊗ ...⊗ e . 1 m p1 pm m! p1 pm pσ(1) pσ(m) σ∈Sm

Hence, if (p1,...,pm) is a permutation of the multiset containing αi entries equal to i for i =1,...,N, then the corresponding diagonal matrix element of the operator equals α ! ...α ! 1 N λ ...λ . m! p1 pm Taking into account the number of such permutations, we arrive at (2.21). Corollary 2.1.6. Each of the families (m) (m) tr1,...,m A E1 ...Em and tr1,...,m H E1 ...Em

glN with m =1,...,N is algebraically independent and generates the algebra S(glN ) . Proof. This follows from Proposition 2.1.5 since the elementary and complete symmetric polynomials are algebraically independent generators of the algebra of symmetric polynomials in N variables. 28 2. INVARIANTS OF SYMMETRIC ALGEBRAS

As another consequence of Proposition 2.1.5, a formula for the coeﬃcients of the characteristic polynomial follows: (m) Cm =tr1,...,m A E1 ...Em,m=1,...,N. It is also implied by the identity (N) det(u + E)=tr1,...,N A (u + E1) ...(u + EN ) whose generalization (3.29) will be proved below.

2.2. Invariants in types B,C and D

Deﬁne the orthogonal Lie algebras oN with N =2n+1 and N =2n (simple Lie algebras of types Bn with n 1andDn with n 3, respectively) and symplectic Lie algebra spN with N =2n (simple Lie algebra of type Cn), as subalgebras of glN spanned by the elements Fij,

(2.23) Fij = Eij − Ej i and Fij = Eij − εi εj Ej i , − respectively, for oN and spN . As before, we use the notation i = N i +1,andin the symplectic case we set εi =1fori =1,...,n and εi = −1fori = n +1,...,2n. Unless otherwise stated, we will consider the three cases B, C and D simultaneously and use the notation gN for any of the Lie algebras oN or spN .Sowehave gN = {X ∈ MatN (C) | X + X =0}, where we use the transposition deﬁned for any N ×N matrix X =[Xij]withentries in an associative algebra A by Xji in the orthogonal case, (2.24) (X )ij = εi εj Xji in the symplectic case. This is the transposition associated with the symmetric or skew-symmetric bilinear form on CN deﬁned on the basis vectors by

(2.25) ei,ej = gij.

The respective matrices G =[gij] are antidiagonal and given by δij in the symmetric case, (2.26) gij = εi δij in the skew-symmetric case, where N =2n is even in the skew-symmetric case. So (2.24) can be written as X = GXtG−1, t t where X → X is the usual matrix transposition, (X )ij = Xji. When X is regarded as the element (1.59), this transposition can be understood as the anti-automorphism of the algebra End CN deﬁned on the matrix unites by eji in the orthogonal case, (2.27) eij → εi εj eji in the symplectic case. Accordingly, for each a ∈{1,...,m} the a-th transposition on the algebra (1.60) acts as the map (2.27) on the a-th copy of End CN andastheidentitymaponall othertensorfactors. 2.2. INVARIANTS IN TYPES B, C AND D 29

Lemma 2.2.1. Suppose that two elements N N X = eij ⊗ Xij and Y = eij ⊗ Yij i,j=1 i,j=1 of the algebra End CN ⊗A satisfy the property

Xij Ykl = Ykl Xij for all i, j, k, l. Then (XY ) = Y X. Proof. In the orthogonal case we have N N Y X = elk eji ⊗ Ykl Xij = eli ⊗ Yjl Xij. i,j,k,l=1 i,j,l=1 On the other hand, N N XY = eij ekl ⊗ Xij Ykl = eil ⊗ Yjl Xij, i,j,k,l=1 i,j,l=1 so that the application of the transposition yields Y X. The same argument is 2 used in the symplectic case together with the relation εi =1. Applying the transposition (2.27) to the a-th and b-th copies of End CN in (1.60) and using Lemma 2.2.1, we derive from (1.67) that for an arbitrary element (1.59) we have (2.28) Xa Qab = Xb Qab and Qab Xa = Qab Xb for all 1 a

The group GN acts on gN by the rule −1 (2.30) Ad g : X → gXg ,X∈ gN , g ∈ GN . 30 2. INVARIANTS OF SYMMETRIC ALGEBRAS

For the image of the matrix F we obtain from (2.8) that (2.31) Ad g : F → gt F (gt)−1.

t Observe that h = g belongs to GN . Hence the same argument as for type A in Section 2.1 shows that all coeﬃcients of the polynomial det (u + F )belongtothe GN subalgebra of invariants S(gN ) . Furthermore, this subalgebra can be described by

GN (2.32) S(gN ) = C[C1,...,Cn], which follows by calculating the images of the coeﬃcients Cm under the Chevalley isomorphism. To do the calculation, we make a standard choice of the triangular decomposition for gN so that the Cartan subalgebra h is spanned by the elements F11,...,Fnn, while the subalgebras n− and n+ are spanned by the elements Fij with i>jand i

2k GN Tk =trF ∈ S(gN ) . We get an alternative family of generators,

GN S(gN ) = C[T1,...,Tn] and → 2k ··· 2k ς : Tk 2(λ1 + + λn ).

gN The algebra of invariants (2.32) coincides with S(gN ) in types B and C, but gN is properly contained in S(gN ) in type D.IfgN = o2n,then n 2 Cn =detF =(−1) Pf F , where Pf F is the Pfaﬃan deﬁned by1 1 (2.33) Pf F = sgn σ · F ...F − . 2nn! σ(1) σ(2) σ(2n 1) σ(2n) σ∈S2n

Indeed, for the even skew-symmetric matrix [Fij ]=FGwe have the well-known 2 relation det FG = Pf F and note that det G =(−1)n. The Pfaﬃan Pf F is found by the expansion Φ n (2.34) = e ∧···∧e ⊗ Pf F, n! 1 2n with 2n Φ= (ei ∧ ej ) ⊗ Fij ∈ Λ(C ) ⊗ S(o2n), i

1 This is actually the Pfaffian of the skew-symmetric matrix [Fij ]. We abuse notation slightly by introducing the Pfaffian of the matrix which is skew-symmetric with respect to the antidiagonal. 2.2. INVARIANTS IN TYPES B, C AND D 31 where Λ(C2n) is the exterior algebra of the vector space C2n. Under the map (2.31) we have FG → hFGht with h = gt. Hence, by using (2.34) we find that the image of Pf F under this map is given by Pf F → det h · Pf F. This means that Pf F is invariant under the adjoint action of the special orthogonal group SO2n, and hence under the action of o2n. However, Pf F does not belong to the algebra (2.32) for GN =O2n. Under the Chevalley isomorphism we have

ς :Pf F → λ1 ...λn. Thus, the subalgebra of g -invariants can be described by N C[C1,...,Cn]forgN = o2n+1, sp , gN 2n (2.35) S(gN ) = C[C1,...,Cn−1, Pf F ]forgN = o2n, since the Chevalley images of the corresponding elements are algebraically independent generators of the algebra of W -invariants of S(h). To prove the orthogonal and symplectic analogues of Proposition 2.1.1, consider the action of the Brauer algebra on the tensor product space (CN )⊗m as deﬁned in Section 1.5. The action of the group GN on this space is obtained from (2.13) by restriction so that N h : v1 ⊗ ...⊗ vm → hv1 ⊗ ...⊗ hvm,vi ∈ C , h ∈ GN , which we write as h → h1 ...hm. The action of any element s ∈Bm(ω)withthe respective values ω = N or ω = −N on the vector space (CN )⊗m commutes with that of any element h ∈ GN . Indeed, this was already veriﬁed for the generators saa+1 in (2.14), whereas for the action of the generators aa+1 we have Qaa+1ha ha+1 = Qaa+1ha+1ha+1 = Qaa+1 = ha haQaa+1 = ha ha+1Qaa+1 where we used (2.29) and the properties Qaa+1ha = Qaa+1ha+1 and haQaa+1 = ha+1Qaa+1 implied by (2.28). Therefore,

Qaa+1 h1 ...hm = h1 ...hm Qaa+1.

We will return to the commuting actions of the Brauer algebra and the group GN in the context of the Brauer–Schur–Weyl duality in Section 5.2. Let S denote the image of s ∈Bm(ω) under the respective map (1.69) or (1.73). We regard S as an element of the algebra CN ⊗ ⊗ CN ⊗ (2.36) End ... End S(gN ) m by identifying it with S ⊗ 1. Also, we regard the matrix F as the element N N F = eij ⊗ Fij ∈ End C ⊗ S(gN ). i,j=1

Proposition 2.2.2. For any s ∈Bm(ω) with ω = N or ω = −N, respectively, the element

(2.37) tr1,...,m SF1 ...Fm

GN belongs to the algebra of invariants S(gN ) .Ifm is odd, then the element is zero. 32 2. INVARIANTS OF SYMMETRIC ALGEBRAS

Proof. Use the same argument as for the proof of Proposition 2.1.1. It relies on the veriﬁed above property that the operator S commutes with the product h1 ...hm for any h ∈ GN . All the generators Ck in (2.32) are homogeneous polynomials of even degree in the variables Fij. Since the invariant (2.37) is a homogeneous polynomial of an odd degree, it must be zero. We will now use Proposition 2.2.2 for some particular choices of the element s.LetS(m) and A(m) denote the respective images of the symmetrizer s(m) and (m) anti-symmetrizer a in the Brauer algebra Bm(ω) (deﬁned by equivalent formulas in Section 1.2) under its actions (1.69) (with ω = N) or (1.73) (with ω = −N). Consider the algebra (2.36) and employ the tensor notation of Section 1.4. We have the following analogue of Proposition 2.1.5.

Proposition 2.2.3. Under the Chevalley isomorphism for gN = oN we have (2k) → − k 2 2 ς :tr1,...,2k A F1 ...F2k ( 1) ek(λ1,...,λn), and for gN = sp2n we have (2k) → 2 2 (2.38) ς :tr1,...,2k A F1 ...F2k hk(λ1,...,λn). Proof. To calculate the traces, we will regard F as the diagonal matrix diag[λ1,...,λn, −λn,...,−λ1]ifN =2n, (2.39) F = diag[λ1,...,λn, 0, −λn,...,−λ1]ifN =2n +1. Hence, by (2.20), in the orthogonal case it will be sufficient to verify the relation − − − k 2 2 e2k(λ1,λ2,..., λ2, λ1)=( 1) ek(λ1,...,λn). It is immediate from the identity N N−m u em(λ1,λ2,...,−λ2, −λ1)=(u + λ1)(u + λ2) ...(u − λ2)(u − λ1). m=0 In the symplectic case the operator A(m) on the tensor product space coincides with the action of the symmetrizer h(m) defined by (1.65), because of the signs in (1.73). Therefore, the claim follow from (2.21) due to the generating function identity ∞ 1 tm h (λ ,λ ,...,−λ , −λ )= m 1 2 2 1 (1 − tλ )(1 − tλ ) ...(1 + tλ )(1 + tλ ) m=0 1 2 2 1 1 = , − 2 2 − 2 2 (1 t λ1) ...(1 t λn) thus completing the proof. The calculation of the Chevalley images of the invariants associated with the symmetrizer S(m) will require some extra work. Set ω + m − 2 (2.40) γ (ω)= m ω +2m − 2 and consider the orthogonal case first.

Proposition 2.2.4. If gN = oN then under the Chevalley isomorphism we have (2k) → 2 2 (2.41) ς : γ2k(N)tr1,...,2k S F1 ...F2k hk(λ1,...,λn). 2.2. INVARIANTS IN TYPES B, C AND D 33

Proof. Assuming that F =diag[λ1,...,λN ] is a diagonal matrix of the form (2.39) and m is a positive integer we will calculate the trace

(m) (2.42) tr1,...,m S F1 ...Fm ⊗ ⊗ by applying the operator to a basis vector ep1 ... epm . Using the expression (1.41) for the symmetrizer, we can write

m/2 − (−1)r N/2+m − 2 1 (2.43) S(m) = H(m) Q Q ...Q , 2 r r! r a1b1 a2b2 ar br r=0 ai

(2.44) Qa1b1 Qa2b2 ...Qar br

⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ (2.45) e1 ... e1 e1 ... e1 ... en ... en en ... en

q1 q1 qn qn

··· with q1 + + qn = m. The product Qa1b1 Qa2b2 ...Qar br occurring in (2.44) annihilates this vector, unless for each pair (ai,bi)theai-th and bi-th tensor factors are, respectively, el and el for some l. For such a pair we have N Q ...⊗ e ⊗ ...⊗ e ⊗ ... = ...⊗ e ⊗ ...⊗ e ⊗ ... . aibi l l li li li=1

Suppose that s1,...,sn are nonnegative integers such that s1 + ···+ sn = r and suppose that for each l =1,...,n the number of the pairs (ai,bi) corresponding to l is sl. Then the number of the products Qa1b1 Qa2b2 ...Qar br with this property can be evaluated as n ql ql sl!. sl sl l=1 Furthermore, due to the subsequent application of the operator H(m) in (2.43), all permutations of the factors in the tensor products of the basis vectors of CN give the same contribution to the diagonal matrix element. The same observation r ⊗ → ⊗ applies to the 2 swaps of the tensor factors eli el el eli for the values i i li =1,...,n. Clearly, we may ignore the values li = n + 1 in the case N =2n +1. Thus, taking into account the number of permutations r!

s1! ...sn! of an r-multiset with the multiplicities s1,...,sn, we ﬁnd that the diagonal matrix element of the operator (m) H Qa1b1 Qa2b2 ...Qar br F1 ...Fm

q1! q1 ! ...qn! qn ! m! q1 q1 qn qn q × r 1 − q1 qn − qn 2 r! ... λ1 ( λ1) ...λn ( λn) , s1 s1 sn sn summed over all n-tuples of nonnegative integers sl such that s1 + ···+ sn = r. Note that the number of different basis vectors obtained by permuting the tensor factors in (2.45) equals m! . q1! q1 ! ...qn! qn ! The trace (2.42) will be found by taking the sum of the diagonal matrix elements corresponding to all basis vectors and by taking the sum over the parameter r with the coefficients occurring in (2.43). Let us look at the coefficient of a monomial of m1 mn the form λ1 ...λn in the resulting expression for the trace. The sum qi qi (2.46) (−1)qi si si qi+qi =mi with a fixed value of si can be calculated as follows. Recalling the generating function ∞ q zs zq = s (1 − z)s+1 q=0 we find that the generating function of the coefficients (2.46) is given by

zsi (−z)si (−1)si z2si = . (1 − z)si+1 (1 + z)si+1 (1 − z2)si+1

This formal power series is even so that the sum (2.46) is zero for odd mi. Therefore, if m is odd then the trace (2.42) is zero, in agreement with Proposition 2.2.2, because at least one power mi must be odd. If m =2k is even, then we may assume that m1 mn all powers mi of the monomial λ1 ...λn are even, say mi =2ki. The sum (2.46) ki − si 2k1 2kn then simpliﬁes to ( 1) . Hence the coeﬃcient of λ1 ...λn in the trace si (2.42) with m =2k equals k −1 N/2+2k − 2 k1 kn (−1)r (−1)s1 ...(−1)sn r s1 sn r=0 s1+···+sn=r which clearly coincides with

− k N/2+2k − 2 1 k (2.47) . r r r=0 By the recurrence relation for the binomial coeﬃcients we have the identity k x − r x +1 = k − r k r=0 2.2. INVARIANTS IN TYPES B, C AND D 35 which holds for an arbitrary variable x. By multiplying both sides by the inverse x of we get k − k x 1 k x +1 (2.48) = . r r x − k +1 r=0 Hence the sum (2.47) equals N +4k − 2 = γ (N)−1 N +2k − 2 2k as required. Corollary 2.2.5. (i) The family (2k) (2.49) γ2k(N)tr1,...,2k S F1 ...F2k

oN with k =1,...,n is algebraically independent and generates the algebra S(oN ) for N =2n +1. (ii) The family (2.49) with k =1,...,n− 1 together with Pf F is algebraically oN independent and generates the algebra S(oN ) for N =2n. Proof. Both claims follow from Proposition 2.2.4, since the Chevalley images of the families are algebraically independent generators of the respective algebras of W -invariants S(h)W .

In the symplectic case the symmetrizer S(m) is well-defined for the values of m satisfying m n+1, while S(m) is not defined for m n+2 as some denominators in the formulas defining s(m) vanish for ω = −2n.Infact,S(n+1) =0,ascanbe seen from the Howe duality interpretation of the symmetrizer (see Proposition 2.3.2 below) or from the Brauer–Schur–Weyl duality (see decomposition (5.13) in Sec- tion 5.2). Nonetheless, the definition of the invariants (2.37) with S = S(m) can be extended to all values 1 m 2n + 1 by a certain kind of ‘analytic continuation’ as we demonstrate below in Proposition 2.2.8. We keep using the notation (2.40). Proposition . 2.2.6 If gN = sp2n and 2k n then under the Chevalley isomorphism we have − (2k) → − k 2 2 (2.50) ς : γ2k( 2n)tr1,...,2k S F1 ...F2k ( 1) ek(λ1,...,λn).

Proof. Assume that F =diag[λ1,...,λ2n] is a diagonal matrix of the form (2.39) and for a positive integer m n calculate the trace (m) (2.51) tr1,...,m S F1 ...Fm by applying the operator to basis vectors of the tensor product space. We let H(m) (m) denote the image of the element h of the Brauer algebra Bm(−2n) under the action (1.73). Note that H(m) acts as the anti-symmetrization operator on the tensor product space. Using the expression (1.41) for the symmetrizer, we can write m/2 − 1 −n + m − 2 1 (2.52) S(m) = H(m) Q Q ...Q , 2 rr! r a1b1 a2b2 ar br r=0 ai

(2.53) Qa1b1 Qa2b2 ...Qar br

(2.54) e ⊗ e ⊗ ...⊗ e ⊗ e ⊗ e ⊗ ...⊗ e ,p=0, 1,...,m/2 , c1 c1 cp cp d1 dm−2p where 1 c1 < ···

n Q − (e ⊗ e )= (e ⊗ e − e ⊗ e ). 2i 12i ci ci l l l l l=1

By applying the product of H(m) and the operator (2.53) to a basis vector of the form (2.54) we find that the coefficient of this vector in the expansion equals r r! p 2 .Sinceλ = −λ , the contributions of the basis vectors with m − 2p>0 m! r i i to the trace in (2.51) add up to zero. In particular, this shows that if m is odd, then the trace (2.51) is zero, in agreement with Proposition 2.2.2. Now suppose that m =2k is even. We may assume that p = k in (2.54), so that the coefficient of the monomial λ2 ...λ2 in the trace equals c1 ck

− k −n +2k − 2 1 k −n +2k − 1 (−1)k =(−1)k =(−1)k γ (−2n)−1, r r −n + k − 1 2k r=0 due to the identity (2.48), as required.

Now we would like to extend the construction of the invariants in S(sp2n) associated with the symmetrizer S(m) to all values 1 m 2n +1.

Lemma 2.2.7. If m n then for the m-th partial trace we have

− n − m +1 m1 tr S(m)F = − F S(m−1). m m m(n − m +2) a a=1

Proof. (m) The left hand side also equals trm Fm S by the cyclic property of trace. Take the image of the recurrence relation (1.53) under the homomorphism (1.73) to get

− 1 m1 S(m) = −1+ P − Q m(−2n +2m − 4) am am a=1 m−1 (m−1) × 2n − m +3+ Pam − Qam S . a=1 2.2. INVARIANTS IN TYPES B, C AND D 37

Now take into account (1.75) to calculate the partial trace

m−1 − − (2.55) trm Fm 1+ Pam Qam a=1 m−1 (m−1) × 2n − m +3+ Pam − Qam S . a=1

We have trm Fm = 0, and using (1.67), (1.68), (1.76) and (2.28) we ﬁnd − trm Fm Pam =trm PamFa = Fa and trm Fm Qam =trm Fa Qam = Fa. Furthermore, if a, b ∈{1,...,m− 1} are distinct indices, then − − − trm Fm Pam Qam Pbm =trm PbmFb Pab Qab = Fb Pab Qab , wherewesetPab = Pba and Qab = Qba for a>b. Similarly, tr F P − Q Q =tr F Q − P Q m m am am bm m m ab ab bm − − =trm Qab Pab Fm Qbm = Pab Qab Fb. By the properties (1.31) of the symmetrizer, we have (m−1) (m−1) Fb Pab − Qab S = −Fb S , whereas (m−1) (m−1) (m−1) Pab − Qab Fb S = Fa Pab − QabFb S = −Fa S . The last relation holds since (m−1) (m−1) (m−1) (m−1) QabFb S = −QabPabFb S = QabFa S = −QabFb S , (m−1) so that QabFb S = 0. Thus, collecting all the terms, we conclude that (2.55) equals m−1 (m−1) 2(n − m +1) Fa S a=1 completing the proof.

Proposition 2.2.8. The expression − (m) (2.56) γm( 2n)tr1,...,m S F1 ...Fm

sp2n admits an equivalent form representing a well-defined element of S(sp2n) for all 1 m 2n +1. Proof. Suppose that m n. By calculating the m-th partial trace with the use of Lemma 2.2.7, and applying the cyclic property of trace, we will write (2.56) as a linear combination of expressions of the form − (m−1) 2 − γm−1( 2n)tr1,...,m−1 S F1 ...Fa ...Fm−1,a=1,...,m 1. All coefficients of this linear combination are well-defined for all 1 m 2n +1. By (1.31) and (1.73), we have

(m−1) a−1 (m−1) S =(−1) S Pσ 38 2. INVARIANTS OF SYMMETRIC ALGEBRAS for the cycle σ =(1, 2,...,a). Hence, (m−1) 2 tr1,...,m−1 S F1 ...Fa ...Fm−1 − a−1 (m−1) 2 =( 1) tr1,...,m−1 S Pσ Fa F1 ...Fa−1 Fa+1 ...Fm−1 − a−1 (m−1) 2 =( 1) tr1,...,m−1 S F1 F2 ...Fa Fa+1 ...Fm−1 Pσ which equals (m−1) 2 tr1,...,m−1 S F1 F2 ...Fm−1 by the cyclic property of trace. Therefore, (2.56) is proportional to the expression − (m−1) 2 (2.57) γm−1( 2n)tr1,...,m−1 S F1 F2 ...Fm−1. Now apply Lemma 2.2.7 again for the (m − 1)-th partial trace. Repeating the previous calculation we will represent (2.57) as a linear combination of two expressions − (m−2) 3 γm−2( 2n)tr1,...,m−2 S F1 F2 ...Fm−2 and − (m−2) 2 2 γm−2( 2n)tr1,...,m−2 S F1 F2 F3 ...Fm−2 so that the coeﬃcients are well-deﬁned for all 1 m 2n + 1. Continuing with the (m − 2)-th partial trace, etc., these calculations can be performed l times in total, with any l satisfying 2l m + 1 to write (2.56) as a linear combination of expressions of the form

− k − − (m l) k1 m l (2.58) γm−l( 2n)tr1,...,m−l S F1 ...Fm−l . Observe that (2.58) is well-defined for m − l n. Hence, if n +1 m 2n +1, then taking l = m − n we get a required expression defined for these values of m. sp2n Moreover, every element (2.58) belongs to the subalgebra of invariants S(sp2n) ; cf. Remark 2.1.2. To complete the proof, we need to show that all possible linear combinations representing the expression (2.56) for n+1 m 2n+1, obtained by the described above procedure, coincide, as elements of the symmetric algebra S(sp2n). If the condition m n holds, then by (2.32) the invariant (2.56) is a unique polynomial in the generators C ,...,C , 1 n (m) γ (−2n)tr S(m)F ...F = a Ck1 ...Ckn , m 1,...,m 1 m k1,...,kn 1 n with the summation over nonnegative integers ka subject to the degree condition 2k1 +4k2 +···+2nkn = m. In particular, the invariant is zero if m is odd. Moreover, k = 0 unless 2a m so we may write the coefficients as a(m) = a(m) with a k1,...,kn k1,...,kp p = m/2 . Now suppose that m is fixed and let n vary. The coefficients a(m) k1,...,kp are rational functions in n and so they are determined by infinitely many values of n with n m. The first part of the proof demonstrates that the rational functions a(m) are defined for all values n (m − 1)/2, that is, for n p.Thus,wemay k1,...,kp conclude that since all equivalent expressions for (2.56) obtained in the first part of sp2n the proof coincide, as elements of S(sp2n) for infinitely many values of n,they must coincide for all values of n (m − 1)/2 for which they are defined. We will use expression (2.56) for all values 1 m 2n + 1, with the under- sp2n standing that it represents a well-defined element of S(sp2n) obtained in the proof of Proposition 2.2.8. 2.3. SYMMETRIZER AND EXTREMAL PROJECTOR 39

Example 2.2.9. For m =2andn 2 expression (2.56) takes the form n 1 P Q γ (−2n)tr S(2)F F = tr − 12 − 12 F F . 2 1,2 1 2 n − 1 1,2 2 2 2n 1 2 Since tr F = 0, the properties (1.68) and (1.76) together with (1.67) and (2.28) 1 imply that this expression coincides with − tr F 2. This is a well-deﬁned element 2 sp2n of S(sp2n) for all n 1. Corollary 2.2.10. Proposition 2.2.6 is valid for all values 1 m 2n +1. Moreover, the family − (2k) (2.59) γ2k( 2n)tr1,...,2k S F1 ...F2k

sp2n with k =1,...,nis algebraically independent and generates the algebra S(sp2n) . Proof. As the proof of Proposition 2.2.8 shows, for a ﬁxed m the polynomial in C1,...,Cp representing the element (2.56) is determined by the values n m. Hence the claim follows from Proposition 2.2.6. The second part of the corollary holds since the Chevalley images of the elements (2.59) are algebraically independent generators of the algebra S(h)W .

2.3. Symmetrizer and extremal projector (m) Consider the orthogonal case gN = oN ﬁrst. We regard the operator H on tensor product space CN ⊗ CN ⊗ ⊗ CN (2.60) ... m (m) as the image of the element h ∈Bm(N) under the action (1.65). Denote by PN the space of polynomials in variables z1,...,zN and identify the image of the (m) P m vector space (2.60) under the operator H with the subspace N of homogeneous polynomials of degree m, (m) CN ⊗m ∼ P m (2.61) H ( ) = N via the isomorphism (m) ⊗ ⊗ → (2.62) H (ei1 ... eim ) zi1 ...zim . (m) P m Due to (2.43) we may consider S as an operator on the space N .Thisoperator commutes with the action of the orthogonal group ON on this space and so it can be expressed in terms of the action of the generators of the Lie algebra sl2 provided by a particular case of the Howe duality [67, Section 3.4]. To get such an expression, let {e, f, h} be the standard basis of sl2 with the commutation relations [e, f]=h, [h, e]=2e, [h, f]=−2f.

The action of the basis elements on the space PN is given by 1 N 1 N N N (2.63) e →− ∂ ∂ ,f → z z ,h →− − z ∂ , 2 i i 2 i i 2 i i i=1 i=1 i=1 where ∂i denotes the partial derivative over zi. Recall that the extremal projector p for the Lie algebra sl2 is given by the formula ∞ (−1)r (2.64) p =1+ f rer. r!(h +2)...(h + r +1) r=1 40 2. INVARIANTS OF SYMMETRIC ALGEBRAS

The projector p is an element of an extension of the universal enveloping algebra U(sl2) and it possesses the properties ep = pf =0.

The center of the universal enveloping algebra U(sl2) is generated by the Casimir element h(h +2) C = fe+ . 4 One easily veriﬁes by induction the identity in U(sl2) h(h +2) (h +2)(h +4) (h +2r − 2)(h +2r) f rer = C − C − ... C − 4 4 4 for any r 1. Therefore the extremal projector (2.64) can be written as ∞ (−1)r (2.65) p =1+ r!(h +2)...(h + r +1) r=1 × fe fe− (h +2) ... fe− (r − 1)(h + r) , which is also equivalent to the inﬁnite product expansion ∞ fe (2.66) p = 1 − . r(h + r +1) r=1 P m The basis element h of sl2 acts on the space N as multiplication by the scalar −N/2 − m. Furthermore, this space is annihilated by the action of er with r>m/2. Hence, using (2.64) we may regard p as an operator on PN . The image of this operator coincides with the subspace of sl2-singular vectors. Proposition . (m) P m 2.3.1 The operator S on the space N coincides with the restriction of the action of the extremal projector p to this space. Proof. We will regard S(m) as the image of the symmetrizer s(m) given by the formula (1.44) under the action (1.69) of the Brauer algebra with ω = N.We have the identity for operators on the space P m, N Qab = −2fe. a

2.4. Bibliographical notes In Sections 2.2 and 2.3 we followed [110]. Some generalizations of relations (2.41) and (2.50) where the symmetrizer S(2k) is replaced by the idempotents in the Brauer algebra associated with standard tableaux, were proved in [117]ina diﬀerent way. The formula (2.64) for the extremal projector was generalized by Asherova, Smirnov and Tolstoy [8] to arbitrary simple Lie algebras. Its properties were further studied by Zhelobenko [157], [159]. For an interpretation of the inﬁnite product formula (2.66) and its generalizations to the Kac–Moody algebras see his paper [158].

CHAPTER 3

Manin matrices

This chapter is devoted to a special class of matrices whose origins go back to Manin [106], [107]. Some generator matrices associated with the Lie algebra glN , the aﬃne Kac–Moody algebra glN and the Yangian Y(glN ) turn out to belong to this class. Consequently, general properties of Manin matrices will be useful in those particular cases. Orthogonal and symplectic analogues of Manin matrices will be discussed below in Section 5.6.

3.1. Deﬁnition and basic properties Definition 3.1.1. A square matrix ⎡ ⎤ M11 ... M1N ⎢ . . ⎥ M = ⎣ . . ⎦ MN1 ... MNN with entries in an associative algebra A over C is called a Manin matrix if the following relations hold

(3.1) Mij Mkl − Mkl Mij = Mkj Mil − Mil Mkj for all i, j, k, l ∈{1,...,N}. If the algebra A is commutative, then any matrix M satisﬁes (3.1). Therefore, the properties of Manin matrices which we discuss below generalize those of the usual numerical matrices. Observe also that the relations (3.1) imply that the entries of each column of a Manin matrix pairwise commute. Consider the tensor product algebra (3.2) End CN ⊗ End CN ⊗A and use the notation of Section 1.4. Lemma 3.1.2. Each of following relations provides an equivalent deﬁnition of Manin matrices:

(3.3) (1 − P )M1 M2 (1 + P )=0,

(3.4) (1 − P )(M1 M2 − M2 M1)=0,

(3.5) (M1 M2 − M2 M1)(1+P )=0. Proof. Note ﬁrst that the three relations are equivalent to each other by (1.67). We have N M1 M2 = eij ⊗ ekl ⊗ Mij Mkl. i,j,k,l=1

43 44 3. MANIN MATRICES

Hence, using formula (1.64) for the element P we get N PM1 M2 = ekj ⊗ eil ⊗ Mij Mkl, i,j,k,l=1 N M1 M2 P = eil ⊗ ekj ⊗ Mij Mkl, i,j,k,l=1 and N PM1 M2 P = ekl ⊗ eij ⊗ Mij Mkl. i,j,k,l=1

Therefore, taking the coeﬃcient of the basis vector eij ⊗ ekl on the left hand side of (3.3) we recover (3.1).

For any m 1 consider the tensor product algebra CN ⊗ ⊗ CN ⊗A (3.6) End ... End . m Asbefore,wedenotebyH(m) and A(m) the elements of the algebra (3.6) which are the respective images of the symmetrizer h(m) and anti-symmetrizer a(m) deﬁned in (1.17) and (1.19) under the map (1.65). Lemma 3.1.3. If M is a Manin matrix, then we have the identities in the algebra (3.6),

(m) (m) (m) (3.7) A M1 ...Mm A = A M1 ...Mm and

(m) (m) (m) (3.8) H M1 ...Mm H = M1 ...MmH .

Proof. To prove (3.7) it suﬃces to show that for any element σ ∈ Sm we have

(m) (m) (3.9) A M1 ...Mm Pσ =sgnσ · A M1 ...Mm, where Pσ denotes the image of σ under the map (1.65). Since the group Sm is generated by the adjacent transpositions, it is enough to verify (3.9) for the elements σ = sa with a =1,...,m− 1. Hence we only need to consider the case m = 2. However, the relation (3.9) with σ = s1 reads 1 − P 1 − P M M P = − M M 2 1 2 2 1 2 which an equivalent form of (3.3). Similarly, the proof of (3.8) reduces to checking that for any σ ∈ Sm (m) (m) (3.10) Pσ M1 ...Mm H = M1 ...MmH . This follows again from (3.3) written in the form 1+P 1+P PM M = M M . 1 2 2 1 2 2 3.2. IDENTITIES AND INVERTIBILITY 45

3.2. Identities and invertibility We point out some useful recurrence formulas for the symmetrizer and anti- symmetrizer. Note that by (1.20)

(m) 1 (m−1) A = A (1 − P −···−P − ). m 1 m m 1 m Multiply both sides by A(m−1) from the right and use the relations

(m) (m−1) (m) (m−1) (m−1) (m−1) (m−1) A A = A and A PamA = A Pm−1 m A for 1 a

(3.16) H(r)A{r+1,..., m} r(m − r +1) (r +1)(m − r) → H(r)A{r,..., m} + H(r+1)A{r+1,..., m}. m m 46 3. MANIN MATRICES

Indeed, by (3.11) and (3.12) the right hand side of (3.16) equals − r(m r +1) (r) 1 {r+1,..., m} H − A m m r +1 m − r − A{r+1,..., m}P A{r+1,..., m} m − r +1 rr+1 (r +1)(m − r) 1 r + H(r) + H(r)P H(r) A{r+1,..., m}. m r +1 r +1 rr+1 To simplify this expression note that permuting H(r) with A{r+1,..., m} and using the cyclic property of the trace we obtain the relation

(r) {r+1,..., m} {r+1,..., m} tr1,..., mH A Prr+1A M1 ...Mm (r) {r+1,..., m} {r+1,..., m} =tr1,..., mH Prr+1A M1 ...MmA . Now apply (3.7) to write this element as (r) {r+1,..., m} tr1,..., mH Prr+1A M1 ...Mm. Similarly, by the cyclic property of the trace, we get

(r) (r) {r+1,..., m} tr1,..., mH Prr+1H A M1 ...Mm {r+1,..., m} (r) (r) =tr1,..., mPrr+1A H M1 ...MmH . Hence,by(3.8)thisequals

{r+1,..., m} (r) tr1,..., mPrr+1A M1 ...MmH (r) {r+1,..., m} =tr1,..., mH Prr+1A M1 ...Mm. We have thus veriﬁed that the replacement (3.16) does not aﬀect the left hand side of (3.15). On the other hand, (3.15) vanishes after this replacement since we get a telescoping sum equal to zero. The summands in (3.13) and (3.14) can be written explicitly in terms of the matrix elements of the matrix M as follows. Proposition 3.2.2. For any 1 m N we have

(3.17) tr A(m)M ...M 1,...,m 1 m · = sgn σ Miσ(1) i1 ...Miσ(m) im .

1i1<···

(m) (3.18) tr1,...,m H M1 ...Mm 1 = Mim iσ(m) ...Mi1 iσ(1) , α1! ...αN ! 1i1···imN σ∈Sm where αi is the multiplicity of index i ∈{1,...,N} in the multiset {i1,...,im}. Proof. Write (3.19) A(m)M ...M = e ⊗ ...⊗ e ⊗ M i1... im , 1 m i1j1 imjm j1... jm I,J 3.2. IDENTITIES AND INVERTIBILITY 47 summed over all m-tuples of indices I =(i1,...,im)andJ =(j1,...,jm)from {1,...,N},whereM i1... im ∈A.Foreacha =1,...,m− 1wehave j1... jm (m) (m) (m) Paa+1A M1 ...Mm = −A M1 ...Mm = A M1 ...MmPaa+1, where the second equality holds due to (3.9). This implies that the matrix elements M i1... im are skew-symmetric with respect to permutations of the upper indices and j1... jm of the lower indices. Hence tr A(m)M ...M = M i1... im = m! M i1... im . 1,...,m 1 m i1... im i1... im I 1i1<··· ··· >im 1 and those in (3.18) by N i1 ··· im 1 for the respective formulas to remain valid. Consider now the particular case m = N of (3.7). The anti-symmetrizer A(N) projects the space (CN )⊗N to its one-dimensional subspace spanned by the vector (3.21) sgn σ · eσ(1) ⊗ ...⊗ eσ(N).

σ∈SN This implies that (N) (N) (N) A M1 ...MN A = A cdet M for a uniquely determined element cdet M ∈Acalled the column-determinant of the matrix M. By (3.7) we also have (N) (N) (3.22) A M1 ...MN = A cdet M.

Applying both sides to the vector e1 ⊗ ...⊗ eN and comparing the coeﬃcients of the vector (3.21) on both sides we get the formula (3.23) cdet M = sgn σ · Mσ(1) 1 ...Mσ(N) N .

σ∈SN

Due to (1.67), for any element s ∈ SN we have (N) (N) (3.24) A Ms(1) ...Ms(N) = A cdet M 48 3. MANIN MATRICES which follows by multiplying both sides of (3.22) by Ps from the right. This leads to a generalization of (3.23), cdet M =sgns sgn σ · Mσ(1) s(1) ...Mσ(N) s(N).

σ∈SN Hence, the column-determinant of a Manin matrix shares the usual properties of determinant: if two rows or two columns are swapped then it changes the sign. Remark 3.2.4. The column-determinant of any square matrix A over a ring coincides with the row-determinant (3.25) rdet B = sgn σ · B1 σ(1) ...BNσ(N)

σ∈SN of the transposed matrix B = At. Taking ω = N in Lemma 1.3.3 and using (1.72), we ﬁnd that N − m +1 (3.26) tr A(m) = A(m−1). m m This implies (N) (3.27) tr1,...,N A =1 so that we recover the particular m = N case of (3.17) by taking trace in (3.22):

(N) (3.28) tr1,...,N A M1 ...MN =cdetM. Now let u be a formal variable. Corollary 3.2.5. We have the identities N m (m) (3.29) cdet(1 + uM)= u tr1,...,m A M1 ...Mm, m=0 ∞ − −1 m (m) (3.30) cdet(1 uM) = u tr1,...,m H M1 ...Mm. m=0 Proof. By the MacMahon Master Theorem (Theorem 3.2.1), identities (3.29) and (3.30) are equivalent. Identity (3.29) holds since the coeﬃcient of um on the left hand side equals the sum of the column-determinants of all principle m × m submatrices of M, which coincides with the corresponding coeﬃcient on the right hand side due to (3.17).

Deﬁne the comatrix for a Manin matrix M as the matrix M with the entries in the algebra A deﬁned by

i+j ji (3.31) Mij =(−1) cdet M , where M ji is the matrix obtained from M by deleting row j and column i. Lemma 3.2.6. We have the relation (3.32) MM = cdet M 1, where 1 denotes the identity matrix. 3.2. IDENTITIES AND INVERTIBILITY 49

Proof. First observe that the deﬁnition (3.31) of the comatrix can be written equivalently in the matrix form as

(N) (N) (3.33) A M1 ...MN−1 = A MN . Indeed, (3.7) implies

(N) (N) (N−1) A M1 ...MN−1 = A M1 ...MN−1 A so that the matrix relation (3.33) is equivalent to the equality of the matrix coeﬃ- cients corresponding to the basis vectors of the form

(3.34) e1 ⊗ ...⊗ ei ⊗ ...⊗ eN ⊗ ej ,i,j∈{1,...,N}, where the hat indicates that the corresponding tensor factor should be skipped. Apply both sides of (3.33) to such a vector and compare the coeﬃcients of the vector (3.21). Using notation (3.19), we get the relation (−1)N−j M 1... j...N =(−1)N−i M 1...i...N ij which is equivalent to (3.31). Now, using (3.22) and (3.33) we obtain

(N) (N) (N) A cdet M = A M1 ...MN = A MN MN . On applying both sides to the vectors (3.34) we get (3.32).

If the algebra A is not commutative, then we need to distinguish between the left and right inverses of elements of A, as well as between the left and right inverses of matrices over A. Lemma 3.2.6 implies that if cdet M has a left inverse l(M) ∈A, then M is a left-invertible matrix with its left inverse found by l(M) M .Onthe other hand, if the matrix M has a right inverse M ∨, then the element cdet M is right-invertible which can be seen as follows. Consider relation (3.24) with the permutation s such that s(i)=N − i +1foralli =1,...,N,

(N) (N) (3.35) A MN ...M1 = A cdet M. It implies (N) (N) ∨ ∨ A =cdetMA M1 ...MN . Multiply both sides by A(N) from the right and note that (N) ∨ ∨ (N) (N) (3.36) A M1 ...MN A = A r(M) for a certain element r(M) ∈A. ThisimpliescdetM · r(M)=1thusverifyingthat cdet M is right-invertible. Note also that due to the associativity of matrix multiplication, if a matrix M has both left and right inverses, then they coincide. Hence, they are equal to a uniquely determined (two-sided) inverse M −1 of the matrix M. Proposition 3.2.7. If a Manin matrix M is right-invertible and cdet M is left-invertible, then M is invertible and M −1 is a Manin matrix. Proof. By the above arguments, the assumptions imply that M is an invertible matrix and cdet M is an invertible element of the algebra A. By (3.35) we have (assuming N 2)

−1 (N) (N) −1 −1 (cdet M) A MN ...M3 = A M1 M2 50 3. MANIN MATRICES so that the right hand side is unchanged after the multiplication by −P12 from the right. Hence, using (1.67) we come to (N) −1 −1 − −1 −1 (3.37) A (M1 M2 M2 M1 )=0.

Using (3.26) and taking the partial trace tr3,...,N in (3.37) we get (2) −1 −1 − −1 −1 A (M1 M2 M2 M1 )=0 so that M −1 is a Manin matrix by (3.4). Corollary 3.2.8. Under the assumptions of Proposition 3.2.7 we have −1 cdet(M −1)= cdet M . Proof. Indeed, the right inverse element r(M) deﬁned in (3.36) coincides with cdet(M −1), since M ∨ = M −1. Corollary 3.2.9. If M is a Manin matrix, then for any nonnegative integer r the following identity holds: − k l (3.38) (1 P ) [M1 ,M2 ]=0, k+l=r where k and l run over nonnegative integers. Proof. Clearly, the matrix 1 − tM with entries in the algebra A⊗C[[t]] of formal power series in a variable t is also a Manin matrix. By Proposition 3.2.7, so is its inverse, and by (3.4) we have the relation −1 −1 (1 − P ) (1 − tM1) , (1 − tM2) =0. Taking the coeﬃcient of tr we get (3.38). The next theorem provides the Newton identity for Manin matrices. As before, M denotes an arbitrary Manin matrix with entries in a unital associative algebra A. Theorem 3.2.10. We have the identity ∞ m m+1 (3.39) ∂t cdet(1 + tM) = cdet(1 + tM) (−t) tr M . m=0 Proof. By (3.22) we have (N) (N) A (1 + tM1) ...(1 + tMN )=A cdet (1 + tM). Calculate the derivative of both sides over t: N (N) (N) A (1 + tM1) ...Ma ...(1 + tMN )=A ∂t cdet(1 + tM). a=1 −1 −1 Replace the factor Ma by t (1+tMa)−t , then take the trace of both sides over all N copies of End CN and use (3.27) to get

N −1 − −1 (N) Nt cdet(1 + tM) t tr1,...,N A (1 + tM1) ...(1 + tMa) ...(1 + tMN ) a=1 = ∂t cdet(1 + tM), 3.3. BIBLIOGRAPHICAL NOTES 51 where the hat indicates the factor to be skipped. Observe that for each value of a the corresponding term in the sum coincides with the term for a = N which equals (N) (3.40) tr1,...,N A (1 + tM1) ...(1 + tMN−1). Indeed, taking the cycle permutation s =(a, a +1,...,N) we ﬁnd that the trace (3.40) equals · (N) sgn s tr1,...,N A Ps (1 + tM1) ...(1 + tMN−1) · (N) =sgns tr1,...,N A (1 + tM1) ...(1 + tMa) ...(1 + tMN )Ps · (N) =sgns tr1,...,N Ps A (1 + tM1) ...(1 + tMa) ...(1 + tMN ) (N) =tr1,...,N A (1 + tM1) ...(1 + tMa) ...(1 + tMN ), where we used (1.67) and the cyclic property of trace. Furthermore, using (3.26) and (3.33) we ﬁnd that (3.40) equals the trace of the comatrix associated with the matrix 1 + tM. Lemma3.2.6impliesthatthisequalscdet(1+tM)tr(1+tM)−1 and so we come to the identity −1 −1 −1 cdet(1 + tM) Nt − t tr(1 + tM) = ∂t cdet(1 + tM). Itcanbewrittenintheform ∞ m m+1 cdet(1 + tM) (−t) tr M = ∂t cdet(1 + tM), m=0 as required. The following equivalent form of the Newton identity holds. Corollary 3.2.11. We have ∞ −1 m m+1 −1 −∂t cdet(1 + tM) = (−t) tr M · cdet(1 + tM) . m=0 Proof. By the Leibniz rule, 0=∂ cdet(1 + tM)cdet(1+tM)−1 t −1 −1 = ∂t cdet(1 + tM) cdet(1 + tM) +cdet(1+tM) ∂t cdet(1 + tM) so that the desired identity is implied by Theorem 3.2.10.

3.3. Bibliographical notes Definition 3.1.1 goes back to the work of Manin [106], [107]. Such matrices are also known in the literature as the right-quantum matrices (with q =1);see[54]. A detailed account of their algebraic properties and applications with an extensive list of references can be found in [20]and[21]. The MacMahon Master Theorem for Manin matrices (Theorem 3.2.1) originates in [54]. More proofs and versions can be found in [43], [44], [64], [92]and[114]. Our proof follows [114], where its super- version is given. All other results, including Lemma 3.1.3, Proposition 3.2.7 and Theorem 3.2.10, are contained in [20]and[21]; see also [22]fortheirq-analogues. Some of the arguments were simplified by taking advantage of the matrix form of the definition of Manin matrices provided by Lemma 3.1.2. The assumption on cdet M in Proposition 3.2.7 was erroneously omitted in [21, Theorem 1, Section 4.3]; cf. [22, Theorem 4.7].

CHAPTER 4

Casimir elements for glN

In the following two chapters we construct ‘quantum analogues’ of the invariants in the symmetric algebras considered in Chapter 2. Those invariants will be ‘lifted’ to the universal enveloping algebra U(g) thus providing its central elements. As in Chapter 2, we will focus on such elements produced with the use of idempotents for the symmetric group and Brauer algebra. However, keeping in mind aﬃne counterparts of these constructions, we will depart from our reliance on the adjoint action of the associated group. Instead, we will develop matrix techniques by writing the commutation relations of Lie algebras in a matrix form. This will allow us to work within the Lie algebra settings. The same approach will then be applied to aﬃne Kac–Moody algebras in Chapters 7 and 8. We will rely on some standard facts about simple Lie algebras and their representations which can be found in the books by Dixmier [29], Goodman and Wal- lach [59] and Humphreys [70]. We start by recalling matrix presentations of Lie algebras.

4.1. Matrix presentations of simple Lie algebras Suppose that g is a finite-dimensional Lie algebra over C equipped with a nondegenerate symmetric invariant bilinear form , . Choose a basis J 1,...,Jd of k g and let J1,...,Jd be its dual with respect to the form so that Ji,J = δik.Let π be a faithful representation of g afforded by a finite-dimensional vector space V ,

π : g → End V.

Introduce the elements

d i G = π(J ) ⊗ Ji ∈ End V ⊗ U(g) i=1 and d i Ω= π(J ) ⊗ π(Ji) ∈ End V ⊗ End V. i=1 It is easy to verify that G and Ω are independent of the choice of the basis J i.In particular,

d i (4.1) Ω = π(Ji) ⊗ π(J ). i=1

53 54 4. CASIMIR ELEMENTS FOR glN

Consider the tensor product algebra End V ⊗ End V ⊗ U(g) and identify Ω with the element Ω ⊗ 1. Introduce elements of this algebra by

d d i i G1 = π(J ) ⊗ 1 ⊗ Ji and G2 = 1 ⊗ π(J ) ⊗ Ji. i=1 i=1 Write the commutation relations for g,

d k (4.2) [Ji,Jj ]= cij Jk k=1

k with structure coefficients cij. We will regard the universal enveloping algebra U(g) as the associative algebra with generators Ji subject to the defining relations (4.2), where the left hand side is understood as the commutator JiJj − Jj Ji. Proposition 4.1.1. The defining relations of U(g) are equivalent to the matrix relation

(4.3) G1 G2 − G2 G1 = −Ω G2 + G2 Ω. Proof. The left hand side of (4.3) reads

d i j π(J ) ⊗ π(J ) ⊗ (JiJj − Jj Ji). i,j=1 For the right hand side we have d i k (4.4) − π(J ) ⊗ π [Ji,J ] ⊗ Jk. i,k=1 By the invariance of the form, we ﬁnd k − k − k [Ji,J ],Jj = J , [Ji,Jj ] = cij. Hence (4.4) equals d k i ⊗ j ⊗ cij π(J ) π(J ) Jk. i,j,k=1 Since the representation π is faithful, we conclude that (4.3) is equivalent to the deﬁning relations (4.2) of U(g).

The deﬁning relations (4.3) can be written in an equivalent form

(4.5) G1 G2 − G2 G1 =ΩG1 − G1 Ω, which is easily veriﬁed with the use of (4.1). The element G canberegardedas an n × n matrix (n =dimV ) with entries in U(g) so that Proposition 4.1.1 gives a matrix presentation of U(g). As we will see below in Proposition 4.2.1, single matrix relations (4.3) or (4.5), encoding the commutation relations of g, are convenient for constructing central elements of U(g). 4.2. HARISH-CHANDRA ISOMORPHISM 55

4.2. Harish-Chandra isomorphism The center Z(g) of the universal enveloping algebra U(g) is defined by Z(g)={z ∈ U(g) | zu = uz for all u ∈ U(g)}. Any element of the center is called a Casimir element for g.SinceU(g) is generated by basis elements of g, for any z ∈ U(g)wehave z ∈ Z(g) if and only if zx = xz for all x ∈ g. This condition can be restricted further to a subset of elements x ∈ g which generate g as a Lie algebra. A family of Casimir elements (often called Gelfand invariants following [55]) can be produced with the use of the matrix presentations introduced in Section 4.1. Proposition 4.2.1. All elements tr Gk with k 1 belong to the center of U(g). Proof. Relation (4.3) implies k k − k r−1 − k−r − k k G1 G2 G2 G1 = G2 ( Ω G2 + G2 Ω)G2 = Ω G2 + G2 Ω. r=1 By taking trace over the second copy of End V and using its cyclic property k (Lemma 1.4.1), we get [G1, tr2 G2] = 0 as required. The Casimir elements tr Gk are widely used in representation theory, especially for the Lie algebras g of classical types. In those cases one usually takes V to be the first fundamental (or vector) representation; see Section 4.8 and Example 5.3.3 below. Now suppose that g is a simple Lie algebra over C. Choose a Cartan subalgebra h of g and a triangular decomposition g = n− ⊕ h ⊕ n+ as in Chapter 2. Recall that the Harish-Chandra homomorphism (4.6) U(g)h → U(h) is the projection of the h-centralizer U(g)h in the universal enveloping algebra to h U(h) whose kernel is the two-sided ideal U(g) ∩ U(g)n+. This ideal coincides with h U(g) ∩ n− U(g). The restriction of the homomorphism (4.6) to the center Z(g)of U(g) yields an isomorphism (4.7) χ :Z(g) → U(h)W called the Harish-Chandra isomorphism,whereU(h)W denotes the subalgebra of invariants in U(h) with respect to a certain (shifted) action of the Weyl group W of g. This leads to a description of the center Z(g) as an algebra of polynomials, C ◦ ◦ Z(g)= [P1 ,...,Pn ], ◦ ◦ for certain algebraically independent central elements P1 ,...,Pn ,wheren is the rank of g. The universal enveloping algebra U(g) is equipped with a canonical filtration so that the associated graded algebra gr U(g) is isomorphic to the symmetric algebra S(g). Moreover, the associated graded algebra gr Z(g) is isomorphic to the subalgebra of g-invariants in S(g) as defined in (2.1). For each i denote by Pi the ◦ symbol of the element Pi , that is, the image of Pi in the corresponding graded component of S(g). Then relation (2.2) holds, and the respective degrees d1,...,dn ◦ ◦ of the elements P1 ,...,Pn coincide with the exponents of g increased by 1. 56 4. CASIMIR ELEMENTS FOR glN

Now recall the general linear Lie algebra glN deﬁned by the commutation relations (2.4). We will regard the basis elements Eij of glN as generators of the universal enveloping algebra U(glN ). So we will think of U(glN ) as the associative algebra with these generators subject to the deﬁning relations

(4.8) Eij Ekl − Ekl Eij = δkj Eil − δil Ekj, i,j,k,l ∈{1,...,N}. A Taking =U(glN ) in (1.59), combine the elements Eij into the matrix E so that N ⊗ ∈ CN ⊗ E = eij Eij End U(glN ). i,j=1 Consider the tensor product algebra CN ⊗ CN ⊗ (4.9) End End U(glN ) and use the notation of Section 1.4. We identify the permutation operator (1.64) with the element P ⊗ 1 of the algebra (4.9). Proposition . 4.2.2 The deﬁning relations of the algebra U(glN ) can be written in the form

(4.10) E1 E2 − E2 E1 =(E1 − E2)P. Proof. This can be derived from Proposition 4.1.1 and it is also easy to check directly. Namely, compare the coeﬃcients of the basis vectors eij ⊗ ekl ⊗ 1onboth sides of (4.10). This is equivalent to applying the operators on both sides to the N N basis vector ej ⊗ el of C ⊗ C and then comparing the coeﬃcients of the vector ei ⊗ ek. For the left hand side we have N ei ⊗ ek ⊗ Eij Ekl − Ekl Eij , i,k=1 while for the right hand side we get N N (E1 − E2)P (ej ⊗ el)=(E1 − E2)(el ⊗ ej )= ei ⊗ ej ⊗ Eil − el ⊗ ek ⊗ Ekj. i=1 k=1

Equating the coeﬃcients of ei ⊗ ek we recover the deﬁning relations (4.8).

The advantage of having the defining relations for U(glN ) written as a single relation (4.10) for the matrix E will be apparent in Theorem 4.5.1 below, where we produce a family of Casimir elements for the Lie algebra glN . Remark 4.2.3. The matrix form of the defining relations given in Proposi- tion 4.2.2 leads to the definition of the degenerate affine Hecke algebra Hm as follows; see [31]. First note the relations

(4.11) PabEc = Ec Pab and PabEa = Eb Pab, A which hold in the algebra (1.60) with =U(glN )for1 a

(4.12) Ea Eb − Eb Ea = Ea Pab − PabEa.

It was pointed out in [3]thatHm is isomorphic to the algebra generated by (abstract) elements E1,...,Em and the group algebra C[Sm] subject to the relations (4.11) and (4.12), where Pab is understood as the transposition sab ∈ Sm.To 4.2. HARISH-CHANDRA ISOMORPHISM 57 see the connection with the deﬁnition of [31], set ua = Ea − xa for a =1,...,m, where xa is the Jucys–Murphy element deﬁned in (1.10). The elements ua pairwise commute, ua ub = ub ua, while

sa ua = ua+1 sa +1 and sb ua = ua sb if b = a − 1,a.

The map taking Ea to ua + xa and identical on C[Sm] provides an isomorphism between the two presentations of Hm.

Remark 4.2.4. The adjoint action of the group GLN of all invertible N × N C matrices over on the Lie algebra glN is deﬁned by (2.7). This extends to a unique action of the group GLN on U(glN ) so that each element of the group acts as an automorphism. The center coincides with the subalgebra of invariants under this GLN action, Z(glN )=U(glN ) . Example 4.2.5. It is an easy calculation to verify directly or with the use of Remark 4.2.4 that N N Eii and Eij Eji i=1 i,j=1 are Casimir elements for glN .

Given an N-tuple of complex numbers λ =(λ1,...,λN ), the corresponding irreducible highest weight representation L(λ)oftheLiealgebraglN is generated by a nonzero vector ξ ∈ L(λ) (the highest vector) such that

(4.13) Eij ξ =0 for 1 i

(4.14) Eii ξ = λi ξ for 1 i N. ∈ Any element z Z(glN )actsinL(λ) by multiplying each vector by a scalar χ(z). When regarded as a function of the highest weight, χ(z) is a symmetric polynomial in the variables l1,...,lN ,whereli = λi − i + 1. This provides an equivalent interpretation of the Harish-Chandra isomorphism (4.7) as the mapping z → χ(z) deﬁnes an algebra isomorphism

→ C SN (4.15) χ :Z(glN ) [l1,...,lN ] ,

SN where C[l1,...,lN ] denotes the algebra of SN -invariant (symmetric) polynomials in l1,...,lN and li is identiﬁed with the element Eii − i +1 ∈ U(h). Thus, the commutative algebra Z(glN ) can be regarded as an algebra of polynomials in N variables. The preimages of any family of algebraically independent generators of the algebra of symmetric polynomials are algebraically independent generators of Z(glN ).

Remark 4.2.6. The shifts in the deﬁnition of the variables li are determined by the half-sum of the positive roots for the simple Lie algebra slN . Clearly, given any constant a ∈ C, one could set li = λi − i + a for i =1,...,N to get an alternative deﬁnition of the isomorphism (4.15). This extra freedom is explained by the fact that the reductive Lie algebra glN is isomorphic to the direct sum of slN and the one-dimensional center spanned by the scalar matrices. Apart from the value a =1 which we use in (4.15), some other common choices of a in the literature include (N +1)/2andN. 58 4. CASIMIR ELEMENTS FOR glN

Example 4.2.7. For the Harish-Chandra images of the Casimir elements of Example 4.2.5 we have N N N χ : E → l + , ii i 2 i=1 i=1 N N N N χ : E E → l2 +(N − 1) l + . ij ji i i 3 i,j=1 i=1 i=1 They are found by the application of the Casimir elements to the highest vector ξ in the representation L(λ)ofglN .

4.3. Factorial Schur polynomials Certain particular families of symmetric polynomials will occur as Harish- Chandra images of central elements for the simple Lie algebras of all classical types. To describe them, consider the algebra of symmetric polynomials in the independent variables x1,...,xn over C and ﬁx a sequence a =(a1,a2,...) of complex numbers. The factorial elementary and complete symmetric polynomials are deﬁned by the respective formulas

(4.16) e (x ,...,x |a) k 1 n − − − = (xp1 ap1 )(xp2 ap2−1) ...(xpk apk−k+1),

1p1<···

(4.17) h (x ,...,x |a) k 1 n − − − = (xp1 ap1 )(xp2 ap2+1) ...(xpk apk+k−1),

1p1···pkn so that ek(x1,...,xn |a)=0fork>n. These polynomials are particular cases of the factorial (or double) Schur polynomials sμ(x1,...,xn |a) deﬁned as follows. Suppose that μ is a diagram with at most n rows; see Section 1.1. Then | − (4.18) sμ(x1,...,xn a)= xT (α) aT (α)+c(α) , sh(T )=μ α∈μ summed over semistandard tableaux T of shape μ with entries in {1,...,n},where T (α) denotes the entry of the tableau T at the box α =(i, j)ofμ and c(α)=j−i is the content of this box. In the particular cases μ =(1k)andμ =(k) the polynomial (4.18) coincides with (4.16) and (4.17), respectively. When a is specialized to the sequence of zeros, then (4.16) and (4.17) become the respective elementary and complete symmetric polynomials ek(x1,...,xn)and hk(x1,...,xn), and sμ(x1,...,xn |a)becomestheSchurpolynomialsμ(x1,...,xn). These homogeneous polynomials also coincide with the top degree components of the respective factorial counterparts. This implies, in particular, that the factorial Schur polynomials with μ running over all diagrams with at most n rows form a Sn basis of the algebra of symmetric polynomials C[x1,...,xn] . Some other equivalent deﬁnitions of the factorial Schur polynomials (4.18) and their basic properties are discussed in the book by Macdonald [104, Section I.3]. Here we derive their vanishing and characterization properties to be used below. 4.3. FACTORIAL SCHUR POLYNOMIALS 59

We will suppose that the sequence a is multiplicity-free, that is ak = al for all k = l.Setx =(x1,...,xn) and for any partition λ =(λ1,...,λn) such that (λ) n introduce the n-tuple of elements of a by

(4.19) aλ =(aλ1+n,...,aλn+1). As before we let λj denote the number of boxes in the column j of the diagram λ.

Proposition 4.3.1. If μ ⊂ λ then sμ(aλ |a)=0. Moreover, (4.20) s (a |a)= a − − a − . μ μ μi+n i+1 n μj +j (i,j)∈μ

Proof. The polynomials sμ(x|a) are symmetric in x, so replacing x with (xn,...,x1), we may rewrite (4.18) in the form | − (4.21) sμ(x a)= xn−T (α)+1 aT (α)+c(α) , sh(T )=μ α∈μ with the summation over semistandard μ-tableau T with entries from {1,...,n}. Suppose that sμ(aλ |a) = 0. Then at least one summand in (4.21) does not vanish for x = aλ, (4.22) a − a =0 . λn−T (α)+1+T (α) T (α)+c(α) α∈μ Since the sequence a is multiplicity free, this implies that

(4.23) λn−T (α)+1 = c(α) for all α ∈ μ. For the entries of the ﬁrst row of the tableau T we have

T (1, 1) ··· T (1,μ1).

Note that c(1, 1) = 0 so that using (4.23) with α =(1, 1) we obtain λn−T (1,1)+1 1. Furthermore, c(1, 2) = 1 and since

λn−T (1,2)+1 λn−T (1,1)+1 1, by using (4.23) with α =(1, 2) we get λn−T (1,2)+1 2. Continuing in the same manner, we conclude that λn−T (1,i)+1 i for all i =1,...,μ1. On the other hand, for the entries of the i-th column of T we have T ··· T (1,i) < < (μi,i). Therefore,

(4.24) λ −T ··· λ −T i. n (μi,i)+1 n (1,i)+1 This means that the diagram λ has at least μi rows of length at least i so that ⊂ λi μi. Hence, μ λ thus proving the ﬁrst part of the proposition. Now take λ = μ and suppose that (4.22) holds for a certain tableau T .By the above argument, the inequalities (4.24) hold for λ = μ which implies that such T T − tableau is determined uniquely by (k, i)=n μi + k for k =1,...,μi.Thus, there is a unique nonzero summand in (4.21) with x = aμ. By using the values of the tableau, we come to (4.20). 60 4. CASIMIR ELEMENTS FOR glN

The vanishing properties of the factorial Schur polynomials given by Proposi- tion 4.3.1 are characteristic in the sense of the next proposition. We suppose that μ is a diagram with at most n rows. Proposition 4.3.2. Let f(x) be a symmetric polynomial of degree not exceeding |μ|. If the top degree component of f(x) coincides with sμ(x) and f(aλ)=0for all λ such that |λ| < |μ|,thenf(x)=sμ(x|a).

Proof. Represent the difference f(x) − sμ(x|a) as a linear combination of the basis polynomials, (4.25) f(x) − sμ(x|a)= cν sν (x|a), ν∈S where cν ∈ C and the sum is taken over the set S of the diagrams ν with |ν| < |μ|. Let λ run over the same set S.Thenμ ⊂ λ and sμ(aλ |a) = 0 by Proposition 4.3.1. Hence putting x = aλ in (4.25) for all such λ, we obtain a system of linear equations on the coefficients c of the form ν cν sν (aλ |a)=0. ν∈S Equip the set S with any linear ordering ≺ such that |λ| < |ν| implies λ ≺ ν. By arranging the system of equations according to the ordering ≺ we find that the corresponding matrix is triangular due to the vanishing properties of Propo- sition 4.3.1. The diagonal entries of the matrix are the values sν (aν |a). All the − − numbers νi + n i +1andn νj + j are distinct and so the product in (4.20) is nonzero since the sequence a is multiplicity-free. Thus sν (aν |a) = 0 and the system has only the trivial solution cν =0whichmeansthatf(x)=sμ(x|a). 4.4. Schur–Weyl duality

Recall the action of the symmetric group Sm on the tensor product space N ⊗m (C ) as deﬁned in Section 1.5. The general linear group GLN acts on this space diagonally; see (2.13). The corresponding action of the Lie algebra glN is given by m ⊗ ⊗ → ⊗ ⊗ ⊗ ⊗ ∈ CN ∈ X : v1 ... vm v1 ... Xva ... vm,vi ,XglN . a=1 In line with our tensor notation, these actions are written as

h → h1 ...hm and X → X1 + ···+ Xm, where ⊗(a−1) ⊗(m−a) ⊗(a−1) ⊗(m−a) ha =1 ⊗ h ⊗ 1 and Xa =1 ⊗ X ⊗ 1 .

As we veriﬁed in Section 2.1, the action of any element s ∈ Sm on the vector N ⊗m space (C ) commutes with that of any element h ∈ GLN (and hence with ∈ the action of any element X glN ). By the classical Schur–Weyl duality,these actions of Sm and GLN centralize each other. This leads to the multiplicity-free decomposition as a representation of the group S × GL , m N N ⊗m ∼ (4.26) (C ) = Vλ ⊗ L(λ), λm, (λ)N where Vλ and L(λ) are the respective irreducible representations of Sm and GLN associated with a Young diagram λ which contains |λ| = m boxes, and the number 4.5. A GENERAL CONSTRUCTION OF CENTRAL ELEMENTS 61 of nonzero rows (λ) does not exceed N.AsaglN -module, L(λ) is the highest weight representation with the highest weight λ =(λ1,...,λN ), where one sets λi =0foralli = (λ)+1,...,N. Let U be a standard tableau of shape λ m and let eU ∈ C[Sm]bethe associated primitive idempotent; see Section 1.1. Denote by EU the image of eU N ⊗m under the action of the symmetric group Sm given by (1.65). The space EU (C ) is an irreducible representation of GLN isomorphic to L(λ). Therefore, the trace E tr1,...,m U h1 ...hm coincides with the character of the representation L(λ) evaluated at the element h. This value is given by the Weyl character formula so that the trace equals the Schur polynomial sμ evaluated at the eigenvalues h1,...,hN of the matrix h, E (4.27) tr1,...,m U h1 ...hm = sμ(h1,...,hN ); see also (2.19). In particular, the dimension dim L(λ) coincides with the trace E tr1,...,m U = sμ(1,...,1). By the correspondence (1.72), the trace formula for the idempotents given in Propo- sition 1.3.5 implies the Robinson hook dimension formula 1 (4.28) dim L(λ)= (N + j − i). h(λ) (i,j)∈λ

4.5. A general construction of central elements

Now suppose that s ∈ C[Sm] is an arbitrary element and let S denote its image under the map (1.65). We regard S as an element of the algebra CN ⊗ ⊗ CN ⊗ (4.29) End ... End U(glN ) m by identifying it with S ⊗ 1.

Theorem 4.5.1. For any s ∈ C[Sm] and u1,...,um ∈ C the element

(4.30) tr1,...,m S (u1 + E1) ...(um + Em) belongs to the center Z(glN ). Proof. Consider the tensor product CN ⊗ CN ⊗m ⊗ (4.31) End End ( ) U(glN ) with an additional copy of the endomorphism algebra End CN . We will label the copies of this algebra respectively by 0, 1,...,m. It will be suﬃcient to show that the following commutator in the algebra (4.31) is zero, (4.32) E0, tr1,...,m S (u1 + E1) ...(um + Em) =0. To this end, note that by (1.67) and Proposition 4.2.2 we can write

[E0,ua + Ea]=P0 a(ua + Ea) − (ua + Ea)P0 a 62 4. CASIMIR ELEMENTS FOR glN so that E0,S(u1 + E1) ...(um + Em) m = S (u1 + E1) ... P0 a(ua + Ea) − (ua + Ea)P0 a ...(um + Em) a=1 m m = S P0 a (u1 + E1) ...(um + Em) − S (u1 + E1) ...(um + Em) P0 a, a=1 a=1 whereweusedtheobservationthatE0 S = SE0 and that P0 a commutes with Eb for b = a. Furthermore, the sum of the permutation operators P0 a commutes with the image of any element of Sm under the map (1.65) so that m tr1,...,m S P0 a (u1 + E1) ...(um + Em) a=1 m =tr1,...,m P0 a S (u1 + E1) ...(um + Em) a=1 m =tr1,...,m S (u1 + E1) ...(um + Em) P0 a, a=1 where the last equality holds by the cyclic property of trace; see Lemma 1.4.1. Its application relies on the fact that two elements of the algebra (4.31) of the form X ⊗ 1⊗m ⊗ 1and1⊗ 1⊗m ⊗ y ∈ CN ∈ commute for any X End and y U(glN ). This proves (4.32). Remark 4.5.2. A slight modiﬁcation of the above argument shows that all elements of the form

tr1,...,m SEa1 ...Eak ∈{ } with arbitrary parameters ai 1,...,m also belong to Z(glN ).

In what follows we consider particular choices of the parameters u1,...,um and the element s ∈ C[Sm] in Theorem 4.5.1. The next lemma will allow us to rely on the properties of Manin matrices discussed in Chapter 3. Lemma 4.5.3. Suppose that α and β are elements of a certain unital associative algebra D which satisfy the relation (4.33) αβ − βα = β2. ⊗D Then the matrix α + Eβ with entries in the algebra U(glN ) is a Manin matrix. Proof. Set M = α + Eβ (we abbreviate notation by omitting the tensor product signs). Using Proposition 4.2.2 we ﬁnd M1 M2 − M2 M1 = α + E1 β α + E2 β − α + E2 β α + E1 β

2 2 =(E1 E2 − E2 E1)β − (E1 − E2)(αβ − βα)=(E1 − E2)(P − 1)β . Hence, on multiplying this element from the right by 1 + P we get 0, and the claim follows by using the equivalent deﬁnition (3.5) of Manin matrices. 4.6. CAPELLI DETERMINANT 63

Examples of elements α and β satisfying (4.33) are provided by

−1 −∂u −∂u α = −∂t,β= t and α = ue ,β= e . In the first example we can take D to be the algebra of polynomial differential operators of the form ··· k a0 + a1 ∂t + + ak ∂t ,k0, where each coefficient ai is a Laurent polynomial in t. In the second example we − take D to be generated by all polynomials in u and an additional element e ∂u , subject to the relations − − (4.34) e ∂u f(u)=f(u − 1)e ∂u for any polynomial f(u).

Lemma 4.5.4. Let u1,...,um be complex parameters and let S be the image of an element s of the center of the group algebra C[Sm] under the homomorphism (1.65). For any permutations σ, τ ∈ Sm we have the identity

tr1,...,m S (Eσ(1) + uτ(1)) ...(Eσ(m) + uτ(m))=tr1,...,m S (E1 + u1) ...(Em + um). Proof. By the cyclic property of trace, the right hand side equals −1 tr1,...,m PσS (E1 + u1) ...(Em + um)Pσ −1 =tr1,...,m SPσ (E1 + u1) ...(Em + um)Pσ

=tr1,...,m S (Eσ(1) + u1) ...(Eσ(m) + um), where we also used (1.67). Thus, it suﬃces to verify the identity in the case where σ is the identity permutation. We may assume that τ is an adjacent transposition sa =(aa+ 1). By Proposition 4.2.2,

(Ea + ua+1)(Ea+1 + ua) − (Ea+1 + ua)(Ea + ua+1)=Paa+1 Ea+1 − Ea+1 Paa+1.

Since SPaa+1 = Paa+1 S, the claim follows from the cyclic property of trace and theﬁrstpartoftheproof. In particular, Lemma 4.5.4 holds for the symmetrizer and anti-symmetrizer S = H(m) and S = A(m).

4.6. Capelli determinant Take S = A(m) with 1 m N in Theorem 4.5.1 and specialize the parameters by ua = u − a +1,a=1,...,m, for a variable u. Then the element (4.30) becomes a polynomial in u of degree m, (m) − (4.35) tr1,...,m A (u + E1) ...(u + Em m +1), whose coeﬃcients are Casimir elements for glN . Proposition 4.6.1. The Harish-Chandra images of the coeﬃcients of the polynomial (4.35) are found by

χ :tr A(m) (u + E ) ...(u + E − m +1) 1,...,m 1 m → ··· − (u + λi1 ) (u + λim m +1).

1i1<···

Proof. Use Lemma 4.5.3 to apply Proposition 3.2.2 to the Manin matrix − M =(u + E)e ∂u . By (4.34) the left hand side of (3.17) can be written as − − tr A(m) u + E e ∂u ... u + E e ∂u 1,...,m 1 m − (m) − m∂u =tr1,...,m A u + E1 ... u + Em m +1 e .

Now consider the product Miσ(1) i1 ...Miσ(m) im which occurs as a summand in the right hand side of (3.17). It is clear from (4.13) that its application to the highest vector ξ of a highest weight representation L(λ) yields zero unless σ is the identity permutation in Sm. In that case, by (4.14) − − M ...M ξ = u + E e ∂u ... u + E e ∂u ξ i1 i1 im im i1i1 imim − − m∂u = u + λi1 ... u + λim m +1 e ξ.

Taking the sum over the indices 1 i1 < ···

1i1<···

χ : C(u) → (u + l1) ...(u + lN ). ◦ ◦ In particular, the elements C1 ,...,CN are algebraically independent generators of the center Z(glN ) of the universal enveloping algebra U(glN ). 4.7. PERMANENT-TYPE ELEMENTS 65

The last property holds since the elementary symmetric polynomials are alge- SN braically independent generators of the algebra C[l1,...,lN ] .

4.7. Permanent-type elements Now take S = H(m) with m 1 in Theorem 4.5.1 and specialize the parameters by

ua = u + a − 1,a=1,...,m, for a variable u. The element (4.30) becomes a polynomial in u of degree m: (m) − (4.39) tr1,...,m H (u + E1) ...(u + Em + m 1).

Its coeﬃcients are Casimir elements for glN . Proposition 4.7.1. The Harish-Chandra images of the coeﬃcients of the polynomial (4.39) are found by

χ :tr H(m) (u + E ) ...(u + E + m − 1) 1,...,m 1 m → ··· − (u + λi1 ) (u + λim + m 1).

1i1···imN Proof. We use again Proposition 3.2.2 and Lemma 4.5.3. The left hand side − of (3.18) can be written for the Manin matrix M =(u + E + m − 1)e ∂u as − − tr H(m) u + E + m − 1 e ∂u ... u + E + m − 1 e ∂u 1,...,m 1 m − =tr H(m) u + E + m − 1 ... u + E e m∂u 1,...,m 1 m − (m) − m∂u =tr1,...,m H u + E1 ... u + Em + m 1 e , where the second equality holds due to Lemma 4.5.4. For σ ∈ Sm consider the product Mim iσ(m) ...Mi1 iσ(1) which occurs as a summand on the right hand side of (3.18). By (4.13) its application to the highest vector ξ of a highest weight representation L(λ) yields zero unless σ belongs to the stabilizer of the multiset {i1,...,im}. In this case, by (4.14) we ﬁnd − − M ...M ξ = u + E + m − 1 e ∂u ... u + E + m − 1 e ∂u ξ im im i1 i1 imim i1i1 − − m∂u = u + λim + m 1 ... u + λi1 e ξ.

Taking into account the number α1! ...αN ! of permutations which stabilize the multiset {i1,...,im}, we obtain the desired formula.

Due to Proposition 4.7.1 all coeﬃcients of the polynomial in u given by ··· − (u + λi1 ) (u + λim + m 1)

1i1···imN are symmetric polynomials in the variables li = λi − i +1 withi =1,...,N.In particular, taking u = 0 we get the factorial complete symmetric polynomial | − ··· − hm(l1,...,lN a)= li1 + i1 1 lim + im + m 2 ,

1i1···imN associated with the parameter sequence a =(ai), ai = −i + 1; see (4.17). 66 4. CASIMIR ELEMENTS FOR glN

4.8. Gelfand invariants

Take S = Pσ in Theorem 4.5.1, where σ =(m, m − 1,...,1) is a long cycle in the symmetric group Sm.Wehave

(4.40) Pσ = Pm−1 m ...P23P12.

We also set ua =0foralla =1,...,m. Then by (1.68) and the cyclic property of trace, (4.30) becomes the Casimir element tr Em known as the Gelfand invariant. Indeed, note that by (1.67)

tr1,...,m E1 ...Em Pm−1 m ...P23P12

=tr1,...,m E1 ...Em−1 Pm−1 m Em−1 Pm−2 m−1 ...P23P12.

Since trmPm−1 m = 1, applying the partial trace trm we bring the expression to the form 2 tr1,...,m−1 E1 ...Em−1 Pm−2 m−1 ...P23P12 and then continue by induction to get m (4.41) tr1,...,m Pσ E1 ...Em =trE which is therefore a Casimir element for glN for any m 1. A more direct way to come to this conclusion is to use the following straightforward consequence of Proposition 4.2.2: m − m m − m E1 E2 E2 E1 = P12 E2 E2 P12, then take the partial trace tr2 and apply Lemma 1.4.1; cf. Proposition 4.2.1. The next theorem is the Newton identity for the Gelfand invariants. The argument will be similar to the proof of Theorem 3.2.10 but we need to use a diﬀerence operator instead of the derivative. We use the Capelli determinant C(u); see (4.38). Theorem 4.8.1. We have the identity ∞ (−1)m tr Em C(u +1) 1+ = . (u − N +1)m+1 C(u) m=0 Proof. By Corollary 4.6.2, (N) − tr1,...,N A (u + E1) ...(u + EN N +1)=C(u). Hence, by Lemma 4.5.4 we also have (N) − tr1,...,N A (u + E1) ...(u + EN−1 N +2)(u + EN +1)=C(u +1). Therefore, − (N) − C(u +1) C(u)=N tr1,...,N A (u + E1) ...(u + EN−1 N +2)

(N) − −1 = N tr1,...,N A C(u)(u + EN N +1) .

Applying the conjugation by P1N we can write the right hand side as (N) − −1 N tr1,...,N A C(u)(u + E1 N +1) .

Finally, calculate the partial trace tr2,...,N by using (3.26) to get ∞ (−1)m tr Em C(u +1)− C(u)=C(u) , (u − N +1)m+1 m=0 as required. 4.9. QUANTUM IMMANANTS 67

Corollary 4.8.2. The Harish-Chandra images of the Gelfand invariants are found by ∞ (−1)m χ(tr Em) N u + l +1 (4.42) 1 + = i . (u − N +1)m+1 u + l m=0 i=1 i ¯ Equivalently, setting li = λi − i + N we have

N (¯l − ¯l +1)...(¯l − ¯l +1) (4.43) χ(tr Em)= ¯l m 1 k N k , k (¯l − ¯l ) ...∧ ...(¯l − ¯l ) k=1 1 k N k where the symbol ∧ indicates that the zero factor is skipped. Proof. Relation (4.42) follows from Corollary 4.6.2 and Theorem 4.8.1. Set- ting v = u − N + 1 we can write this relation in the form ∞ N v + ¯l +1 1+ (−1)m χ(tr Em) v−m−1 = i . v + ¯l m=0 i=1 i Now use a partial fraction decomposition for the right hand side,

N v + ¯l +1 a a i =1+ 1 + ···+ N . v + ¯l v + ¯l v + ¯l i=1 i 1 N ¯ ¯ The constants ak are found by multiplying both sides by v +lk and setting v = −lk. This gives (4.43) by expanding the rational functions into power series in v−1.

The Harish-Chandra images of certain modiﬁed Gelfand invariants will be calculated in Corollary 13.4.3 below.

4.9. Quantum immanants As in Section 1.1, suppose that U is a standard tableau of shape μ m and let eU ∈ C[Sm] be the associated primitive idempotent. By ca = ca(U)wedenote the content c(α)=j − i of the box α =(i, j)inU occupied by a ∈{1,...,m}.As before, we let EU denote the image of eU under the action of the symmetric group Sm given by (1.65). The operator EU is zero if the length (μ)ofμ exceeds N.So we will assume that (μ) N. Now specialize the parameters in Theorem 4.5.1 by

ua = u + ca,a=1,...,m for a variable u and take S = EU . Denote the corresponding Casimir element by S E (4.44) μ(u)=tr1,...,m U (u + E1 + c1) ...(u + Em + cm).

Proposition 4.9.1. The polynomial S μ(u) is independent of the choice of the standard tableau U of shape μ. Moreover, the Harish-Chandra images of its coeﬃ- cients are found by χ : S μ(u) → u + λT (α) + c(α) , sh(T )=μ α∈μ summed over semistandard tableau T of shape μ with entries in {1,...,N}. 68 4. CASIMIR ELEMENTS FOR glN

We postpone the proof till Chapter 10 as Proposition 4.9.1 will follow from a more general result on Yangian characters; see Remark 10.1.5(i). In the particular cases where μ is the column or row-diagram, the statement reduces to Proposi- tions 4.6.1 and 4.7.1, as S μ(u) coincides with the polynomials (4.35) and (4.39), respectively. The Harish-Chandra image of the polynomial S μ(u) is a symmetric polynomial in the variables li = λi − i +1 with i =1,...,N. The Casimir elements S μ = S μ(0) given by S E (4.45) μ =tr1,...,m U (E1 + c1) ...(Em + cm) are called the quantum immanants. Their images are the factorial Schur polynomials sμ(l1,...,lN |a)= lT (α) + T (α)+c(α) − 1 , sh(T )=μ α∈μ associated with the parameter sequence a =(ai), ai = −i + 1; see (4.18). The following is a quantum version of Proposition 2.1.3.

Corollary 4.9.2. The quantum immanants S μ with μ running over all diagrams with at most N rows form a basis of Z(glN ).

Proof. The factorial Schur polynomial sμ(l1,...,lN |a) is a non-homogeneous symmetric polynomial in l1,...,lN whose top degree component coincides with the Schur polynomial sμ(l1,...,lN ). Since the Schur polynomials form a basis of the algebra of symmetric polynomials so do their factorial counterparts. The claim follows due to the Harish-Chandra isomorphism (4.15). Remark 4.9.3. Our definition of the quantum immanants (4.45) is slightly different from the original one due to Okounkov [128], where they are given by E − − (4.46) tr1,...,m U (E1 c1) ...(Em cm). These elements possess certain stability properties which motivated their definition. To make a connection between the two families, consider the automorphism of the algebra U(glN ) defined by

(4.47) φ : Eij →−Eji, so that φ takes E to the negative transpose matrix −E t.Then S → E − t − t φ : μ tr1,...,m U ( E1 + c1) ...( Em + cm).

It follows by induction from (1.14) that the element EU is stable under the transposition applied simultaneously to all m copies of End CN in (4.29). Since this operation does not aﬀect the trace, we obtain S → − m E − − φ : μ ( 1) tr1,...,m U (E1 c1) ...(Em cm).

On the other hand, by twisting the action of glN on the ﬁnite-dimensional irreducible representation L(λ1,...,λN ) by the automorphism (4.47) we obtain a representation isomorphic to L(−λN ,...,−λ1). Hence, the shifted variables li = λi−i+1 get mapped by →− − li lN−i+1 N +1,i=1,...,N. The conclusion is that the Harish-Chandra image of the Casimir element (4.46) coincides with the polynomial sμ(l1 + N − 1,...,lN + N − 1|a); see also [109, Theorem 7.4.6]. 4.10. BIBLIOGRAPHICAL NOTES 69

4.10. Bibliographical notes Matrix presentations of simple Lie algebras were used by Gould [60] in relation with characteristic identities. In a different way, they also emerge in the work of Drinfeld [32] in the context of quantum groups and can be obtained by restricting the R-matrix presentations of the Yangians Y(g) to the subalgebras U(g). A detailed exposition of properties of the factorial Schur functions can be found e.g. in the paper by Macdonald [103]. Propositions 4.3.1 and 4.3.2 are due to Ok- ounkov [128]. More details on the use of the idempotents for the derivation of the Robinson formula (4.28) can be found in [117]; see also [34]and[104, Sec- tion I.3, Example 4] for other proofs. The first direct proof that the coefficients of the Capelli determinant C(u) in (4.38) are Casimir elements was given by Howe and Umeda [68]. The images of the Gelfand invariants under the Harish-Chandra isomorphism (Corollary 4.8.2) were first found by Perelomov and Popov [133]. This corollary implies the Newton identity of Theorem 4.8.1. Direct proofs of this theorem were given by Umeda [148]andItoh[76]; see also [109, Chapter 7] for its derivation with the use of the Yangians and more references. The matrices of the form α + Eβ as in Lemma 4.5.3 are among principle examples of Manin matrices; see [21]. The quantum immanants and related higher Capelli identities are due to Okounkov [128]; see also Nazarov [124] and Okounkov and Olshanski [129].

CHAPTER 5

Casimir elements for oN and spN

Extending the arguments of Chapter 4, we will now use the Brauer–Schur–Weyl duality between the Brauer algebra and the classical groups in types B, C and D to construct Casimir elements for the orthogonal and symplectic Lie algebras. As with type A, our approach will be based on matrix presentations of these Lie algebras provided by Proposition 4.1.1.

5.1. Harish-Chandra isomorphism

Recall from Section 2.2 that the Lie subalgebra of glN spanned by the elements

(5.1) Fij = Eij − Eji ,i,j=1,...,N, is the orthogonal Lie algebra oN . As before, we use the notation i = N − i +1.If N =2n + 1 is odd with n 1, then this is a simple Lie algebra of type Bn,andif N =2n is even with n 3, then this is a simple Lie algebra of type Dn.TheLie algebra o4 is semi-simple, it is isomorphic to the direct sum sl2 ⊕ sl2.Inbothodd and even cases, (4.8) implies the commutation relations

(5.2) Fij Fkl − Fkl Fij = δkj Fil − δil Fkl − δki Fjl + δjl Fki , for all i, j, k, l ∈{1,...,N}. Note also the symmetry relation

Fij + Fji =0,i,j∈{1,...,N}.

Similarly, if N =2n is even, then the Lie subalgebra of gl2n spanned by the elements

(5.3) Fij = Eij − εi εj Eji ,i,j=1,...,2n, is the symplectic Lie algebra sp2n which is a simple Lie algebra of type Cn.Weset εi =1fori =1,...,nand εi = −1fori = n +1,...,2n. We have the commutation relations (5.4) Fij Fkl − Fkl Fij = δkj Fil − δil Fkl − εi εj δki Fjl − δjl Fki for all i, j, k, l ∈{1,...,2n}. Note also the symmetry relation

Fij + εi εj Fji =0,i,j∈{1,...,2n}.

We will use the uniform notation gN for the Lie algebra oN (with N =2n or N =2n +1)or spN (with N =2n). Combine the elements Fij into the matrix F =[Fij] and identify it with the element N N (5.5) F = eij ⊗ Fij ∈ End C ⊗ U(gN ). i,j=1 Consider the tensor product algebra N N End C ⊗ End C ⊗ U(gN )

71 72 5. CASIMIR ELEMENTS FOR oN AND spN and use the operators P = P12 and Q = Q12 deﬁned in (1.64), (1.70) and (1.74) for the respective cases. Observe that the application of the transposition (2.27) to the operator P with respect to either copy of the endomorphism algebra yields the operator Q.

Proposition 5.1.1. The deﬁning relations of the algebra U(gN ) can be written in the form

(5.6) F1 F2 − F2 F1 =(P − Q) F2 − F2 (P − Q) together with the relation F + F =0. Proof. Clearly, F + F = 0 is equivalent to the symmetry relations. One way to proceed with (5.6) is to derive it from the general Proposition 4.1.1. We will use Lemma 2.2.1 instead. Write (4.10) in the form

E1 E2 − E2 E1 = PE2 − E2 P. The application of the transposition (2.27) to the ﬁrst copy of End CN in (4.9) gives − − E1 E2 E2 E1 = QE2 E2 Q, while its application to the second copy with the use of Lemma 2.2.1 gives − − E1 E2 E2 E1 = E2 Q QE2. Now applying the transposition to the ﬁrst copy of End CN in the latter relation we come to − − E1 E2 E2 E1 = E2 P PE2. These four relations imply − − − − − − F1 F2 F2 F1 =(E1 E1)(E2 E2) (E2 E2)(E1 E1) − − − − = PE2 E2 P QE2 + E2 Q E2 Q + QE2 + E2 P PE2

=(P − Q) F2 − F2 (P − Q), as claimed. By (2.28), for the matrix F we have

(5.7) QF1 + QF2 =0 and F1 Q + F2 Q =0. Together with (1.67) this implies an equivalent form of relations (5.6),

F1 F2 − F2 F1 = F1 (P − Q) − (P − Q) F1.

Remark 5.1.2. The matrix form of the defining relations for U(gN )asprovided by Proposition 5.1.1 leads to the definition of a Brauer-type analogue of the degenerate affine Hecke algebra; cf. Remark 4.2.3. Consider the algebra Fm(ω) generated by the Brauer algebra Bm(ω)andthe(abstract)elementsF1,...,Fm subject to the relations

PabFc = Fc Pab,QabFc = Fc Qab, where 1 a

PabFa = Fb Pab,QabFa + QabFb =0 and Fa Qab + Fb Qab =0, together with

Fa Fb − Fb Fa = Fa (Pab − Qab) − (Pab − Qab)Fa, 5.1. HARISH-CHANDRA ISOMORPHISM 73 where Pab and Qab are understood as the respective elements sab and ab of Bm(ω). The quotient of Fm(ω) by some additional relations is known as the Nazarov–Wenzl algebra and was originally introduced in [123].

Given any n-tuple of complex numbers λ =(λ1,...,λn), the corresponding irreducible highest weight representation L(λ)oftheLiealgebragN is generated by a nonzero vector ξ ∈ L(λ) (the highest vector) such that

Fij ξ =0 for 1 i

Fii ξ = λi ξ for 1 i n.

Asbefore,wedenotebyZ(gN ) the center of the universal enveloping algebra U(gN ). Any element z ∈ Z(gN )actsinL(λ) by multiplying each vector by a scalar χ(z). When regarded as a function of the highest weight, χ(z)isapolynomial in λ1,...,λn with certain symmetry properties arising from the action of the corresponding Weyl group W . Introduce shifted variables l1,...,ln by li = λi + ρi, where the ρi are the coordinates of the half-sum of the positive roots of gN so that ρ = n − i + ε with i ⎧ ⎪ ⎨0forgN = o2n, (5.8) ε = 1 for g = o , ⎩⎪ 2 N 2n+1 1forgN = sp2n.

In types Bn and Cn the polynomial χ(z)inl1,...,ln is invariant under any permutations of the variables and under the changes of signs li →−li of any subset of the variables. In type Dn the polynomial χ(z)inl1,...,ln is invariant under any permutations of the variables and under the changes of signs li →−li of any subset containing an even number of the variables. In all three cases, the mapping z → χ(z) defines an algebra isomorphism W (5.9) χ :Z(gN ) → C[l1,...,ln] to the algebra of invariant polynomials. This provides an equivalent interpretation of the Harish-Chandra isomorphism defined in (4.7), where li is identified with the element Fii + ρi ∈ U(h). Note also that in types Bn and Cn the image of the isomorphism coincides with the algebra of symmetric polynomials in the variables 2 2 l1,...,ln, C W C 2 2 Sn [l1,...,ln] = [l1,...,ln] . In type Dn, the image of the isomorphism is generated by the algebra of symmetric 2 2 polynomial in the variables l1,...,ln and the polynomial l1 ...ln; cf. (2.35). Recall from Section 2.2 that the classical group GN =ON or GN =SpN is defined as the group of complex matrices preserving the respective form (2.25). The adjoint action of the group GN on gN is defined by (2.30). This extends to a unique action of GN on U(gN ) so that each element of the group acts as an automorphism. GN The subalgebra of invariants U(gN ) under this action coincides with the center Z(gN )intypesBn and Cn, and is properly contained in the center in type Dn.In all three cases, the restriction of (5.9) induces the isomorphism

GN → C 2 2 Sn (5.10) χ :U(gN ) [l1,...,ln] . We will need the factorial elementary and complete symmetric polynomials as given in (4.16) and (4.17) for the particular sequences a deﬁned by (5.11) a =(ε2, (ε +1)2, (ε +2)2,...), 74 5. CASIMIR ELEMENTS FOR oN AND spN

2 where ε is introduced in (5.8), so that ai =(ε + i − 1) . GN Note that any element z ∈ U(gN ) is uniquely determined by the eigenvalues χ(z) in the irreducible modules L(λ), where λ =(λ1,...,λn) runs over the set of partitions with (λ) n. Hence, we come to the following characterization properties of Casimir elements.

GN Proposition 5.1.3. Suppose that k is a nonnegative integer and z ∈ U(gN ) is an element of degree 2k with respect to the canonical ﬁltration, which vanishes in each representation L(λ) with |λ|

5.2. Brauer–Schur–Weyl duality

Finite-dimensional irreducible representations of the orthogonal group ON are parameterized by all diagrams λ with the property λ1 + λ2 N,whereλj denotes the number of boxes in the column j of λ. The corresponding representation will be denoted by L(λ). Let λ∗ be the diagram obtained from λ by replacing the first − column with the column containing N λ1 boxes. If N =2n+1 and λ1 n then the associated representation of the Lie algebra oN in the space L(λ) is irreducible and isomorphic to the representation L(λ) whose highest weight (λ1,...,λn)coincides ∗ with λ;ifλ1 >nthen the associated representation of oN is isomorphic to L(λ ). If N =2n and λ1 nthe ∗ associated representation of oN is isomorphic to L(λ ). If N =2n and λ1 = n then the associated representation of oN in L(λ) is isomorphic to the direct sum of two ◦ ◦ irreducible representations L(λ)andL(λ ) with λ =(λ1,...,λn−1, −λn). Finite-dimensional irreducible representations of the symplectic group SpN with N =2n are parameterized by partitions λ whose lengths do not exceed n.The associated representation of the Lie algebra spN is irreducible and isomorphic to L(λ). As GN is a subgroup of GLN , the action of GN on the tensor product space N ⊗m (C ) can be defined by the restriction of the diagonal action of GLN defined in (2.13). The centralizer of this action in the endomorphism algebra End (CN )⊗m coincides with the homomorphic image of the Brauer algebra Bm(ω) with the parameter ω specialized to N and −N, respectively, in the orthogonal and symplectic case, where the homomorphism is defined in Section 1.5. This implies the tensor product decomposition analogous to (4.26): for GN =ON we have m/2 N ⊗m ∼ (5.12) (C ) = Vλ ⊗ L(λ), f=0 λm−2f λ1+λ2 N 5.2. BRAUER–SCHUR–WEYL DUALITY 75 where Vλ and L(λ) are the respective irreducible representations of Bm(N)andON associated with the diagram λ. Similarly, in the symplectic case with N =2n, m/2 N ⊗m ∼ (5.13) (C ) = Vλ ⊗ L(λ ), f=0 λm−2f λ1n where Vλ and L(λ ) are the respective irreducible representations of Bm(−N)and SpN associated with λ and λ . Now suppose that λ is a partition of m with λ1 n in the orthogonal case and λ1 n in the symplectic case. Let U be a standard tableau of shape λ which we also regard as an updown tableau. We let eU be the associated primitive idempotent of the respective Brauer algebra Bm(N)orBm(−N); see Section 1.2. Denote by EU the image of eU under the respective action of the Brauer algebra as defined in Section 1.5. Denote by V the standard tableau obtained from U by deleting the box occupied by m and let μ be the shape of V. By the decompositions (5.12) and N ⊗m (5.13), the subspace EU (C ) is an irreducible representation of GN isomorphic to L(λ) in the orthogonal case and to L(λ) in the symplectic case. Therefore, E the trace tr1,...,m U equals the dimension of the respective representation L(λ)or L(λ). Thus, by calculating the trace we can recover the well-known hook dimension formulas for these representations; cf. the proof of (4.28). We omit the calculations and only state the formulas. For GN =ON we set 1 D(λ)= N − 1+d(i, j) , h(λ) (i,j)∈λ where λ + λ − i − j +1 if i j, d(i, j)= i j − − − λi λj + i + j 1ifi>j. We have the relation D(λ) tr E = E m U V D(μ) which implies (5.14) dim L(λ)=D(λ).

Similarly, if GN =SpN , for any diagram ρ with at most n rows set 1 D(ρ)= N +1+d(i, j) , h(ρ) (i,j)∈ρ where the parameters d(i, j) are now deﬁned by ρ + ρ − i − j +1 if i>j, d(i, j)= i j − − − ρi ρj + i + j 1ifi j. Now we have D(λ) tr E = E m U V D(μ) so that (5.15) dim L(ρ)=D(ρ). 76 5. CASIMIR ELEMENTS FOR oN AND spN

5.3. A general construction of central elements

Suppose that s ∈Bm(ω) is an arbitrary element, where the parameter ω of the Brauer algebra is specialized by N for g = o , (5.16) ω = N N − N for gN = spN . We let S denote the respective image of s under the map (1.69) or (1.73), which is also regarded as the element S ⊗ 1 of the algebra CN ⊗ ⊗ CN ⊗ (5.17) End ... End U(gN ). m The following is an analogue of Theorem 4.5.1 for the orthogonal and symplectic Lie algebras and it is proved by a similar argument.

Theorem 5.3.1. For any s ∈Bm(ω) and u1,...,um ∈ C the element

(5.18) tr1,...,m S (u1 + F1) ...(um + Fm) belongs to the center Z(gN ). Proof. Take the tensor product

N N ⊗m (5.19) End C ⊗ End (C ) ⊗ U(gN ) with the copies of the endomorphism algebra End CN labeled by 0, 1,...,m.Itis suﬃcient to show that (5.20) F0, tr1,...,m S (u1 + F1) ...(um + Fm) =0 in the algebra (5.19). By Proposition 5.1.1 we can write

[F0,ua + Fa]=(P0 a − Q0 a)(ua + Fa) − (ua + Fa)(P0 a − Q0 a) so that m F0,S(u1 + F1) ...(um + Fm) = S (P0 a − Q0 a)(u1 + F1) ...(um + Fm) a=1 m − S (u1 + F1) ...(um + Fm) (P0 a − Q0 a). a=1 m − By the properties of the Jucys–Murphy elements (1.26), the sum a=1(P0 a Q0 a) commutes with S, so that the trace tr1,...,m of the above commutator vanishes due to the cyclic property of trace.

Remark 5.3.2. Although we will not use it below, we point out a more general family of Casimir elements for gN . Using the assumptions of Theorem 5.3.1, consider the element

(5.21) tr1,...,m S (u1 + F1)R1(u2 + F2) ...Rm−1(um + Fm), where

Ra = ra + raa+1 Qaa+1 + ···+ ramQam,a=1,...,m− 1, 5.3. A GENERAL CONSTRUCTION OF CENTRAL ELEMENTS 77 for some complex numbers ra and rab. Applying the same approach as in the proof of Theorem 5.3.1, we can see that the element (5.21) belongs to the center Z(gN ). One extra step in the argument is the use of the relations

Qca(P0 a − Q0 a)=−Qca(P0 c − Q0 c)and

(P0 a − Q0 a)Qab = −(P0 b − Q0 b)Qab whichholdforc

(5.22) tr1,...,m SFa1 ...Fak with arbitrary parameters ai ∈{1,...,m} also belong to Z(gN ).

Example 5.3.3. As in Section 4.8, take S = Pσ in Theorem 5.3.1, where σ =(m, m−1,...,1) is a long cycle (see (4.40)) and set ua =0foralla =1,...,m. Repeating the argument of Section 4.8 we get

m tr1,...,m Pσ F1 ...Fm =trF which is therefore a Casimir element for gN for any m 1. It is known as the Gelfand invariant for the Lie algebra gN . In fact, it belongs to the algebra of GN m invariants U(gN ) . The Harish-Chandra image χ(tr F ) is a symmetric polyno- 2 2 mial in l1,...,ln; see (5.10). If m =2k is even, then the top degree component of 2k 2k ··· 2k the polynomial χ(tr F )coincideswith2(l1 + + ln ). Therefore, the Gelfand invariants tr F 2, tr F 4,...,tr F 2n

GN are algebraically independent generators of the algebra U(gN ) . There are analogues of the Newton identities (cf. Theorem 4.8.1) relating the Gelfand invariants with Capelli-type determinants. However, the known proofs of these identities rely on some other methods as compared to Theorem 4.8.1; see Section 5.7 for references.

The following is an analogue of Lemma 4.5.4 for the orthogonal and symplectic Lie algebras.

Lemma 5.3.4. Let u1,...,um be complex parameters and let S be the image of an element s of the center of the Brauer algebra Bm(ω)(with ω specialized as in (5.16)), under the respective homomorphism (1.69) or (1.73). For any permutations σ, τ ∈ Sm we have the identity

tr1,...,m S (Fσ(1) + uτ(1)) ...(Fσ(m) + uτ(m))=tr1,...,m S (F1 + u1) ...(Fm + um). Proof. We argue as in the proof of Lemma 4.5.4, except that the commutation relations are now implied by Proposition 5.1.1 to give

(Fa + ua+1)(Fa+1 + ua) − (Fa+1 + ua)(Fa + ua+1)

=(Paa+1 − Qaa+1)Fa+1 − Fa+1 (Paa+1 − Qaa+1). This leads to the desired relation. 78 5. CASIMIR ELEMENTS FOR oN AND spN

5.4. Symmetrizer and anti-symmetrizer for oN Here we formulate analogues of Proposition 4.6.1 and 4.7.1 for the Lie algebras gN = oN with N =2n +1 or N =2n (the symplectic case will be considered in the next section). Unlike glN , the proofs of these analogues are more involved due to the lack of Manin matrix techniques. Although there are certain analogues of Manin matrices in types B, C and D (see Section 5.6 below), they do not seem to suit well the calculations of Harish-Chandra images. Our proofs will instead rely on the properties of factorial symmetric polynomials; see Proposition 5.1.3. Recall that the symmetrizer s(m) and anti-symmetrizer a(m) in the Brauer (m) (m) algebra Bm(ω) are defined Section 1.2, and S and A are the respective images of these elements under the action of the corresponding Brauer algebra in the tensor spaces, defined by (1.69) with ω = N. We will use the function γm(ω)ofthe parameter ω defined by (2.40). The factorial elementary and complete symmetric polynomials defined in (4.16) and (4.17) will be associated with the sequence of parameters a given in (5.11). Take even values m =2k and set S = S(2k) with k 1 in Theorem 5.3.1. By the cyclic property of trace we can place S(2k) as the last factor in (5.18). We choose a particular specialization of the parameters u1,...,u2k as follows. Proposition 5.4.1. For any k 1 the image of the Casimir element − − (2k) ∈ (5.23) γ2k(N)tr1,...,2k (F1 + k 1) ...(F2k k) S Z(oN ) 2 2 | under the Harish-Chandra isomorphism coincides with hk(l1,...,ln a). Proof. It will be convenient for the arguments below to interpret the tensor notation in a certain dual way. Namely, we will write the tensor factors in (5.17) in a different order, by placing U(gN )first: ⊗ CN ⊗ ⊗ CN (5.24) U(gN ) End ... End . m Accordingly, we swap the factors in (5.5) so that now

N ⊗(a−1) ⊗(m−a) (5.25) Fa = Fij ⊗ 1 ⊗ eij ⊗ 1 . i,j=1 This does not aﬀect the notation for the action of the Brauer algebra. In particular, Theorem 5.3.1 holds in the same form. Denote the Casimir element (5.23) by Dk. We will use Proposition 5.1.3 and start by showing that Dk vanishes in all representations L(λ)ofoN ,wherethe partitions λ satisfy |λ|

Fij →−eji + eij , 1 i, j N, so that this representation is isomorphic to L(1, 0,...,0). The space of tensors N ⊗r (C ) then also becomes a representation of oN . With the notation (5.25), under the corresponding homomorphism

N ⊗2k N ⊗r N ⊗2k ϕ :U(oN ) ⊗ End (C ) → End (C ) ⊗ End (C ) 5.4. SYMMETRIZER AND ANTI-SYMMETRIZER FOR oN 79 we have r (5.26) ϕ(Fa)= −P br+a + Q br+a ,a=1,...,2k. b=1 By (5.12), the decomposition of (CN )⊗r into a direct sum of irreducible representations of oN contains L(λ) with a nonzero multiplicity. Hence, the desired vanishing condition of Proposition 5.1.3 will follow if we show that (2k) (5.27) ϕ(F1)+k − 1 ... ϕ(F2k) − k S =0, (2k) where S denotes the image of the symmetrizer in the Brauer algebra B2k(N) acting on the last 2k copies of the tensor product space (CN )⊗(r+2k). Due to (5.26) and relations (1.31), the desired identity (5.27) can be written in the form (2k) (5.28) −Yr+1 + k − 1 ... −Yr+2k + k − 1 S =0, where Yr+b with b =1,...,2k are the images of the respective Jucys–Murphy elements yr+b − (ω − 1)/2 given in (1.26) under the action of the Brauer algebra N ⊗(r+2k) Br+2k(N)in(C ) . To prove the identity (5.28) note that the two operators (2k) N ⊗(r+2k) (−Yr+1 + k − 1) ...(−Yr+2k + k − 1) and S on the vector space (C ) commute with the action of ON . We will show that their images have zero intersection. To describe the image of the ﬁrst operator, represent the vector space as the direct sum of irreducible representations of ON , r +k 2 N ⊗(r+2k) N ⊗(r+2k) (C ) = EU (C ) , l=0 νr+2k−2l U where the last sum is taken over all updown tableaux U =(Λ1,...,Λr+2k)ofshape ν associated with the Brauer algebra Br+2k(N). We claim that if U is an updown tableau of shape ν =Λr+2k with ν1 k then (5.29) −Yr+1 + k − 1 ... −Yr+2k + k − 1 EU =0.

Indeed, since r

Proof. Observe that by the definition of trace, Dm(u) is stable under the transposition (2.24) applied simultaneously to each of the m copies of the algebra End CN . This simultaneous transposition is an anti-automorphism of the algebra N ⊗m End (C ) which preserves all elements Pab and Qab. Hence the symmetrizer S(m) is also stable under this transposition, as can be easily seen from the formulas of Section 1.2, for instance, (1.46). On the other hand, since F = −F ,using Lemma 5.3.4, for the image of Dm(u) under this anti-automorphism we find m − 1 m − 1 D (u)=γ (N)tr −F + u + ... −F + u − S(m) m m 1,...,m 1 2 m 2 m − 1 m − 1 =(−1)m γ (N)tr F − u − ... F − u + S(m) m 1,...,m 1 2 m 2 m − 1 m − 1 =(−1)m γ (N)tr F − u + ... F − u − S(m) m 1,...,m 1 2 m 2 m which coincides with (−1) Dm(−u). This shows that the polynomials D2k(u)are even, while the polynomials D2k−1(u) are odd; in particular, D2k−1(0) = 0. The value χ D2k(1/2) = χ D2k(−1/2) is found in Proposition 5.4.1. These values agree with the Harish-Chandra images provided in the statement of the corollary. Hence, the proof will be completed if we show that the polynomials Dm(u)andthe 5.4. SYMMETRIZER AND ANTI-SYMMETRIZER FOR oN 81 polynomials which are claimed to be their Harish-Chandra images satisfy the same recurrence relations. We show first that

(5.31) Dm(u +1/2) − Dm(u − 1/2) = (N + m − 2) Dm−1(u) for all m 1, where we set D0(u) = 1. By Lemma 5.3.4 we can write m − Dm(u +1/2) = γm(N)tr1,...,m F1 + u + 1 2 m m (m) ×···× F − + u − − 1 F + u + S , m 1 2 m 2 so that D (u +1/2) − D (u − 1/2) m m m m (m) = mγ (N)tr F + u + − 1 ... F − + u − − 1 S . m 1,...,m 1 2 m 1 2 Due to Lemma 1.3.2, the partial trace of the symmetrizer is found by (N + m − 3)(N +2m − 2) (5.32) tr S(m) = S(m−1) m m (N +2m − 4) thus verifying (5.31). It is straightforward to check that the same relation is satisﬁed by the alleged images χ Dm(u) as stated in the corollary. To prove the second part, note that by the relation m −1 −1 z −∂z + F1 z ... −∂z + Fm z = −z∂z + F1 + m − 1 ... −z∂z + Fm the polynomial D (u)canbewrittenintheform m m − −1 − −1 (m) z γm(N)tr1,...,m ∂z + F1 z ... ∂z + Fm z S with the replacement of −z∂z by u − (m − 1)/2. Similarly, for any k 0, k −1 k z −∂z + rz = −z∂z + r + k − 1 ... −z∂z + r , so that the second relation follows from the ﬁrst. Now we will prove analogues of Proposition 5.4.1 and Corollary 5.4.2, where theroleofthesymmetrizerS(m) is taken by the anti-symmetrizer A(m).First,set S = A(2k) in Theorem 5.3.1. Proposition 5.4.3. For any 1 k n the image of the Casimir element − (2k) ∈ (5.33) tr1,...,2k (F1 k +1)...(F2k + k) A Z(oN ) − k 2 2 | under the Harish-Chandra isomorphism coincides with ( 1) ek(l1,...,ln a).

Proof. Denote the Casimir element (5.33) by Ck. We use Proposition 5.1.3 and show that Ck vanishes in all representations L(λ)ofoN , where the partitions λ satisfy |λ|

To verify this relation, we show exactly as in the proof of Proposition 5.4.1 that the image of the operator −Yr+1 − k +1 ... −Yr+2k − k +1 on the vector N ⊗(r+2k) space (C ) is contained in a direct sum of representations L(ν)ofON with (ν)

m 2 N − 2r m−2r → (−1)r e (l2,...,l2 |a) z−2r −∂ + rz−1 . m − 2r r 1 n z r=0 Proof. We argue as in the proof of Corollary 5.4.2. Using Lemma 5.3.4 m we show ﬁrst that Cm(u)=( −1) Cm(−u)sothatC2k−1(0) = 0. Moreover, χ C2k(1/2) = χ C2k(−1/2) is found from Proposition 5.4.3. As a next step, we verify that

(5.35) Cm(u +1/2) − Cm(u − 1/2) = (N − m +1)Cm−1(u),m 1, where C0(u) = 1. This follows easily from Lemma 5.3.4, and the formula (3.26) for the partial trace of the anti-symmetrizer, thus verifying (5.35). The same relation is satisﬁed by the polynomials which are claimed to be the images χ Cm(u) as stated in the corollary. The second part follows from the ﬁrst; cf. the proof of Corollary 5.4.2.

In the case gN = o2n introduce the (noncommutative) Pfaﬃan Pf F as the element of the universal enveloping algebra U(o2n) deﬁned by the formula 1 (5.36) Pf F = sgn σ · F ...F − . 2nn! σ(1) σ(2) σ(2n 1) σ(2n) σ∈S2n

The Pfaﬃan belongs to the center Z(o2n) and its image under the Harish-Chandra isomorphism is found by χ :Pf F → l1 ...ln. 1 Example 5.4.5. Since Fij + Fji =0,forn =1wehave

Pf F = F11,χ:PfF → l1.

1We thus use an obvious version of the Harish-Chandra isomorphism for the abelian Lie algebra o2. 5.5. SYMMETRIZER AND ANTI-SYMMETRIZER FOR spN 83

For n =2weobtain 1 Pf F = (F F − F F + F F + F F − F F + F F ) 2 13 31 12 21 11 22 31 13 21 12 22 11 = F11 F22 − F21 F12 + F31 F13 + F22 and χ :PfF → λ1 λ2 + λ2 so that the image coincides with l1 l2.

Corollary 5.4.6. If gN = o2n+1 then the elements of each family (5.23) and (5.33) with k =1,...,n are algebraically independent generators of the center Z(o2n+1). If gN = o2n then Pf F together with the elements of either family (5.23) or (5.33) with k =1,...,n− 1 are algebraically independent generators of the center Z(o2n). Proof. By Propositions 5.4.1 and 5.4.3, the images of the elements of each family under the isomorphism (5.9) are algebraically independent generators of the W algebra of invariant polynomials C[l1,...,ln] .

5.5. Symmetrizer and anti-symmetrizer for spN

Now we let gN = spN with N =2n. Note that the image of the symmetrizer (m) h ∈ C[Sm] under the action (1.73) coincides with the anti-symmetrization operator on the tensor space (C2n)⊗m. Therefore, the image of h(m) is the zero operator for all m>2n. On the other hand, as we pointed out in Section 2.2, the image (m) (m) S of the symmetrizer s ∈Bm(ω) under the action (1.73) with ω = −2n is well-defined provided that m n +1andS(n+1) = 0; see Proposition 2.3.2. The operator S(m) is not defined for m n+2 since some denominators in the formulas defining s(m) vanish for ω = −2n. Consider the expression − (m) (5.37) γm( 2n)tr1,...,m S (u1 + F1) ...(um + Fm) obtained by taking S = S(m) in (5.18) and multiplying it by the coefficient (2.40) with ω = −2n. For any fixed m regard (5.37) as a function of n which is defined for all integer values n m. Our next goal is to show that (5.37) is well-defined for all integer values n (m − 1)/2 (that is, it has ‘removable singularities’ at the extra values on n). In other words, for a given n, the expression (5.37) can be evaluated for all m such that 1 m 2n + 1; cf. Proposition 2.2.8. Lemma 5.5.1. Suppose that n m. For any 1 a

Fa Fb − Fb Fa = Fa (Pab − Qab) − (Pab − Qab)Fa. Therefore, both relations follow from (1.31), (1.67) and (1.73) with the use of the identities (m) (m) (5.40) S Fa Qab =0 and QabFa S =0. 84 5. CASIMIR ELEMENTS FOR oN AND spN

The latter hold due to (1.75) and (5.7); cf. the proof of Lemma 2.2.7:

(m) (m) (m) (m) (m) S Fa Qab = −S Fb Qab = S Fb PabQab = S PabFa Qab = −S Fa Qab, and the same calculation applies to verify the second identity in (5.40).

Lemma 5.5.2. Suppose that n m.Forthem-th partial trace in the algebra (5.17) we have

− n − m +1 m1 tr S(m)F = − F S(m−1). m m m(n − m +2) a a=1 Proof. Apply the same argument as for the symmetric algebra counterpart of this relation; see Lemma 2.2.7.

Proposition 5.5.3. The expression (5.37) admits an equivalent form which − represents a well-deﬁned element of Z(sp2n) for all n (m 1)/2. Proof. Expanding the product u1 + F1 ... um + Fm ,wecometoverifying the claim for expressions of the form 1 (5.41) tr S(m)F ...F , n − m +1 1,...,m a1 ak where 1 a1 < ···

l +1 a1 < ···

Calculating the partial trace trm−l by using this relation, we can write (5.42) as 2n − m + l +3 tr S(m−l−1)F ...F F ...F . (m − l)(n − m + l +2) 1,...,m−l−1 b1 bp a1 ak Up to the constant factor, this expression is of the form (5.42) with the parameter l replaced by l + 1. Hence, the claim follows by the induction hypothesis. Now 5.5. SYMMETRIZER AND ANTI-SYMMETRIZER FOR spN 85 suppose that ak = m − l. Applying Lemma 5.5.2 together with the cyclic property of trace, we can write (5.42) as 1 − tr S(m−l−1)F ...F (m − l)(n − m + l +2) 1,...,m−l−1 b1 bp m−l−1 × Fa1 ...Fak−1 Fa. a=1 We will use Lemma 5.5.1 to show that this expression equals a linear combination of expressions of the form (5.42) with the parameter l replaced by l + 1 so that the induction hypothesis could be applied. For the summands with a =1,...,l this can be done as follows. First apply the cyclic property of trace to move the factor S(m−l−1) to the right-most position, then transform the product (m−l−1) Fa1 ...Fak−1 Fa S by making a repeated use of (5.39) to move Fa to the leftmost position and do the same with all the terms of the form Fa occurring as a result of the use of relation

(5.39). Note that this relation is applicable to permute the factors Fai and Fa because the symmetrizer can be written as the product S(m−l−1) = S(h)S(m−l−1) for an appropriate value h

Namely, we use either the cycle σ =(a1,a2,...,ai,l+1) if a1 >l+1, or the cycle σ =(a2,...,ai,l+1) if a1 = l+1. Then, arguing as above, use (5.38) repeatedly to move Fl+1 to the position next after Fbp and do the same with all the terms of the form Fl+1 occurring as a result of the use of relation (5.38). Finally, use the cyclic property of trace again and transform the resulting products of the form (m−l−1) Fb1 ...Fbp ...Fak−1 Fl+1 S by using (5.39) repeatedly to move Fl+1 and all terms of the form Fl+1 occurring in the process to the position just after the term of the form Fl+1 or to the position next after Fbp if the term of the form Fl+1 is not present. To show that the possible expressions representing (5.37) by the above procedure are well-defined for n +1 m 2n + 1, we need to verify that they coincide, as elements of U(sp2n). First observe that all these expressions belong to the center Z(sp2n); see (5.22) in Remark 5.3.2. If the condition m n holds, then the central element (5.37) is a unique polynomial in the Gelfand invariants T1,...,Tn with T =trF 2k (see Example 5.3.3) and so equals k (m) a T k1 ...Tkn , k1,...,kn 1 n summed over nonnegative integers ka such that 2k1 +4k2 + ···+2nkn m,where the coefficients a(m) are polynomials in u ,...,u .Inparticular,k = 0 unless k1,...,kn 1 m a 2a m so we may write the coefficients as a(m) = a(m) with p = m/2 . k1,...,kn k1,...,kp 86 5. CASIMIR ELEMENTS FOR oN AND spN

Now suppose that m is fixed and let n vary. By the definition of the symmetrizer S(m),eachcoefficienta(m) is a rational function in n. Hence, it is determined k1,...,kp by infinitely many values of n with n m. Due to the first part of the proof, the rational functions a(m) are defined for all values n p. Since all equivalent k1,...,kp expressions for (5.37) obtained in the first part of the proof coincide for infinitely many values of n, they must coincide for all values of n (m − 1)/2 for which they are defined.

In what follows, we will use notation (5.37) for all values 1 m 2n +1, assuming that it represents a well-defined element of Z(sp2n) obtained in the proof of Proposition 5.5.3. Recall the factorial symmetric functions (4.16) and (4.17) which are associated 2 with the sequence a =(ai) defined in (5.11), so that in the symplectic case ai = i 2 2 | for i 1. The factorial elementary symmetric polynomials ek(l1,...,ln a) with C 2 2 Sn k =1,...,nare algebraically independent generators of the algebra [l1,...,ln] . Therefore, the Harish-Chandra image of the Casimir element (5.37) can be written as a unique linear combination p (m) c e (l2,...,l2 |a)kp , k1,...,kp r 1 n r=1 with the summation taken over nonnegative integers ka satisfying the condition 2k +4k +···+2pk m,wherep = m/2 and the coefficients c(m) are poly- 1 2 p k1,...,kp nomials in u1,...,um. The coefficients of these polynomials are rational functions in n. Since a rational function is uniquely determined by its values at infinitely many points, the Harish-Chandra image of the Casimir element (5.37) is uniquely determined by its values for n m. Take even values m =2k and set S = S(2k) with k 1 in Theorem 5.3.1 and consider a particular specialization of the parameters u1,...,u2k as follows. Proposition 5.5.4. For any 1 k n the image of the Casimir element − − (2k) ∈ (5.44) γ2k( 2n)tr1,...,2k (F1 k +1)...(F2k + k) S Z(sp2n) − k 2 2 | under the Harish-Chandra isomorphism coincides with ( 1) ek(l1,...,ln a). Proof. As pointed out above, it will be sufficient to prove the statement for any fixed k under the assumption that n 2k. We will use the same argument as in the proof of Proposition 5.4.1. Let Dk denote the Casimir element (5.44). Relying on Proposition 5.1.3 we will show that Dk vanishes in all representations | | | | L(λ)ofSp2n, where the partitions λ satisfy λ

Fij →−eji + εi εj eij , 1 i, j 2n. Under the corresponding representation in the tensor space ⊗ C2n ⊗2k → C2n ⊗r ⊗ C2n ⊗2k ϕ :U(sp2n) End ( ) End ( ) End ( ) we have r (5.45) ϕ(Fa)= −P br+a + Q br+a ,a=1,...,2k. b=1 5.5. SYMMETRIZER AND ANTI-SYMMETRIZER FOR spN 87

The vanishing condition of Proposition 5.1.3 will be veriﬁed if we show that if |λ|

(2k) where S denotes the image of the symmetrizer in the Brauer algebra B2k(−2n) acting of the last 2k copies of the tensor product space (C2n)⊗(r+2k). Due to (5.45) and relations (1.31), the desired identity (5.46) can be written in the form (2k) (5.47) Yr+1 − k +1 ... Yr+2k − k +1 S =0, where Yr+b with b =1,...,2k are the images of the respective Jucys–Murphy elements yr+b − (ω − 1)/2 given in (1.26) under the action (1.73) of the Brauer 2n ⊗(r+2k) algebra Br+2k(−2n)in(C ) . The identity (5.47) is verified by the same argument as in the proof of Proposition 5.4.1. The leading term of χ(Dk)coincides − k 2 2 with ( 1) ek(λ1,...,λn) as shown in Proposition 2.2.6; see (2.50). A different proof of Proposition 5.5.4 will follow from the results of Section 13.4. For 1 m 2n + 1 consider the polynomials in a variable u whose coefficients are Casimir elements for sp2n given by − − − m 1 m 3 Dm(u)=γm( 2n)tr1,...,m F1 + u + F2 + u + 2 2 m − 1 ×···× F + u − S(m). m 2 Corollary 5.5.5. For the images under the Harish-Chandra isomorphism we have

m m−2r−1 2 2n − 2r +1 m − 1 χ : D (u) → (−1)r e (l2,...,l2 |a) u − + r + i m m − 2r r 1 n 2 r=0 i=0 and − − −1 − −1 (m) χ : γm( 2n)tr1,...,m ∂z + F1 z ... ∂z + Fm z S

m 2 2n − 2r +1 m−2r → (−1)r e (l2,...,l2 |a) z−2r −∂ + rz−1 . m − 2r r 1 n z r=0 Proof. Exactly as in the proof of Corollary 5.4.2, we use Lemma 5.3.4 to m verify the relations Dm(u)=(−1) Dm(−u)and

Dm(u +1/2) − Dm(u − 1/2) = (2n − m +2)Dm−1(u),m 1, with D0(u) = 1. Since the same relation is satisﬁed by the polynomials which are the desired images χ Dm(u) , the statement follows from Proposition 5.5.4.

Now let S = A(2k) in Theorem 5.3.1 with k 1. Proposition 5.5.6. For any k 1 the image of the Casimir element − − (2k) ∈ (5.48) tr1,...,2k (F1 + k 1) ...(F2k k) A Z(sp2n) 2 2 | under the Harish-Chandra isomorphism coincides with hk(l1,...,ln a). 88 5. CASIMIR ELEMENTS FOR oN AND spN

Proof. Denote the Casimir element (5.48) by Ck. We use Proposition 5.1.3 and show that Ck vanishes in all representations L(λ)ofSp2n, where the partitions λ satisfy |λ|

Now consider the polynomials in u given by m − 1 m − 3 m − 1 C (u)=tr F + u + F + u + ... F + u − A(m). m 1,...,m 1 2 2 2 m 2

Their coeﬃcients are Casimir elements for sp2n. Corollary 5.5.7. For the images under the Harish-Chandra isomorphism we have m − − 2 2n + m − 1 m 2r 1 m − 1 χ : C (u) → h (l2,...,l2 |a) u − + r + i m m − 2r r 1 n 2 r=0 i=0 and − −1 − −1 (m) χ :tr1,...,m ∂z + F1 z ... ∂z + Fm z A m 2 2n + m − 1 m−2r → h (l2,...,l2 |a) z−2r −∂ + rz−1 . m − 2r r 1 n z r=0 m Proof. Lemma 5.3.4 implies Cm(u)=(−1) Cm(−u)sothatfortheodd values m =2k −1wehaveC2k−1(0) = 0. The values χ C2k(1/2) = χ C2k(−1/2) are provided by Proposition 5.5.6. Furthermore, we verify that

(5.49) Cm(u +1/2) − Cm(u − 1/2) = (2n + m − 1) Cm−1(u),m 1, where C0(u) = 1. This follows from Lemma 5.3.4 together with the relation for the partial trace of the operator A(m) implied by (1.54) and (1.77), 2n + m − 1 tr A(m) = A(m−1). m m This verifies (5.49). The same relation is satisfied by the polynomials which are claimed to be the images χ Cm(u) as stated in the corollary. The second part follows from the first as in the proof of Corollary 5.4.2.

Corollary 5.5.8. The elements of each family (5.44) and (5.48) with k = 1,...,n are algebraically independent generators of the center Z(sp2n). Proof. The images of the elements of each family under the Harish-Chandra isomorphism (5.9) are algebraically independent generators of the algebra of sym- C 2 2 Sn metric polynomials [l1,...,ln] . 5.6. MANIN MATRICES IN TYPES B, C AND D 89

5.6. Manin matrices in types B, C and D Our calculations of the Harish-Chandra images in Propositions 4.6.1 and 4.7.1 were based on properties of Manin matrices described in Proposition 3.2.2. On the other hand, the orthogonal and symplectic counterparts of these results stated in Sections 5.4 and 5.5 were derived in a different way, by using the Brauer–Schur– Weyl duality. Nevertheless, it is possible to define natural analogues of Manin matrices in types B, C and D and make a connection with the generator matrices F similar to Lemma 4.5.3. In the following definition which we make by analogy with Lemma 3.1.2, we use tensor product algebra (3.2) and assume that M is a N ×N matrix with entries in an associative algebra A. As before, we suppose that even values N =2n are associated with types Cn or Dn, whereas odd values N =2n + 1 are associated with type Bn. Definition 5.6.1. AmatrixM is called a Manin matrix of type B or D if it satisfies 1+P Q (5.50) 1 − P M M − =0. 1 2 2 N AmatrixM is called a Manin matrix of type C if it satisfies 1 − P Q (5.51) − M M 1+P =0. 2 2n 1 2

Note that if n = 1 in the symplectic case, then P + Q = 1 so that condition (5.51) holds trivially for any matrix M. We point out the following as an analogue of Lemma 3.1.3 which is veriﬁed in the same way. As before, we denote by H(m) and S(m) the respective images of the element h(m) deﬁned in (1.17) and the element s(m) given by equivalent formulas in Section 1.2, with respect to the homomorphism (1.69) in the orthogonal case and the homomorphism (1.73) in the symplectic case. The identities hold in the algebra (3.6). Lemma 5.6.2. If M is a Manin matrix of type B or D then (m) (m) (m) (5.52) H M1 ...Mm S = M1 ...Mm S .

Moreover, for any permutation σ ∈ Sm we have (m) (m) (5.53) Mσ(1) ...Mσ(m) S = M1 ...Mm S . If M is a Manin matrix of type C then (m) (m) (m) (5.54) S M1 ...Mm H = S M1 ...Mm.

Moreover, for any permutation σ ∈ Sm we have (m) (m) (5.55) S Mσ(1) ...Mσ(m) = S M1 ...Mm.

Proof. The proof of (5.52) and (5.53) reduces to checking that for any σ ∈ Sm (m) (m) (5.56) Pσ M1 ...Mm S = M1 ...MmS .

It is suﬃcient to verify this property for the adjacent transpositions σ = sa with a =1,...,m− 1 and so it will follow from the particular case m =2.Inthiscase (2) (2) (5.56) follows from the deﬁnition (5.50) which reads PM1 M2 S = M1 M2 S . 90 5. CASIMIR ELEMENTS FOR oN AND spN

Similarly, for the proof of (5.54) and (5.55) we need to verify that (m) (m) S M1 ...Mm Pσ =sgnσ · S M1 ...Mm, for any σ ∈ Sm. This reduces to the particular case m = 2 in the same way as for types B and D. In that case the property is equivalent to the deﬁnition (5.51) (2) (2) written in the form S M1 M2 P = −S M1 M2. Suppose that α and β are elements of a unital associative algebra D which satisfy the relation (4.33) and recall the matrix F deﬁned in (5.5) which is associated with the Lie algebra gN of type B, C or D.

Lemma 5.6.3. The matrix α + Fβ with entries in the algebra U(gN ) ⊗D is a Manin matrix of type B, C or D, respectively. Proof. Set M = α + Fβ. Using Proposition 5.1.1 and (1.67) we ﬁnd M1 M2 − M2 M1 = α + F1 β α + F2 β − α + F2 β α + F1 β =(F F − F F )β2 − (F − F )(αβ − βα) 1 2 2 1 1 2 2 = F1 (P − 1) + F2 (Q − P +1)− QF2 β . (2) In the cases B and D, we need to verify that (M1 M2 − M2 M1)S =0. This (2) follows from (1.31) and the relation QF2 S = 0. The latter holds since (2) (2) (2) (2) − (2) QF2 S = QP F2 S = QF1 S = QF2 S = QF2 S , where we used (2.28). Similarly, in the case C we write 2 M1 M2 − M2 M1 = F2 Q +(P − Q +1)F2 − (P +1)F1 β (2) and verify that S (M1 M2 − M2 M1)=0. As we will see in Chapter 8, certain generator matrices for the aﬃne Kac–Moody algebras of types B, C and D are Manin matrices of the respective types. It would be interesting to extend the general algebraic properties described in Chapter 3 to the Manin matrices of those types.

5.7. Bibliographical notes For general results outlined in Sections 5.1 and 5.2 see [29], [59], [70]and[153]. The hook dimension formulas for representations of ON and SpN weregivenin[34]; another proof can be found in [117] together with a version of the characteristic map for the Brauer algebra. The exposition in Sections 5.4 and 5.5 follows [73]. Propo- sitions 5.4.1 and 5.5.4 were also proved in [111] in a diﬀerent way. A Capelli-type determinant for oN was given by Wachi [150]; see also Itoh [77] for its relationship with the Casimir elements (5.33) and Itoh [78] for the corresponding results concerning the permanent-type Casimir elements (5.48) in the symplectic case. A determinant of diﬀerent type was found as an application of the Olshanski twisted Yangians; see [109, Chapter 7] for a detailed exposition and more constructions of Casimir elements, including analogues of the Newton identities of Theorem 4.8.1 and their equivalence to the Perelomov–Popov formulas for the Harish-Chandra images of the Gelfand invariants. These formulas were also proved in [25]withthe use of the Schur–Weyl duality. CHAPTER 6

Feigin–Frenkel center

Now we introduce the principal object of the book, the center of the affine vertex algebra at the critical level. Its structure was described by a theorem of Feigin and Frenkel [39] and hence is also known as the Feigin–Frenkel center.We begin by reviewing basic definitions of the vertex algebra theory and establish key properties of the center following the books by Frenkel [46], Frenkel and Ben- Zvi [47] and Kac [87]. Our goal will be to extend the R-matrix techniques developed in Chapters 4 and 5 to the affine Kac–Moody algebras and apply these techniques to construct explicit generators of the center for all classical types. This will allow us to give a direct proof of the Feigin–Frenkel theorem in those cases. The vertex algebra structure on the center will then be used to construct Sugawara operators for the affine Kac–Moody algebras g. These operators are affine analogues of the Casimir elements for the classical Lie algebras. They generate the center of a completed universal enveloping algebra of g at the critical level.

6.1. Center of a vertex algebra Let V be a vector space over C. Consider formal Laurent series in a variable z with coefficients in the endomorphism algebra End V . A series of the form −n−1 −1 cnz ∈ End V [[z,z ]] n∈Z is called a field, if for any v ∈ V there exists an integer N 0 such that cn v =0 for all n N. Definition 6.1.1. A vertex algebra is a vector space V (the space of states) with the additional data (Y,T,1), where 1 is the vacuum vector 1 ∈ V ,thetranslation T is an operator T : V → V and the state-field correspondence Y is a linear map Y : V → End V [[z,z−1]] such that the image of any element a ∈ V is a field, Y : a → a(z), −n−1 a(z)= a(n)z ,a(n) ∈ End V. n∈Z These data must satisfy the following axioms: (1) 1(z)=idV , ∈ (2) a(z)1 is a power series in z and a(z)1 z=0 = a for each a V , (3) T 1 =0, (4) T,a(z) = ∂z a(z)foreacha ∈ V , (5) for any states a, b ∈ V there exists a nonnegative integer N such that (z − w)N [a(z),b(w)] = 0.

91 92 6. FEIGIN–FRENKEL CENTER

The field a(z)isoftenwrittenintheforma(z)=Y (a, z), and the endomorphisms a(n) are called the Fourier coefficients of a(z). The span in End V of all Fourier coefficients a(n) of all fields a(z)isaLiesubalgebraofEndV . The commutator is given by m (6.1) a ,b = a b . (m) (k) n (n) (m+k−n) n0

The axioms imply that the operator T acts by the rule T (a)=a(−2)1 so it is determined by the map Y . This implies a more general formula n (6.2) T (a)=n!a(−n−1)1,n 0.

For every integer n the corresponding n-th product (a, b) → a(n)b equips V with an algebra structure which need not be associative or commutative. A subalgebra of a vertex algebra V is a subspace U of V containing the vacuum vector 1, such that a(n)U ⊂ U for all a ∈ U and all n ∈ Z.Inparticular,U is T -invariant; that is, T (U) ⊂ U. A subalgebra U is itself a vertex algebra; the state-ﬁeld correspondence is obtained by the restriction of the endomorphisms so that −n−1 ∈ a(z)= a(n) U z for any a U. n∈Z

A vertex algebra V is called commutative (or holomorphic)ifa(n) =0forall n 0andalla ∈ V ; that is, each ﬁeld a(z)isapowerseriesinz.Equivalently,V is commutative if and only if [a(z),b(w)] = 0 for all a, b ∈ V . Proposition 6.1.2. Suppose that V is a commutative vertex algebra. Then the (−1)-product makes V into a unital commutative associative algebra.

Proof. Write ab = a(−1)b for a, b ∈ V . The axioms give 1a = a1 = a for any a ∈ V . Furthermore, since [a(z),b(w)] = 0 we have a(z)b(w)c = b(w)a(z)c for any a, b, c ∈ V. Setting z = w =0wegeta(bc)=b(ac). By taking c = 1 we conclude that the product is commutative. Therefore, we also have a(cb)=(ac)b so that the product is associative. Definition 6.1.3. The center of a vertex algebra V is the subspace

z(V )={b ∈ V | a(n) b =0 forall a ∈ V and all n 0}. Equivalently, b ∈ z(V ) if and only if [Y (a, z),Y(b, w)] = 0 for all a ∈ V . The equivalence of the deﬁnitions is implied by the commutator formula (6.1). Proposition 6.1.4. The center z(V ) of a vertex algebra V is a commutative vertex algebra. Proof. We show ﬁrst that z(V )isasubalgebraofV . Clearly, 1 ∈ z(V ). Furthermore, suppose that b, c ∈ z(V )anda ∈ V .Thenforn 0 and arbitrary m ∈ Z we have a(n)b(m)c = b(m)a(n)c +[a(n),b(m)]c =0.

Therefore, b(m)c ∈ z(V )sothatz(V ) is a subalgebra of V . Finally, if a ∈ z(V ), then a(n) =0foralln 0, as an element of End z(V ). Hence the vertex algebra z(V )is commutative. 6.2. AFFINE VERTEX ALGEBRAS 93

In particular, z(V )isT -invariant which can also be easily checked directly. Due to Proposition 6.1.2, the center of any vertex algebra is a unital commutative associative algebra. Definition 6.1.5. A Lie conformal algebra is a C[∂]-module R endowed with a C-linear map

R ⊗ R → C[λ] ⊗ R, a ⊗ b → [aλb], called the λ-bracket, which satisﬁes the following axioms:

(1) [∂aλb]=−λ[aλb], [aλ∂b]=(λ + ∂)[aλb](sesquilinearity), − (2) [bλa]= [a−λ−∂b]( skewsymmetry ), (3) aλ[bμc] = [aλb]λ+μc + bμ[aλc] (Jacobi identity). It can be derived from the axioms listed in Deﬁnition 6.1.1 that a vertex algebra V is a Lie conformal algebra with ∂ = T and the λ-bracket ∞ λn (6.3) [a b]= a b, a, b ∈ V. λ n! (n) n=0 In terms of the λ-bracket on V , Deﬁnition 6.1.3 can be written as

z(V )={b ∈ V | [aλb] = 0 for all a ∈ V }.

6.2. Affine vertex algebras Let g be a simple Lie algebra over C equipped with a standard symmetric invariant bilinear form , defined as the normalized Killing form 1 (6.4) X, Y = tr adX adY , 2h∨ where h∨ is the dual Coxeter number for g. The normalization is determined by the condition that the square length of the longest root equals 2. The corresponding affine Kac–Moody algebra g is defined as the central extension (6.5) g = g[t, t−1] ⊕ CK, where g[t, t−1] is the Lie algebra of Laurent polynomials in t with coefficients in g. For any r ∈ Z and X ∈ g we will write X[r]=Xtr. The commutation relations of the Lie algebra g have the form X[r],Y[s] =[X, Y ][r + s]+rδr,−sX, Y K, X, Y ∈ g, and the element K is central in g. For any κ ∈ C introduce the universal enveloping algebra Uκ(g) at the level κ as the quotient of U(g) by the ideal generated by K − κ. Similarly, we will say that a g-module V is at the level κ if K acts on V as multiplication by the scalar κ.The vacuum module at the level κ over g is the quotient

(6.6) Vκ(g)=Uκ(g)/ I, whereIistheleftidealofUκ(g) generated by g[t]. By the Poincaré–Birkhoff–Witt theorem, the vacuum module is isomorphic to the universal enveloping algebra U t−1g[t−1] , as a vector space. We will equip this vector space with a vertex algebra structure by using its g-module structure; see Definition 6.1.1. Introduce 94 6. FEIGIN–FRENKEL CENTER the data (Y,T,1) as follows. The vacuum vector is 1 ∈ Vκ(g) and the translation operator

(6.7) T : Vκ(g) → Vκ(g), is determined by the properties T :1 → 0and T,X[r] = −rX[r − 1],X∈ g,r<0, where X[r] is understood as the operator of multiplication by X[r]. The state-field correspondence Y is defined by setting Y (1,z) = id, (6.8) Y (J a[−1],z)= J a[r]z−r−1, r∈Z 1 d a where J ,...,J is a basis of g,andJ [r] is regarded as an element of End Vκ(g) via the action of g.ThemapY is extended to the whole of Vκ(g) with the use of normal ordering. The normally ordered product of fields −r−1 −r−1 a(z)= a(r)z and b(w)= b(r)w r∈Z r∈Z is the formal Laurent series

(6.9) : a(z)b(w): = a(z)+ b(w)+b(w)a(z)−, where −r−1 −r−1 a(z)+ = a(r)z and a(z)− = a(r)z . r<0 r0 This definition extends to an arbitrary number of fields with the convention that the normal ordering is read from right to left. For instance, : a(z)b(w)c(v): = : a(z) : b(w)c(v): :. Then, denoting the field in (6.8) by J a(z)wehave

a1 am (6.10) Y (J [−r1 − 1] ...J [−rm − 1],z) 1 r1 a1 rm am = : ∂z J (z) ...∂z J (z): r1! ...rm! for any r1,...,rm 0. One can verify that the map Y is well defined and that all axioms in Definition 6.1.1 are satisfied by the data. So Vκ(g) is a vertex algebra which is called the (universal) affine vertex algebra. Its Lie conformal algebra structure is determined by the λ-bracket (6.3). In particular, X[−1]λY [−1] =[X, Y ][−1] + λX, Y κ, X, Y ∈ g. By the results of Section 6.1, the center z Vκ(g) is a unital commutative associative algebra. It can be shown that this algebra is trivial (coincides with C 1) unless the level is critical, κ = −h∨. In what follows we will be concerned with the structure of the center at the critical level. Definition 6.2.1. The Feigin–Frenkel center z(g) is the center of the vertex algebra V−h∨ (g). Any element of z(g)iscalledaSegal–Sugawara vector. By the definition of the state-field correspondence map, z(g) coincides with the subspace of g[t]-invariants on the vacuum module,

z(g)={v ∈ V−h∨ (g) | g[t]v =0}. 6.2. AFFINE VERTEX ALGEBRAS 95 −1 −1 Identifying the vector space V−h∨ (g)withU t g[t ] , we get a vector space embedding z(g) → U t−1g[t−1] so that Segal–Sugawara vectors can be viewed as elements of U t−1g[t−1] .It is immediate from the axioms of vertex algebra that the (−1)-product on z(g) coincides with the product in the universal enveloping algebra. In other words, z(g) can be regarded as a subalgebra of U t−1g[t−1] . Moreover, by Propositions 6.1.2 and 6.1.4 this subalgebra is commutative. The same algebra structure on z(g) can be deﬁned equivalently as follows. As before, let I denote the left ideal of U−h∨ (g) generated by g[t]andletNormIbe its normalizer,

Norm I = {v ∈ U−h∨ (g) | I v ⊂ I}.

The normalizer is a subalgebra of U−h∨ (g), and I is a two-sided ideal of Norm I. The Feigin–Frenkel center z(g) coincides with the associative algebra deﬁned as the quotient z(g)=NormI/ I . Another way to interpret the algebra structure on z(g)isprovidedbytheiso- morphism

∼ ∨ z(g) = End g V−h (g). Namely, any element a ∈ z(g) gives rise to the g-endomorphism of the vacuum module V−h∨ (g) which is determined by the condition 1 → a1. Example 6.2.2. The canonical Segal–Sugawara vector S ∈ z(g)isgivenby d a (6.11) S = Ja[−1]J [−1], a=1 1 d where J1,...,Jd is the basis of g dual to J ,...,J with respect to the form (6.4). The element S is independent of the choice of the dual bases. Since g =[g, g], to verify that S is a Segal–Sugawara vector it suﬃces to show that Ji[0]S =0and Ji[1]S =0foralli. The ﬁrst condition is essentially equivalent to the property that d a C = Ja J a=1 is a Casimir element for g. The second condition is implied by the fact that the eigenvalue of C in the adjoint representation equals 2h∨;thatis, d a ∨ Ja, [J ,X] =2h X a=1 for any X ∈ g.

Example 6.2.3. Take g = sl2 with its standard basis e, f, h and the commutation relations (6.12) [e, f]=h, [h, e]=2e, [h, f]=−2f. The only nonzero values of the form (6.4) on the basis elements are given by e, f =1 and h, h =2. 96 6. FEIGIN–FRENKEL CENTER

Hence the dual basis is f,e,h/2 and so the Segal–Sugawara vector (6.11) is written as 1 S = e[−1]f[−1] + f[−1]e[−1] + h[−1]2. 2 All translations T r S with r 0 turn out to be algebraically independent, and every element of the Feigin–Frenkel center z(sl2) is a polynomial in the Segal–Sugawara vectors T r S. This is a particular case of the general result (6.15); cf. Example 7.1.6 in the next chapter. Remark 6.2.4. By a theorem of Rybnikov [140], the commutative subalgebra z(g)ofU t−1g[t−1] coincides with the centralizer of the canonical Segal–Sugawara vector (6.11). In particular, this subalgebra is maximal commutative.

6.3. Feigin–Frenkel theorem We will regard the translation T on the affine vertex algebra as the derivation of the algebra U t−1g[t−1] whose action on the generators is given by (6.13) T : X[r] →−rX[r − 1],X∈ g,r<0. The algebra U t−1g[t−1] is equipped with the grading where the homogeneous components are eigenspaces of another derivation D defined on the generators by (6.14) D : X[r] →−rX[r],X∈ g,r<0. The subalgebra z(g)ofU t−1g[t−1] is invariant with respect to each of the derivations T and D. In particular, all homogeneous components of any Segal–Sugawara vector belong to z(g). Let n denote the rank of the simple Lie algebra g. A set of homogeneous elements S1,...,Sn ∈ z(g) is called a complete set of Segal–Sugawara vectors if all r elements T Sl are algebraically independent, and every Segal–Sugawara vector is a polynomial in these elements. In other words, z(g) is then the algebra of polynomials r (6.15) z(g)=C[T Sl | l =1,...,n, r 0]. We can now state the Feigin–Frenkel theorem in the following form. Theorem 6.3.1. A complete set of Segal–Sugawara vectors exists for any g. We aim to produce complete sets of homogeneous Segal–Sugawara vectors for the simple Lie algebras g of the classical types A, B, C and D and hence give a direct proof of Theorem 6.3.1 in those cases. As a first step, consider a ‘classical limit’ of the vacuum module V−h∨ (g) and its g[t]-invariants. Regard V−h∨ (g)asa g[t]-module obtained by restriction of the action of g to the subalgebra g[t]. Use the canonical filtration on the universal enveloping algebra U t−1g[t−1] to equip the vector space V−h∨ (g) with an ascending filtration U0 ⊂ U1 ⊂ ....The subspace Up is spanned by all monomials of the form − − (6.16) Xi1 [ r1] ...Xis [ rs],r1,...,rs 1,sp, ∈ ⊂ with Xi1 ,...,Xis g. Clearly, g[t]Up Up so that the filtration is preserved by the action of g[t]. Therefore, the associated graded space (6.17) gr V−h∨ (g)= Up / Up−1, U−1 = {0}, p0 becomes a g[t]-module. As a vector space, gr V−h∨ (g) will be identified with the symmetric algebra S t−1g[t−1] . The action of g[t]onS t−1g[t−1] is obtained 6.3. FEIGIN–FRENKEL THEOREM 97

− ∼ − − by extending the adjoint representation of g[t]ong[t, t 1]/g[t] = t 1g[t 1]tothe symmetric algebra. More explicitly, take a monomial of the form (6.16), regarding it as an element of the symmetric algebra and let Y [r] ∈ g[t]sothatr 0. Then · − − (6.18) Y [r] Xi1 [ r1] ...Xis [ rs] s − − − = Xi1 [ r1] ... Y,Xia [r ra] ...Xis [ rs]. a=1,r >r a This shows that Y [r]actsasaderivationonthealgebraS t−1g[t−1] .Inparticular, this implies that the subspace g[t] S t−1g[t−1] = v ∈ S t−1g[t−1] | g[t]v =0 of g[t]-invariants in S t−1g[t−1] is closed under multiplication. Such invariants can be produced by the following procedure. Choose a basis J1,...,Jd of g and suppose that

(6.19) P = P (J1,...,Jd) is an element of the symmetric algebra S(g). Introduce formal power series in a variable z with coefficients in t−1g[t−1]by ∞ r Ji(z)= Ji[−r − 1]z ,i=1,...,d. r=0 Now make the substitution into (6.19) and expand into a power series of z, ∞ r P J1(z),...,Jd(z) = P(−r−1) z . r=0 −1 −1 The coefficients P(−r−1) are elements of the symmetric algebra S t g[t ] and they can also be found as follows. Consider the derivation (6.20) T :S t−1g[t−1] → S t−1g[t−1] induced by (6.13) so that it acts on the generators by T : Y [−r] → rY[−r − 1] for r 1andY ∈ g.Then T r (6.21) P − − = P J [−1],...,J [−1] . ( r 1) r! 1 d When written in the equivalent form P J (z),...,J (z) = ezTP J [−1],...,J [−1] , 1 d 1 d this is essentially the statement that the commutative algebra S t−1g[t−1] equipped with the derivation T is a commutative vertex algebra; cf. (6.2). Lemma . ∈ 6.3.2 If the element P S(g) isg-invariant, then all elements P(−r−1) with r 0 are g[t]-invariants in S t−1g[t−1] . Proof. As a Lie algebra, g[t] is generated by elements of the form Y [0] and ∈ Y [1] with Y g. Hence it suffices to show that P(−r−1) vanishes under the action of Y [0] and Y [1]. We have the relations for the derivations in S t−1g[t−1] ,which are easily verified on generators, Y [0]T = TY[0] and Y [1]T = TY[1] + Y [0]. Therefore, the claim follows from (6.21) by induction on r. 98 6. FEIGIN–FRENKEL CENTER

The subalgebra of invariants is described by the following theorem.

Theorem 6.3.3. If P1,...,Pn are algebraically independent generators of the g algebra S(g) of g-invariants in S(g), then the elements P1(−r−1),...,Pn (−r−1) with r =0, 1,... are algebraically independent generators of the algebra of g[t]-invariants in S t−1g[t−1] . Proof. The adjoint g-module g is isomorphic to the coadjoint module g∗.An isomorphism is given by the linear map taking any element X ∈ g to the func- ∗ tional fX ∈ g such that fX (Y )=X, Y for Y ∈ g, with the invariant bilinear −1 −1 form defined in (6.4). Therefore, the g[t]-module S t g[t ] can be identified with S t−1g∗[t−1] and we will work with the latter for notational convenience. − ∗ − ∼ ∗ − ∗ Furthermore, the g[t]-module t 1g [t 1] = g [t, t 1]/g [t] is isomorphic to the (restricted) dual g[t]∗ of the adjoint module g[t]. To verify this claim, keep the basis ∗ ∗ ∗ J1,...,Jd of g and let J1 ,...,Jd be the dual basis of g . We can represent any ∗ ∗ element of g[t] as a finite linear combination of the dual basis vectors Ji[r] with ∗ i =1,...,d and r 0 such that Ji[r] Jj [s] = δij δrs. For any s 0 the isomor- ∗ ∈ ∗ ∗ − − ∈ −1 ∗ −1 phism takes the basis vector Jj [s] g[t] to Jj [ s 1] t g [t ]. Indeed, for theactionoftheLiealgebrag[t]wehave d ∗ − j − ∗ Ji[r]Jj[s] = cik Jk[ r + s] if r s k=1 ∗ k and Ji[r]Jj[s] = 0 otherwise, where cij are the structure constants of g with respect to the basis J1,...,Jd. This agrees with the action of Ji[r] on the basis ∗ − − −1 ∗ −1 vector Jj [ s 1] of t g [t ] and the claim follows. Hence, the g[t]-module S t−1g∗[t−1] is isomorphic to S g[t]∗ . The latter can be naturally identified, as a g[t]-module, with the algebra of polynomial functions Fun g[t]ong[t]. Using this identification, we will be proving the theorem in the following equivalent form. Suppose that Q1,...,Qn are algebraically independent generators of the algebra (Fun g)g of g-invariants in Fun g. Write an arbitrary element X ∈ g in coordinates as X = x1J1 + ···+ xdJd with xi ∈ C so that each generator Qi can be regarded as a polynomial in the variables x1,...,xd. Making the substitutions ∞ r xi(z)= xir z ,i=1,...,d, r=0 for certain variables xir, define polynomials Qi (r) by the expansions ∞ r (6.22) Qi x1(z),...,xd(z) = Qi (r) z . r=0

Each Qi (r) is a polynomial function on the Lie algebra g[t] whose elements are understood as ﬁnite linear combinations d xirJi[r]. i=1 r0 It follows from Lemma 6.3.2 and can also be checked directly, that all polynomials g[t] Qi (r) with i =1,...,n and r 0belongtothealgebra Fun g[t] of g[t]- invariants in Fun g[t]. We will be proving that they are algebraically independent g[t] generators of Fun g[t] . 6.3. FEIGIN–FRENKEL THEOREM 99

Observe that the polynomial Qi (r) does not depend on the variables xjs with s>r. If there is an algebraic dependence between the polynomials Qi (r),itcan only involve finitely many of them. Furthermore, any given g[t]-invariant in the algebra Fun g[t] does not depend on the variables xis with large enough values of s. Therefore, it will be sufficient to prove that for any nonnegative integer m the polynomials Qi (r) with i =1,...,n and r =0,...,m are algebraically independent gm generators of the algebra of invariants (Fun gm) ,wheregm denotes the quotient of g[t] by the ideal tm+1g[t]. Moreover, it is clear from the way the polynomials Qi (r) are defined, that it is enough to establish these properties for one particular g family of generators Q1,...,Qn of (Fun g) . Choose a regular nilpotent element e ∈ g. By the Jacobson–Morozov theorem, it can be completed to an sl2-subalgebra of g with the standard basis elements e, f, h as in (6.12). Identify g with the constant term subalgebra of gm and consider the affine subspace f f f m km = e + g + g t + ···+ g t f of gm,whereg denotes the centralizer of f in g.Let

gm (6.23) ρ :(Fungm) → Fun km be the homomorphism deﬁned by restricting polynomial functions from gm to the aﬃne subspace km. By the classical work of Kostant [94], there exist algebraically g independent generators Q1,...,Qn of (Fun g) and a basis U1,...,Un of the cen- f tralizer g such that for all j =1,...,n and all n-tuples (c1,...,cn)ofcomplex numbers we have

(6.24) Qj (e + c1 U1 + ···+ cn Un)=cj .

The algebra Fun km is freely generated by the coordinate functions pir : u → air, with i =1,...,n and r =0,...,m,where n m r (6.25) u = e + air Ui t ∈ km. i=1 r=0 By the property (6.24) of the generators of (Fun g)g, for any value of t in C we have m Qj (u)=aj0 + aj1 t + ···+ ajm t . On the other hand, the definition (6.22) implies m Qj (u)=Qj (0) + Qj (1) t + ···+ Qj (m) t , where the polynomials Qj (r) are evaluated at the respective coefficients of the powers of t in (6.25). We may thus conclude that all coordinate functions pir are contained in the image of the homomorphism ρ defined in (6.23) and so this homomorphism is surjective. As a final step, we will verify that the homomorphism ρ is injective. Consider gm the adjoint group Gm associated with gm. The algebra (Fun gm) coincides with Gm the algebra of Gm-invariants (Fun gm) . The injectivity of the homomorphism Gm ρ :(Fungm) → Fun km will follow if we show that (Ad Gm)(km)isdenseingm; that is, the map

φ :Gm ×km → gm, (g, x) → (Ad g)(x) is dominant. By [143, Theorem 4.3.6(i)], it will be suﬃcient to verify that the diﬀerential dφ(id,e) at the point (id,e) is surjective. Identifying the tangent spaces 100 6. FEIGIN–FRENKEL CENTER

f f f m to gm and km at e with gm and g + g t + ···+ g t , respectively, we can write the diﬀerential in the form f f f m dφ(id,e) : gm ⊕ g + g t + ···+ g t → gm, (x, v) → [x, e]+v.

However, the decomposition g =[g,e] ⊕ gf is well-known due to Kostant [94]. This implies that f f m gm =[gm,e]+g t + ···+ g t so that the diﬀerential is surjective. The proof is now complete as we showed that the map (6.23) is an isomorphism.

In Chapters 7 and 8 we will construct certain elements S1,...,Sn of the algebra z(g) in an explicit form for each Lie algebra g of classical type. We will prove that the families are complete sets of Segal–Sugawara vectors by relying on Theorem 6.3.3 as follows. Consider the symbols S1,...,Sn in the associated graded space as defined in (6.17). It will be possible to regard them as elements of S(g) via the embedding g → t−1g[t−1] such that X → X[−1]. Then verify that they are algebraically independent generators of S(g)g and apply Theorem 6.3.3. This will imply by r a standard argument that all elements T Sl with l =1,...,n and r 0are algebraically independent and generate z(g). Indeed, if there was an algebraic r r dependence between the elements T Sl, then the symbols T Sl would also be algebraically dependent. Furthermore, suppose that a certain nonzero element ∈ r S z(g) of the minimal possible degree is not a polynomial in the T Sl.Considerits −1 −1 g[t] r symbol S ∈ S t g[t ] and write it as a polynomial P (T Sl) in the generators r r T Sl. The difference S − P (T Sl)isanelementofz(g) whose degree is less than r the degree of S and it is not a polynomial in the T Sl. This makes a contradiction. Remark 6.3.4. Inthecasewhereg is an arbitrary simple Lie algebra, the elements S1,...,S n of any complete set of Segal–Sugawara vectors are ‘liftings’ to the algebra U t−1g[t−1] of certain algebraically independent generators of the algebra S(g)g in the sense described above. In particular, the respective degrees of the homogeneous elements S1,...,Sn must coincide with the degrees d1,...,dn as defined in (2.2); see [46, Proposition 4.3.3]. For the classical types this will follow from our constructions of the Segal–Sugawara vectors. Example . 6.3.5 Theorem 6.3.3 clearly extends to the reductive Lie algebra glN and holds in the same form. Recall the generators of the algebra of glN -invariants in S(glN ) introduced in Section 2.1. Consider the matrix E(z)=[Eij(z)] with ∞ r Eij(z)= Eij[−r − 1]z r=0 and expand N ∞ N N−m r (6.26) det u + E(z) = u + u Cm (−r−1) z , m=1 r=0 where u + E(z) is understood as u1+E(z), where 1 is the N × N identity matrix. Then the elements Cm (−r−1) with m =1,...,N and r 0 are algebraically 6.4. AFFINE SYMMETRIC FUNCTIONS 101 −1 −1 glN [t] independent generators of S t glN [t ] . Similarly, writing ∞ m r (6.27) tr E(z) = Tm (−r−1) z r=0 we obtain another algebraically independent family of generators Tm (−r−1) with m =1,...,N and r 0. Example . 6.3.6 Recall that for g = oN and spN generators of the subalgebra of g-invariants in the symmetric algebra S(g) were produced in Section 2.2. Consider the matrix F (z)=[Fij(z)] with ∞ r Fij(z)= Fij[−r − 1]z r=0 and expand n ∞ N N−2 m r det u + F (z) = u + u Cm (−r−1) z . m=1 r=0

Moreover, in the case of o2n consider the Pfaﬃan Pf F (z) deﬁned by the formula (2.33), where the elements Fij should be respectively replaced by Fij(z), and expand it as a power series in z, ∞ r Pf F (z)= Pf(−r−1) z . r=0

By Theorem 6.3.3, the elements C − − with m =1,...,n and r 0 are alge- m ( r 1) −1 −1 g[t] braically independent generators of S t g[t ] for g = sp2n and g = o2n+1, while the elements C − − with m =1,...,n−1 together with Pf − − for r 0 m ( r 1) ( r 1) −1 −1 o2n[t] are algebraically independent generators of S t o2n[t ] . Similarly, writing ∞ 2 m r tr F (z) = Tm (−r−1) z r=0 we obtain another algebraically independent family of generators Tm (−r−1) with m =1,...,n and r 0inthecasesg = sp2n and g = o2n+1.Inthecaseg = o2n the family is comprised of the elements Tm (−r−1) with m =1,...,n− 1 together with Pf(−r−1) for r 0.

6.4. Affine symmetric functions A natural class of affine analogues of symmetric polynomials emerges by the application of a Chevalley-type projection to elements of the algebra of g[t]-invariants in S t−1g[t−1] . Similar to the Chevalley projection (2.3) applied to the g-invariants in S(g), the images in the affine case for the classical types turn out to be related to these affine symmetric polynomials as implied by Examples 6.3.5 and 6.3.6. Given a symmetric polynomial P (λ) in the family of variables λ =(λ1,...,λN ), replace each variable λi by the respective formal power series ∞ r λi(z)= λi[−r − 1]z , r=0 102 6. FEIGIN–FRENKEL CENTER where the λi[−r − 1] are independent variables, and write ∞ r P λ1(z),...,λN (z) = Pr z , r=0 where the coefficients Pr are polynomials in the variables λi[−r − 1]. Equivalently, Pr is found as the derivative T rP P = , r r! where P = P (λ) is regarded as a polynomial in the variables λi[−1] = λi and the derivation T acts on the variables by the rule

T : λi[−r] → rλi[−r − 1],i=1,...,N, r 1. Definition 6.4.1. Denote by Λaff(N) the subalgebra of the algebra of polynomials in the variables λi[−r − 1] generated by all coefficients Pr associated with all symmetric polynomials P . Any element of Λaff(N)iscalledanaffine symmetric polynomial.

aff It is clear that the algebra Λ (N) is generated by the coefficients Pr associated to any family {P } of generators of the algebra of symmetric polynomials C[λ]SN . Returning to Example 6.3.5, consider the image of the determinant (6.26) under theaffineextensionoftheChevalleyprojection,

ς : Eij[−r − 1] → δij λi[−r − 1]. We have ς :det u + E(z) → u + λ1(z) ... u + λN (z) so that T r ς : C − − → λ [−1] ...λ [−1]. m ( r 1) r! p1 pm 1p1<···

i=1 r1+···+rm=r where the second sum is taken over the m-tuples (r1,...,rm) of nonnegative integers. We get an algebraically independent family of generators of Λaff(N)by restricting m to the values 1,...,N.Thus,intypeA we get the following affine version of the Chevalley isomorphism (2.3): −1 −1 glN [t] → aff ς :S t glN [t ] Λ (N). In this form, its extension to other types is straightforward, although it would be interesting to get an independent description of the image of ς inside the algebra of all polynomials, possibly, as a ‘classical limit’ of the affine Harish-Chandra isomorphism; see Chapter 13. Setting deg λi[−r]=r, r 1, 6.5. FROM SEGAL–SUGAWARA VECTORS TO CASIMIR ELEMENTS 103 defines a grading on the algebra of polynomials in the λi[−r]. The subalgebra Λaff(N) inherits the grading so that we have the direct sum decomposition Λaff(N)= Λaff(N)k, k0 where Λaff(N)k denotes the subspace of Λaff(N) spanned by homogeneous elements aff 0 of degree k and we set Λ (N) := C.WeletHN (q) denote the corresponding Hilbert–Poincaré series ∞ aff k k HN (q)= dim Λ (N) q . k=0 By a plane partition over the N-strip we will understand a finite sequence of Young diagrams (or partitions) λ(1) ⊃···⊃λ(r) such that λ(1) contains at most N rows. Such a plane partition can be viewed as an array formed by unit cubes, the i-th level of the array has the shape λ(i). Proposition 6.4.2. The dimension dim Λaff(N)k coincides with the number of plane partitions over the N-strip containing exactly k unit cubes. Equivalently, the Hilbert–PoincaréseriesHN (q) is given by N ∞ (6.29) (1 − qr)−1 = (1 − qr)− min(r,N ). m=1 rm r=1 Proof. Consider the family of algebraically independent generators of the algebra Λaff(N) given in (6.28). For each m =1,...,N this family contains generators of degrees m, m +1,... so the formula for the Hilbert–Poincaré series HN (q) follows. It is well-known that the generating function for the plane partitions over the N-strip is given by (6.29); see, e.g., [104, Section I.5].

6.5. From Segal–Sugawara vectors to Casimir elements Here we point out a connection between the Feigin–Frenkel center z(g)andthe center Z(g) of the universal enveloping algebra U(g). Given a variable z,consider the evaluation homomorphism (6.30) :U t−1g[t−1] → U(g) ⊗ C[z−1],X[r] → Xzr, for any X ∈ g and r<0, where we suppress the tensor product sign. Recall the derivation T of the algebra U t−1g[t−1] deﬁned in (6.13). The following property is easily veriﬁed on monomials in the generators X[r]: (6.31) (TS)=−∂ (S), z which holds for any S ∈ U t−1g[t−1] . Proposition 6.5.1. The image of the Feigin–Frenkel center z(g) under the homomorphism is contained in the tensor product Z(g) ⊗ C[z−1].

Proof. In the algebra U− ∨ (g)wehave h X[0]S = X[0],S + SX[0] −1 −1 for any X ∈ g and S ∈ U t g[t ] . Taking this relation modulo the left ideal I generated by g[t], we ﬁnd that the condition S ∈ z(g) implies that X[0],S =0. Therefore X, (S) =0inU(g) ⊗ C[z−1]. This shows that (S) is contained in Z(g) ⊗ C[z−1]. 104 6. FEIGIN–FRENKEL CENTER

If the variable z takes a particular nonzero value in C, then the same formula (6.30) deﬁnes a homomorphism −1 −1 → → r (6.32) z :U t g[t ] U(g),X[r] Xz , where we indicate the dependence on z ∈ C. Proposition 6.5.2. Suppose that z is a nonzero complex number. Then the image of the Feigin–Frenkel center z(g) under the homomorphism z coincides with Z(g). Moreover, if S1,...,Sn is a complete set of Segal–Sugawara vectors, then z(S1),...,z(Sn) are algebraically independent generators of Z(g).

Proof. As we pointed out in Remark 6.3.4, elements S1,...,Sn of any complete set of Segal–Sugawara vectors are associated with certain algebraically inde- g pendent generators of the algebra S(g) of the respective degrees d1,...,dn.More precisely, by Proposition 6.5.1 for the images z(S1),...,z(Sn)wemusthave

◦ −di z(Si)=Si z ,i=1,...,n, ◦ ◦ for certain elements S1 ,...,Sn of Z(g). Moreover, the symbols of these elements in the algebra S(g)g are its algebraically independent generators. This shows that ◦ ◦ S1 ,...,Sn are algebraically independent generators of Z(g).Theﬁrstpartofthe proposition also follows. ∈ C In particular, the image of the homomorphism z does not depend on z which can also be derived from (6.31) by using the T -invariance of z(g), independently of Proposition 6.5.2.

6.6. Center of the completed universal enveloping algebra We will now discuss Casimir elements for the affine Kac–Moody algebras g associated with simple Lie algebras g as defined in (6.5). The universal enveloping algebra U(g) needs to be completed to contain such elements. As in Section 6.2, fix an eigenvalue κ ∈ C of the central element K of g and consider the universal enveloping algebra Uκ(g)atthelevelκ. We would like the action of the completed algebra to be defined on a certain natural category of smooth modules. A g-module V at the level κ is called smooth, if for any v ∈ V there exists a nonnegative integer p such that tpg[t]v =0,wheretpg[t] is the subalgebra of g spanned by all elements X[r] with X ∈ g and r p. Introduce a linear topology on Uκ(g) by using the neighborhood basis for 0 p formed by the left ideals Ip of Uκ(g) generated by t g[t] for all p 0. The completed universal enveloping algebra Uκ(g)isthecompletionofUκ(g) with respect to this topology. Equivalently, Uκ(g) can be defined as the inverse limit (6.33) U (g) = lim U (g)/I . κ ←− κ p The description of the center of the completed algebra Uκ(g) is a principal theme of the book by Frenkel [46] which stems from the original work of Feigin and Frenkel [39]. As with the center of the affine vertex algebra Vκ(g), the center ∨ of Uκ(g) turns out to be trivial unless the value of κ is critical, κ = −h ;cf. Section 6.2. We refer the reader to the book [46] for a detailed discussion of the center Z(g) of the algebra U−h∨ (g) at the critical level and its relationship with the geometry of opers. On the algebraic side, central elements of the completed algebra 6.6. CENTER OF THE COMPLETED UNIVERSAL ENVELOPING ALGEBRA 105 U−h∨ (g) are obtained from elements of the Feigin–Frenkel center z(g)byemploying the vertex algebra structure. To outline the construction [46, Sections 3.2 and 4.3], recall from Section 6.2 that the application of the state-field correspondence map Y to an element S ∈ z(g) yields a field −r−1 (6.34) Y (S, z)= S(r) z , r ∈Z where all coefficients S(r) are operators on the vacuum module V−h∨ (g). We will reinterpret (6.34) in such a way that instead of the operators S(r) we will get corresponding elements S[r] of the center Z(g)knownasSugawara operators.Consider the Laurent series J a(z)=Y (J a[−1],z)definedin(6.8), J a(z)= J a[r]z−r−1. r∈Z We will now regard J a(z) as a series with coefficients in the universal enveloping algebra U−h∨ (g). Furthermore, finite linear combinations of the coefficients of the series

r1 a1 rm am : ∂z J (z) ...∂z J (z): can be regarded as elements of the completed universal enveloping algebra U−h∨ (g) defined in (6.33). The definition of the state-field correspondence map is now interpreted in such a way that for any a ∈ V−h∨ (g) the coefficients a[n] of the series −n−1 (6.35) Y [a, z]= a[n]z n∈Z defined by the same rule (6.10), are understood as elements of U−h∨ (g). The commutator formula (6.1) is now replaced by m a ,b = a b . [m] [k] n (n) [m+k−n] n0 This formula implies that if a belongs to the Feigin–Frenkel center z(g), then all coefficients a[n] are Sugawara operators; that is, they belong to the center Z(g)of the completed algebra U−h∨ (g). Furthermore, we have the following.

Proposition 6.6.1. Suppose that S1,...,Sn ∈ z(g) is a complete set of Segal– Sugawara vectors. Then the Sugawara operators

{S1[r],...,Sn [r] | r ∈ Z} are topological generators of Z(g).

Therefore, complete sets of Segal–Sugawara vectors for classical Lie algebras g which we will produce in Theorems 7.1.4, 8.1.9 and 8.3.8 below, will provide respective topological generators of the completed universal enveloping algebras; see Chapters 7 and 8 for the formulas. These Sugawara operators will reappear in Chapter 15, where we will calculate their eigenvalues in the Wakimoto modules over g of the critical level. 106 6. FEIGIN–FRENKEL CENTER

6.7. Bibliographical notes Our proof of the key Theorem 6.3.3 follows Ra¨ıs and Tauvel [134] with some modifications as in the paper by Brown and Brundan [15]. A different proof is due to Beilinson and Drinfeld (see [46, Theorem 3.4.2] for detailed arguments) and Eisenbud and Frenkel [119, Proposition A.1]. A generalization of this theorem involving centralizers of nilpotent elements was proved by Panyushev, Premet and Yakimova [131]. Prior to the description of the algebra of invariants z(g)ofthe vacuum module V−h∨ (g) for an arbitrary simple Lie algebra g in [39], particular cases of the Feigin–Frenkel theorem were proved independently by Goodman and Wallach [58](typeA)andHayashi[65](typesA, B and C). A geometric proof of the theorem for all g was given recently by Raskin [135]. The Feigin–Frenkel theorem was generalized by Arakawa [5] who showed that the center of the W -algebra at the critical level, associated with an arbitrary nilpotent element f ∈ g coincides with z(g). A super version of affine symmetric polynomials was introduced in [112]. Proofs and more details on the center of the completed universal enveloping algebra (Section 6.6) can be found in the book by Frenkel [46]. CHAPTER 7

Generators in type A

In this chapter we will produce a few complete sets of Segal–Sugawara vectors for glN . We will thus get a direct proof of the Feigin–Frenkel theorem (Theo- rem 6.3.1) in type A. We will rely on Theorem 6.3.3 in a way indicated in the comments following the proof of that theorem in Chapter 6. We will also connect the Segal–Sugawara vectors to Casimir elements for glN and reproduce some constructions of Chapter 4 by applying the general approach explained in Section 6.5. Furthermore, we will use Proposition 6.6.1 to give explicit formulas for generators of the center of the completed universal enveloping algebra of glN at the critical level.

7.1. Segal–Sugawara vectors In our constructions of Segal–Sugawara vectors in all classical types, we will use the extended Lie algebra g ⊕ Cτ where the additional element τ satisﬁes the commutation relations (7.1) τ,X[r] = −rX[r − 1], τ,K =0.

Instead of the simple Lie algebra slN of type A we will work with the reductive Lie algebra glN . We extend the form (6.4) to the invariant symmetric bilinear form on glN which can be written as 1 (7.2) X, Y =trXY − trX trY, X,Y ∈ gl . N N Note that the kernel of the form is spanned by the element E11 + ···+ ENN,and its restriction to the subalgebra slN is given by

X, Y =trXY, X,Y ∈ slN . −1 ⊕ C The aﬃne Kac–Moody algebra glN = glN [t, t ] K has the commutation relations δij δkl (7.3) E [r],E [s] = δ E [r + s] − δ E [r + s]+rδ − K δ δ − , ij kl kj il il kj r, s kj il N and the element K is central. The critical level −N coincides with the negative of the dual Coxeter number for slN . With the above deﬁnition of the bilinear form on glN the Feigin–Frenkel center z(glN ) coincides with both the algebra of glN [t]-invariants and slN [t]-invariants of the vacuum module, { ∈ | } { ∈ | } z(glN )= v V−N (glN ) glN [t]v =0 = v V−N (glN ) slN [t]v =0 .

For any r ∈ Z combine the elements Eij[r]intothematrixE[r]sothat N ⊗ ∈ CN ⊗ E[r]= eij Eij[r] End U(glN ). i,j=1

107 108 7. GENERATORS IN TYPE A

For each a ∈{1,...,m} introduce the element E[r]a of the algebra CN ⊗ ⊗ CN ⊗ (7.4) End ... End U m by N ⊗(a−1) ⊗(m−a) (7.5) E[r]a = 1 ⊗ eij ⊗ 1 ⊗ Eij[r], i,j=1 ⊕ C where U stands for the universal enveloping algebra of glN τ. Proposition . 7.1.1 The deﬁning relations of the algebra U(glN ) can be written in the form 1 (7.6) E[r] E[s] − E[s] E[r] = E[r + s] − E[r + s] P + rδ − K P − , 1 2 2 1 1 2 r, s N where both sides are elements of the algebra (7.4) with m =2.

Proof. This is immediate from (7.3) by taking the tensor products of eij ⊗ekl with both sides and then applying the summation over i, j, k, l. Recall equivalent definitions of Manin matrices as given by Lemma 3.1.2. Lemma 7.1.2. The matrix E = τ + E[−1] = δijτ + Eij[−1] with the entries ⊕ C in the universal enveloping algebra of glN τ is a Manin matrix. Proof. This is quite similar to the proof of Lemma 4.5.3. Using Proposi- tion 7.1.1 we find E1 E2 −E2 E1 = τ + E[−1]1 τ + E[−1]2 − τ + E[−1]2 τ + E[−1]1 = E[−1] E[−1] − E[−1] E[−1] − E[−2] + E[−2] 1 2 2 1 1 2 = E[−2]1 − E[−2]2 (P − 1). On multiplying this element from the right by 1+P we get 0, and the claim follows by using the equivalent definition (3.5) of Manin matrices.

As before, we let H(m) and A(m) denote the elements of the algebra (7.4) (with the identity components in U) which are the respective images of the symmetrizer h(m) and anti-symmetrizer a(m) deﬁned in (1.17) and (1.19) under the map (1.65). ∈ ∼ −1 −1 Deﬁne the elements φma,ψma,θma V−N (glN ) = U t glN [t ] by the expansions (m)E E m m−1 ··· (7.7) tr1,...,m A 1 ... m = φm 0 τ + φm 1 τ + + φmm,

(m)E E m m−1 ··· (7.8) tr1,...,m H 1 ... m = ψm 0 τ + ψm 1 τ + + ψmm, and E m m m−1 ··· (7.9) tr = θm 0 τ + θm 1 τ + + θmm. (m) Note that all elements φma are zero for m>Nsince the anti-symmetrizer A vanishes in this case. Since E is a Manin matrix by Lemma 7.1.2, taking m = N in (7.7) and applying (3.28) we get (N)E E E (7.10) tr1,...,N A 1 ... N =cdet , 7.1. SEGAL–SUGAWARA VECTORS 109 where the column-determinant is given by (3.23). Expand it as a polynomial in τ, E N N−1 ··· (7.11) cdet = τ + φ1 τ + + φN so that φNa = φa for a =1,...,N. The identity (3.29) implies the expansion of the noncommutative characteristic polynomial, N N−m (m) cdet (u + E)= u tr1,...,m A E1 ...Em, m=0 where u is a variable. Hence replacing τ by u + τ in (7.11), we obtain more general relations N − a (7.12) φ = φ , 0 a m N. ma m − a a

In particular, φmm = φm for m =1,...,N. Theorem . 7.1.3 All elements φma, ψma and θma belong to the Feigin–Frenkel center z(glN ). Proof. By Lemma 7.1.2, it is suﬃcient to verify the claim for only one of the three families. The corresponding statements for the two remaining families will then follow from the MacMahon Master Theorem (Theorem 3.2.1) and the Newton identity (Theorem 3.2.10) together with the identities of Corollary 3.2.5. Still we will give direct arguments for two families ψma and θma, as they use diﬀerent matrix calculations. The direct proof for the elements φma is quite similar to the one for ψma. Proof for the elements ψma. It will be enough to verify that these elements are annihilated by the operators Eij[0] and Eij[1] with 1 i, j N in the vacuum module V−N (glN ). Consider the tensor product algebra CN ⊗ ⊗ CN ⊗ End ... End U m+1 with the m + 1 copies of End CN labeled by 0, 1,...,m. The required annihilation properties can then be written in the equivalent form

(m) (m) (7.13) E[0]0 tr1,...,m H E1 ...Em =0 and E[1]0 tr1,...,m H E1 ...Em =0 modulo the left ideal of U generated by glN [t]andK + N. To verity the ﬁrst relation, note that by (1.67) and (7.6) we have

(7.14) [E[0]0, Ea]=P0 a Ea −Ea P0 a. Hence m (m) (m) E[0]0 tr1,...,m H E1 ...Em = tr1,...,m H E1 ... P0 a Ea −Ea P0 a ...Em a=1 (m) =tr1,...,m H P01 + ···+ P0 m E1 ...Em (m) − tr1,...,m H E1 ...Em P01 + ···+ P0 m .

(m) The sum P01+ ···+ P0 m commutes with the symmetrizer H so that the expression is equal to zero by the cyclic property of trace. 110 7. GENERATORS IN TYPE A

To verify the second property in (7.13) use the relation implied by (7.6):

(7.15) [E[1]0, Ea]=E[0]0 + P0 a E[0]a − E[0]a P0 a + K (P0 a − 1/N ). We have m (m) (m) E[1]0 tr1,...,m H E1 ...Em = tr1,...,m H E1 ...Ea−1 a=1 × E[0]0 + P0 a E[0]a − E[0]a P0 a + K (P0 a − 1/N ) Ea+1 ...Em. Applying (7.14) we can write this expression as (m) tr1,...,m H E1 ...Ea ... P0 b Eb −Eb P0 b ...Em 1a

Taking ω = N in Lemma 1.3.4 and using (1.72), we ﬁnd that N + m − 1 tr H(m) = H(m−1). m m Furthermore, arguing as in the proof of Lemma 1.3.4, and using (1.68) we ﬁnd

m−1 1 tr H(m)P = tr H(m−1) 1+ P P m 0 m m m am 0 m a=1 m−1 m−1 1 1 = tr H(m−1) P 1+ P = H(m−1) 1+ P . m m 0 m 0 a m 0 a a=1 a=1 (m) By the cyclic property of trace (or a similar calculation), trm P0 m H is given by the same formula. Therefore, − m−1 (m) N + m 1 (m−1) E[1] tr H E ...E = 2a tr − H P E ...E − 0 1,...,m 1 m m 1,...,m 1 0 a 1 m 1 a=1 m−1 (m−1) +(K − m +1)tr1,...,m−1 H 1+ P0 a E1 ...Em−1 a=1 − K(N + m 1) (m−1) − tr − H E ...E − . N 1,...,m 1 1 m 1 Finally, using the property (3.10) of Manin matrices and Lemma 7.1.2 we ﬁnd that for any 1 a

(m) E[1]0 tr1,...,m H E1 ...Em m−1 − (m−1) m 1 =(K + N)tr − H P − E ...E − 1,...,m 1 0 a N 1 m 1 a=1 which is zero, since K + N = 0 at the critical level. Proof for the elements . θma It is suﬃcient to verify that for all i, j m m (7.16) Eij[0] tr E = Eij[1] tr E =0 ⊗ C C in the glN -module V−N (glN ) [τ] with the trivial action on [τ]. Consider the tensor product algebra End CN ⊗ End CN ⊗ End CN ⊗ U with the copies of End CN labeled by 0, 1, 2, where U stands for the universal ⊕ C enveloping algebra of glN τ. Relations (7.16) can now be written in the matrix form as E m E m (7.17) E[0]0 tr1 1 =0 and E[1]0 tr1 1 =0 112 7. GENERATORS IN TYPE A modulo the left ideal of U generated by gl [t]andK + N.Notetheidentity N E m E m −Em (7.18) E[0]0, 1 = P01 1 1 P01,m=0, 1, 2,..., which follows from (7.14): m m E m E r−1 E E m−r E r−1 E E m−r E m E[0]0, 1 = 1 E[0]0, 1 1 = 1 [P01, 1] 1 =[P01, 1 ]. r=1 r=1

Now the ﬁrst relation in (7.17) follows by taking the trace tr1 on both sides of (7.18) and using its cyclic property. For the proof of the second relation in (7.17), use (7.15) to write m K (7.19) E[1] , E m = E i−1 E[0] + P E[0] − E[0] P + KP − E m−i. 0 1 1 0 01 1 1 01 01 N 1 i=1 E m−i E m−i Applying (7.18) and the relation 1 =tr2 2 P12 we can rewrite (7.19) modulo the left ideal of U generated by glN [t]as m mK E[1] , E m = E i−1 P , E m−i + KE i−1P E m−i − E m−1 0 1 1 01 1 1 01 1 N 1 i=1 m E i−1 − E m−i +tr2 1 P01E[0]1 E[0]1P01 2 P12. i=1 Now transform the last summand using (7.18) to get m E i−1 − E m−i tr2 1 P01E[0]1 E[0]1P01 2 P12 i=1 m m E i−1 E m−i − E i−1 E m−i =tr2 1 P01 P12, 2 P12 tr2 1 P12, 2 P01P12. i=1 i=1

Taking into account the relation tr2P02 = 1, we can simplify this to m−1 E i−1 E m−i −Ei−1 E m−i E i−1E m−i − − E m−1 N 1 P01 1 1 P01 tr + 1 0 (m 1) 1 . i=1 Combining all the terms and taking the trace tr we derive 1 m E[1] , tr E m =(K + N) E m−1 − tr E m−1 +(K + N − m +2)E m−1 0 1 1 0 N 0 m−1 m−1 E i−1E m−i − E i−1 E m−i +(K + N +1)tr1 P01 0 1 tr1 0 , 1 . i=2 i=2 Finally, we use Lemma 7.1.2 and Corollary 3.2.9 to write m−1 m−1 E i−1 E m−i E i−1 E m−i 0 , 1 = P01 0 , 1 . i=2 i=2

By calculating the trace tr1 on both sides of this relation we obtain m−1 m−1 E i−1 E m−i E i−1E m−i − − E m−1 tr1 0 , 1 =tr1 P01 0 1 (m 2) 0 . i=2 i=2 7.1. SEGAL–SUGAWARA VECTORS 113

Taking this into account, we get − m m1 E[1] , tr E m =(K + N) 2E m−1 − tr E m−1 +tr P E i−1E m−i . 0 1 1 0 N 1 01 0 1 i=2 This expression vanishes in the vacuum module V−N (glN )asK + N =0. One more proof of Theorem 7.1.3 will be given in Section 10.5. The Feigin– Frenkel theorem (Theorem 6.3.1) for type A is implied by the following. Theorem 7.1.4. Each of the families

φ1,...,φN ,ψ11,...,ψNN and θ11,...,θNN is a complete set of Segal–Sugawara vectors for glN . Proof. Recall that the graded space gr V−N (glN ) deﬁned in (6.17) is identi- −1 −1 ﬁed with the symmetric algebra S t glN [t ] . The symbols of the elements of each family in the symmetric algebra coincide with the images of certain elements → −1 −1 of the symmetric algebra S(glN ) under the embedding S(glN ) S t glN [t ] ∈ − taking X glN to X[ 1]. Due to Theorems 6.3.3 and 7.1.3 we only need to verify that the corresponding elements of S(glN ) are algebraically independent generators glN of the algebra of invariants S(glN ) ; see the arguments following the proof of The- orem 6.3.3. However, this was already done in Section 2.1, where these invariants were produced and their Chevalley images calculated. The respective families of invariants corresponding to the Segal–Sugawara vectors in the formulation of the theorem are given in (2.6), (2.20), (2.21) and (2.11).

Example 7.1.5. We have N − φ1 = Eii[ 1], i=1 N − − − − − − − φ2 = Eii[ 1]Ejj[ 1] Eji[ 1]Eij[ 1] + (i 1) Eii[ 2]. 1i

Multiplying both sides by P12 from the right and taking trace tr12 we get the first relation in (7.20). Similarly, 2 − 2 − E[r]1 E[s]2 E[s]2 E[r]1 = E[r + s]1 E[r + s]2 P12 E[s]2 + E[s]2 E[r + s]1 − E[r + s]2 P12 and so, multiplying by P12 from the right and using the first relation we find

tr E[r]E[s]2 − tr E[s]2E[r]=N tr E[r + s]E[s] − tr E[r + s]trE[s] +trE[s]trE[r + s] − N tr E[s]E[r + s]=0 thus implying the second relation in (7.20). Recall that the Feigin–Frenkel center z(glN )isT -invariant; see (6.7). For k 1wehave k − − T θ11 = k!trE[ k 1]. Furthermore, by (7.20) T tr E[−1]2 =2trE[−1]E[−2], T 2 tr E[−1]2 =4trE[−1]E[−3] + 2 tr E[−2]2, and T tr E[−1]3 =2trE[−1]2E[−2] + tr E[−1]E[−2]E[−1]. Taking the above relations into account, we can conclude that all elements (7.21) tr E[−1], tr E[−1]2, tr E[−1]3, tr E[−1]4 − tr E[−2]2 are Segal–Sugawara vectors for glN . In particular, the following are complete sets of Segal–Sugawara vectors: − − 2 for gl2 :trE[ 1], tr E[ 1] − − 2 − 3 for gl3 :trE[ 1], tr E[ 1] , tr E[ 1] for gl :trE[−1], tr E[−1]2, tr E[−1]3, tr E[−1]4 − tr E[−2]2. 4

7.2. Sugawara operators in type A Introduce the Laurent series in z with coeﬃcients in U−N (glN )by −r−1 (7.22) Eij(z)= Eij[r] z ,i,j=1,...,N. r∈Z

We adopt the matrix notation as in Section 7.1 and combine the series Eij(z)into the matrix E(z)sothat N E(z)= eij ⊗ Eij(z). i,j=1

For a ∈{1,...,m} we extend the notation (7.5) to the elements E(z)a of the algebra (7.4), where U now stands for the algebra of polynomial diﬀerential operators over −1 z with coeﬃcients in U−N (glN )[[z,z ]]. 7.2. SUGAWARA OPERATORS IN TYPE A 115

As explained in Section 6.6, the application of the state-field correspondence map (6.35) to the coefficients of the polynomials in (7.7)–(7.9), leads to the definition of the Laurent series φma(z),ψma(z),θma(z) given by the following expansions with the usual reading of normal ordering from right to left: (m) (7.23) : tr1,...,m A ∂z + E(z)1 ... ∂z + E(z)m : m ··· = φm 0(z) ∂z + + φmm(z), (m) (7.24) : tr1,...,m H ∂z + E(z)1 ... ∂z + E(z)m : m ··· = ψm 0(z) ∂z + + ψmm(z), and m m ··· (7.25) : tr ∂z + E(z) :=θm 0(z) ∂z + + θmm(z).

Moreover, following (7.11) deﬁne the series φa(z)by N N−1 ··· (7.26) : cdet ∂z + E(z) :=∂z + φ1(z) ∂z + + φN (z). By the relations between the Segal–Sugawara vectors pointed out in Section 7.1, we have

φNa(z)=φa(z),a=1,...,N and N − a φ (z)= φ (z), 0 a m N; ma m − a a see (7.12). In particular, φmm(z)=φm(z)form =1,...,N. The next theorem is implied by Proposition 6.6.1 together with Theorems 7.1.3 and 7.1.4. Theorem . 7.2.1 All coeﬃcients of the Laurent series φma(z), ψma(z) and θma(z) belong to the center of the algebra U−N (glN ). Moreover, the coeﬃcients of each family of the Laurent series

φ1(z),...,φN (z),ψ11(z),...,ψNN(z) and θ11(z),...,θNN(z) are topological generators of the center of U−N (glN ). Example . 7.2.2 Consider the ﬁrst few Segal–Sugawara vectors for glN given in (7.21). The corresponding Laurent series will have the form 2 3 4 2 tr E(z), :trE(z) : , :trE(z) : , :trE(z) : − :tr ∂zE(z) : . We have N −r−1 tr E(z)= Eii[r] z r∈Z i=1 ··· so that all elements E11[r]+ + ENN[r] are central in U−N (glN )(infact,these elements are central in Uκ(glN ) for any level κ). Furthermore, N N 2 :trE(z) :=: Eij(z)Eji(z):= Eij(z)+Eji(z)+Eji(z)Eij(z)− . i,j=1 i,j=1 116 7. GENERATORS IN TYPE A

Hence for each p ∈ Z the coeﬃcient of z−p−2 in this series is a central element of U−N (glN )givenby N Eij[r]Eji[p − r]+ Eji[p − r]Eij[r] . i,j=1 r<0 r0 Similar explicit formulas for central elements of U−N (glN ) of higher order can be obtained by taking the coeﬃcients of all powers of z in the expanded normally ordered traces; for instance, N 3 :trE(z) :=: Eij(z)Ejk(z)Eki(z): i,j,k=1 N = Eij(z)+Ejk(z)+Eki(z)+Eij(z)+Eki(z)Ejk(z)− i,j,k=1

+ Ejk(z)+Eki(z)Eij(z)− + Eki(z)Ejk(z)−Eij(z)− .

We conclude this section by pointing out how the Casimir elements for glN constructed in Chapter 4 can be reproduced from elements of the Feigin–Frenkel center z(glN ) (see Section 6.5) or from the above Sugawara operators. Consider the Segal–Sugawara vectors for glN provided by the expansion (7.7); see Theorem 7.1.3. The images of the coefficients of the polynomial (7.7) under the homomorphism defined in (6.30) can be written as (m) −1 −1 (7.27) tr1,...,m A −∂z + E1 z ... −∂z + Em z , where the image of τ is understood as the differential operator −∂z.Usingthe relation ∂z z = z∂z + 1 and setting u = −∂z z we can write (m) −1 −1 m tr1,...,m A −∂z + E1 z ... −∂z + Em z z (m) =tr1,...,m A u + E1 ... u + Em − m +1 thus reproducing the Casimir elements for glN provided by (4.35). In particular, taking the image of the column-determinant (7.11) under we recover the Capelli determinant (4.37): −1 N cdet −∂z + Ez z = C(u). The permanent-type Casimir elements (4.39) and the Gelfand invariants (4.41) are recovered from the respective Segal–Sugawara vectors given in (7.8) and (7.9). By an equivalent approach, take the images of the Sugawara operators given above, in the quotient of the completed universal enveloping algebra by the left ideal generated by glN [t]. These images can be regarded as elements of the Feigin– Frenkel center z(glN ). For instance, for the left hand side of (7.23) we get (m) (7.28) tr1,...,m A ∂z + E(z)+1 ... ∂z + E(z)+ m , whereweset N −r−1 E(z)+ = eij ⊗ Eij(z)+,Eij(z)+ = Eij[r] z . i,j=1 r<0 7.3. BIBLIOGRAPHICAL NOTES 117

∈ C All coefficients of the power series (7.28) are elements of z(glN ). For a nonzero a apply the evaluation homomorphism a defined in (6.32) to these coefficients. We have →− − −1 a : Eij(z)+ Eij(z a) . → Now shift the variable by z z+a so that the image of (7.28) under a will coincide with (7.27), up to a sign. Similarly, by using the Sugawara operators (7.24)–(7.26) we recover the respective Casimir elements considered in Chapter 4.

7.3. Bibliographical notes Explicit formulas for generators of the Feigin–Frenkel center in type A were ﬁrst given in the preprint by Chervov and Talalaev [24] in a form close to (7.26). That work was inspired by Talalaev’s construction of explicit higher Gaudin Hamiltonians [145]. A direct proof of Theorem 7.1.4 for the elements φa was given by Chervov and the author [23]. In the proof of Theorem 7.1.3 we followed the matrix approach which was used in [110] and in the paper by Ragoucy and the author [114]. The latter work is concerned with the Lie superalgebras glm|n and provides a super- analogue of Theorem 7.1.3. A geometric description of the Segal–Sugawara vectors in type A was given in the work by Kamgarpour [90] in the context of the local geometric Langlands correspondence.

CHAPTER 8

Generators in types B, C and D

We will now construct some analogues of the Segal–Sugawara vectors of Chap- ter 7 for the orthogonal and symplectic Lie algebras. As for type A,wethusgeta direct proof of the Feigin–Frenkel theorem (Theorem 6.3.1). Then we give explicit formulas for generators of the center of the completed universal enveloping algebra at the critical level as provided by Proposition 6.6.1.

8.1. Segal–Sugawara vectors in types B and D

We will keep the notation for the generators of the orthogonal Lie algebra oN introduced in Chapter 5, where N =2n or N =2n+1. That is, oN is the subalgebra of glN spanned by the elements Fij defined in (5.1). −1 Now consider the affine Kac–Moody algebra oN = oN [t, t ] ⊕ CK as defined r in (6.5) and set Fij[r]=Fijt for any r ∈ Z. The dual Coxeter number for the Lie algebra oN is given by (8.1) h∨ = N − 2 so that the normalized Killing form (6.4) can be written as 1 X, Y = trXY, X,Y ∈ o . 2 N Remark 8.1.1. Formula (8.1) provides correct values of the dual Coxeter number for oN only for N 5; cf. [86, Exercise 7.10]. Indeed, the Lie algebras o3 and ∨ sl2 are isomorphic, so that for N =3weshouldhaveh =2.Moreover,theLie algebra o4 is isomorphic to the direct sum sl2 ⊕ sl2 and it is not simple. On the other hand, the invariants of the vacuum modules constructed below will still make sense for all values N 3, where the values of h∨ should be redefined as in (8.1) for the cases N =3andN =4.

−1 By (5.2), the aﬃne Kac–Moody algebra oN = oN [t, t ] ⊕ CK has the commutation relations (8.2) Fij[r],Fkl[s] = δkj Fil[r + s] − δil Fkj[r + s] − δki Fjl[r + s]+δjl Fki [r + s]+rδr,−s K δkj δil − δki δjl , where the element K is central and the symmetry property Fij[r]+Fji [r]=0 holds. For any r ∈ Z we regard the N × N matrix F [r]= Fij[r] as the element

N N F [r]= eij ⊗ Fij[r] ∈ End C ⊗ U(oN ). i,j=1

119 120 8. GENERATORS IN TYPES B, C AND D

It has the symmetry property F [r]+F [r] = 0; see (2.24). For each a ∈{1,...,m} introduce the element F [r]a of the algebra CN ⊗ ⊗ CN ⊗ (8.3) End ... End U(oN ) m by N ⊗(a−1) ⊗(m−a) (8.4) F [r]a = 1 ⊗ eij ⊗ 1 ⊗ Fij[r]. i,j=1 By analogy with (5.6), it is immediate from (8.2) that the deﬁning relations of the algebra U(oN ) can be written in the form

(8.5) F [r]1 F [s]2 − F [s]2 F [r]1

=(P − Q) F [r + s]2 − F [r + s]2 (P − Q)+rδr,−s (P − Q)K, where both sides are elements of the algebra (8.3) with m = 2; cf. Proposition 7.1.1. In the case where N =2n is even, by analogy with (5.36) introduce the (noncommutative) Pfaﬃan Pf F [−1] as an element of the universal enveloping algebra −1 −1 U t oN [t ] by the formula 1 (8.6) Pf F [−1] = sgn σ · F [−1] ...F − [−1]. 2nn! σ(1) σ(2) σ(2n 1) σ(2n) σ∈S2n Example 8.1.2. For n =2wehave 1 Pf F [−1] = F13[−1]F31[−1] − F12[−1]F21[−1] + F11[−1]F22[−1] 2

+ F31[−1]F13[−1] − F21[−1]F12[−1] + F22[−1]F11[−1] which simpliﬁes to

Pf F [−1] = F11[−1]F22[−1] − F21[−1]F12[−1] + F31[−1]F13[−1] + F22[−2]

= F11[−1]F22[−1] − F12[−1]F21[−1] + F13[−1]F31[−1] − F22[−2].

The formula (8.6) for the Pfaffian is associated with the presentation of the orthogonal Lie algebra o2n corresponding to the symmetric bilinear form (2.25) with the matrix G =[δij ]; see (2.26). It will be convenient to have an equivalent expression for the Pfaffian in the canonical presentation of o2n by skew-symmetric matrices. To this end, fix a square matrix A of size 2n such that AAt = G. ◦ Introduce new generators Fij of o2n by the formula ◦ −1 ◦ ◦ (8.7) F = A FA, F =[Fij], where F =[Fij] is the generator matrix for o2n which is also regarded as the element (5.5). Similar to Proposition 5.1.1, we have the following defining relations ◦ for U(o2n) in terms of the matrix F : ◦ ◦ − ◦ ◦ − t ◦ − ◦ − t (8.8) F1 F2 F2 F1 =(P P ) F2 F2 (P P ) together with the relation F ◦ + F ◦ t =0,where N t t1 t2 N N P := P = P = eij ⊗ eij ∈ End C ⊗ End C . i,j=1 8.1. SEGAL–SUGAWARA VECTORS IN TYPES B AND D 121

Indeed, observe first that the mapping E → A−1EA defines an automorphism of ◦ −1 U(gl2n) so that (4.10) is satisfied by the matrix E = A EA. Thisiseasilyverified with the use of (1.67). On the other hand, since F = E − E = E − GEtG−1, the definition (8.7) is equivalent to F ◦ = E◦ − E◦ t. Hence F ◦ is skew-symmetric with respect to the standard transposition t, and relation (8.8) follows by the same argument as in the proof of Proposition 5.1.1 where t is used instead of the transposition (2.24). Accordingly, the defining relations for the universal enveloping algebra U(o2n) take the form

◦ ◦ ◦ ◦ (8.9) F [r]1 F [s]2 − F [s]2 F [r]1 t ◦ ◦ t t =(P − P ) F [r + s]2 − F [r + s]2 (P − P )+rδr,−s (P − P )K, together with the symmetry property F ◦[r]+F ◦[r]t =0,whereforallr ∈ Z the generator matrix is given by ◦ −1 ◦ ◦ F [r]=A F [r]A, F [r]= Fij[r] . Ittermsofgenerators,(8.9)iswrittenas ◦ ◦ ◦ − ◦ (8.10) Fij[r],Fkl[s] = δkj Fil[r + s] δil Fkj[r + s] − ◦ ◦ − δki Fjl[r + s]+δjl Fki[r + s]+rδr,−s K δkj δil δki δjl . ◦ ◦ In particular, the elements Fij[r]andFkl[s] commute if the indices i, j, k, l are distinct. Therefore, the Pfaffian of the matrix F ◦[−1] can be defined by the usual formula ◦ − · ◦ − ◦ − (8.11) Pf F [ 1] = sgn σ Fσ(1) σ(2)[ 1] ...Fσ(2n−1) σ(2n)[ 1], σ summed over the elements σ of the subset A2n ⊂ S2n which consists of the permutations with the properties σ(2k − 1) <σ(2k) for all k =1,...,n and σ(1) <σ(3) < ···<σ(2n − 1). Lemma 8.1.3. We have the relation Pf F [−1] = det A · Pf F ◦[−1]. Proof. The Pfaffian Pf F [−1] is found by the expansion Ψ n (8.12) = e ∧···∧e ⊗ Pf F [−1], n! 1 2n with 2n Ψ= (ei ∧ ej ) ⊗ Fij [−1] ∈ Λ(C ) ⊗ U(o2n), i 2. Then σ(3) = 2 and σ(4) > 2. In V−h∨ (o2n)wehave ◦ ◦ − ◦ − ◦ − F12[0] F1 σ(2)[ 1]F2 σ(4)[ 1] ...Fσ(2n−1) σ(2n)[ 1] − ◦ − ◦ − ◦ − = F2 σ(2)[ 1]F2 σ(4)[ 1] ...Fσ(2n−1) σ(2n)[ 1] ◦ − ◦ − ◦ − + F1 σ(2)[ 1]F1 σ(4)[ 1] ...Fσ(2n−1) σ(2n)[ 1]. Set i = σ(2) and j = σ(4). Note that the permutation σ = σ (2 4) also belongs to the subset A2n,andsgnσ = −sgn σ. Applying (8.10) we get − ◦ − ◦ − ◦ − ◦ − ◦ − ◦ − − ◦ − ◦ − F2 i[ 1]F2 j [ 1] + F1 i[ 1]F1 j [ 1] + F2 j [ 1]F2 i[ 1] F1 j [ 1]F1 i[ 1] = 0. ◦ ◦ − This implies that the terms in the expansion of F12[0] Pf F [ 1] corresponding to the pairs of the form (σ, σ) cancel pairwise, thus proving the first relation in (8.13). Now we verify that ◦ ◦ − (8.14) F12[1] Pf F [ 1] = 0.

Consider ﬁrst the summands in (8.11) with σ(1) = 1 and σ(2) = 2. In V−h∨ (o2n) we have ◦ ◦ − ◦ − ◦ − F12[1] F12[ 1]Fσ(3) σ(4)[ 1] ...Fσ(2n−1) σ(2n)[ 1] − ◦ − ◦ − = KFσ(3) σ(4)[ 1] ...Fσ(2n−1) σ(2n)[ 1].

Furthermore, let τ ∈A2n with τ(2) > 2. Then τ(3) = 2 and τ(4) > 2. We have ◦ ◦ − ◦ − ◦ − F12[1] F1 τ(2)[ 1]F2 τ(4)[ 1] ...Fτ(2n−1) τ(2n)[ 1] − ◦ ◦ − ◦ − = F2 τ(2)[0]F2 τ(4)[ 1] ...Fτ(2n−1) τ(2n)[ 1] ◦ − ◦ − = Fτ(2) τ(4)[ 1] ...Fτ(2n−1) τ(2n)[ 1]. 8.1. SEGAL–SUGAWARA VECTORS IN TYPES B AND D 123

Suppose now that σ ∈A2n with σ(1) = 1 and σ(2) = 2 is fixed and calculate the coefficient of the monomial ◦ − ◦ − Fσ(3) σ(4)[ 1] ...Fσ(2n−1) σ(2n)[ 1] ◦ ◦ − in the expansion of F12[1] Pf F [ 1]. A contribution to this coefficient can come from the terms in Pf F ◦[−1] corresponding to permutations τ ∈A of the form 2n τ = 1,σ(2k − 1), 2,σ(2k),σ(3),σ(4),...,σ(2k − 1), σ(2k),...,σ(2n − 1),σ(2n) and τ = 1,σ(2k), 2,σ(2k − 1),σ(3),σ(4),...,σ(2k − 1), σ(2k),...,σ(2n − 1),σ(2n) for k =2, 3,...,n, where the hats indicate the numbers to be skipped. These permutations can be represented as the compositions τ = σ ◦ ρ for the respective odd and even permutations ρ written in the cycle notation as ρ =(2k, 2k − 2,...,4)(2k − 1, 2k − 3,...,3, 2) and ρ =(2k, 2k − 2,...,4, 2k − 1, 2k − 3,...,3, 2). Therefore, sgn τ = −sgn σ and sgn τ =sgnσ in the first and second case, respectively. Hence, taking into account the skew-symmetry property of the matrix F ◦[−1] we find that ◦ ◦ − − − · ◦ − ◦ − F12[1] Pf F [ 1] = K 2n +2 sgn σ Fσ(3) σ(4)[ 1] ...Fσ(2n−1) σ(2n)[ 1], σ summed over σ ∈A2n with σ(1) = 1 and σ(2) = 2. Since K +2n − 2=0atthe critical level, we get (8.14).

(m) (m) We keep the notation S for the image of the symmetrizer s ∈Bm(ω) as defined in Section 1.2, under the action of the Brauer algebra Bm(N)inthe tensor space; see (1.69). We also regard S(m) as an element of the algebra (8.3), (m) identifying it with S ⊗ 1. We will use the constant γm(N) defined in (2.40) so that N + m − 2 γ (N)= . m N +2m − 2 ∨ The critical level of the vacuum module Vκ(oN ) corresponds to κ = −h ;see Remark 8.1.1. We will work with the extended Lie algebra oN ⊕ Cτ where the commutation relations for τ are defined in (7.1). Define the elements φma of the ∼ −1 −1 vacuum module V−h∨ (oN ) = U t oN [t ] by the expansion (m) m m−1 (8.15) γm(N)tr1,...,m S F1 ...Fm = φ τ + φ τ + ···+ φ , m 0 m 1 mm where F = τ + F [−1] = δijτ + Fij[−1] is the N × N matrix with entries in the universal enveloping algebra of oN ⊕ Cτ. Recall the Manin matrices of types B and D (of odd and even sizes, respectively) as introduced in Definition 5.6.1. Lemma 8.1.5. The matrix F is a Manin matrix of type B or D, respectively. Moreover, for any 1 a

F [r]a F [s]b − F [s]b F [r]a =(Pab − Qab) F [r + s]b − F [r + s]b (Pab − Qab). (m) (m) Furthermore, S (Pab − Qab)=S by (1.31), while (m) (m) (m) S F [r + s]b Pab = S Pab F [r + s]a = S F [r + s]a. Now (8.16) is implied by the ﬁrst of the following two general identities which hold for all r ∈ Z: (m) (m) (8.18) S F [r]a Qab =0 and QabF [r]a S =0. To verify (8.18), for the ﬁrst relation write (m) (m) (m) S F [r]a Qab = S F [r]a PabQab = S PabF [r]b Qab

(m) (m) = S F [r]b Qab = −S F [r]a Qab, where we used (1.31) again, together with the ﬁrst of the identities

F [r]a Qab + F [r]b Qab =0 and QabF [r]a + QabF [r]b =0. These are immediate from (2.28) and the skew-symmetry of the matrix F [r]. The proof of (8.17) is quite similar. Finally, we have

F1 F2 −F2 F1 = F [−1]1 F [−1]2 − F [−1]2 F [−1]1 − F [−2]1 + F [−2]2. Hence, applying (8.17) with m =2weget (2) F1F2 −F2F1 S =0 so that F satisfies (5.50), as required. Theorem . 8.1.6 All elements φma belong to the Feigin–Frenkel center z(oN ). Proof. It will be sufficient to verify that for all i, j (m) (m) (8.19) Fij[0] tr1,...,m S F1 ...Fm = Fij[1] tr1,...,m S F1 ...Fm =0 in the oN -module V−h∨ (oN ) ⊗ C[τ] with the trivial action on C[τ]. Consider the tensor product algebra CN ⊗ ⊗ CN ⊗ (8.20) End ... End U m+1 with m+1 copies of End CN labeled by 0, 1,...,m, where U stands for the universal enveloping algebra of the Lie algebra oN ⊕ Cτ. Relations (8.19) can be written in the equivalent form (m) (m) (8.21) F [0]0 tr1,...,m S F1 ...Fm =0 and F [1]0 tr1,...,m S F1 ...Fm =0 ∨ modulo the left ideal of U generated by oN [t]andK + h . To verity the first relation, note that by (8.5) we have

(8.22) [F [0]0, Fa]=Φ0 a Fa −Fa Φ0 a, wherewesetΦ=P − Q. Hence m (m) (m) F [0]0 tr1,...,m S F1 ...Fm = tr1,...,m S F1 ... Φ0 a Fa −Fa Φ0 a ...Fm a=1 8.1. SEGAL–SUGAWARA VECTORS IN TYPES B AND D 125 which equals

m m (m) (m) tr1,...,m S Φ0 aF1 ...Fm − tr1,...,m S F1 ...Fm Φ0 a. a=1 a=1 m (m) The sum a=1 Φ0 a commutes with the action of the symmetrizer S so that the expression is equal to zero by the cyclic property of trace. To prove the second relation in (8.21) use the following consequence of (8.5):

[F [1]0, Fa]=F [0]0 +Φ0 a F [0]a − F [0]a Φ0 a +Φ0 a K.

We have

m (m)F F (m)F F F [1]0 tr1,...,m S 1 ... m = tr1,...,m S 1 ... a−1 a=1

× F [0]0 +Φ0 a F [0]a − F [0]a Φ0 a +Φ0 a K Fa+1 ...Fm.

Applying (8.22) we can write this expression as

(m)F F F −F F tr1,...,m S 1 ... a ... Φ0 b b b Φ0 b ... m 1a

Transforming all remaining sums in a similar way, we get m−1 (m)F F (m) F F F [1]0 tr S 1 ... m = a tr1,...,m S Φ0 a 1 ... m−1 a=1 m−1 − (m)F F a tr1,...,m Φ0 a S 1 ... m−1 a=1 m−1 (m) F F + a tr1,...,m S Φ0 mΦam 1 ... m−1 a=1 m−1 (m)F F + a tr1,...,m Φam Φ0 m S 1 ... m−1 a=1 − − (m) F F + mK m(m 1) tr1,...,m S Φ0 m 1 ... m−1. Furthermore, note that (m) (m) (m) S Φ0 m Pam = S Pam Φ0 a = S Φ0 a, and (m) (8.23) S Φ0 m Qam =0. Relation (8.23) holds since

(m) (m) (m) (m) S Φ0 m Qam = S Φ0 m Pam Qam = S Φ0 a Qam = −S Φ0 m Qam, where we used (2.28) and observed that Φ is skew-symmetric with respect to the transposition (2.27) applied to the ﬁrst of second copy of End CN . Therefore, simplifying the expressions further we get

m−1 (m)F F (m) F F (8.24) F [1]0 tr1,...,m S 1 ... m = 2 a tr1,...,m S Φ0 a 1 ... m−1 a=1 − − (m) F F + mK m(m 1) tr1,...,m S Φ0 m 1 ... m−1.

As a next step, calculate the partial trace trm on the right hand side with respect to the m-th copy of End CN in (8.20) with the use of the following lemma. Lemma 8.1.7. We have − N +2m − 2 m1 tr S(m)Φ = S(m−1) Φ . m 0 m m (N +2m − 4) 0 a a=1 Proof. Taking the image of the recurrence relation (1.53) under the homomorphism (1.69) we get

m−1 m−1 1 S(m) = 1+ Φ N + m − 3+ Φ S(m−1). m(N +2m − 4) am am a=1 a=1 Now use relations (1.31) and (1.71) to bring this to the form − − m1 2 m1 S(m−1) S(m) = 1+ P − Q + P Q . am N +2m − 4 am am bm m a=1 a=1 1a

Using the cyclic property of trace we come to calculating the partial trace − − m1 2 m1 tr Φ 1+ P − Q + P Q S(m−1). m 0 m am N +2m − 4 am am bm a=1 a=1 1a

We have trm Φ0 m = 0, while

trm Φ0 m Pam =Φ0 a and trm Φ0 m Qam = −Φ0 a. Moreover,

trm Φ0 m PamQbm =trm Φ0 m QabPam =trm QabΦ0 m Pam = QabΦ0 a. (m−1) However, QabΦ0 a S = 0, which is a version of (8.23) and is veriﬁed in the (m) same way. So the desired expression for trm S Φ0 m follows.

(m) By using the formula (5.32) for the partial trace trm S and Lemma 8.1.7 we come to the relation N +2m − 2 F [1] tr S(m)F ...F = 0 1,...,m 1 m m (N +2m − 4) m−1 × − (m−1) F F (N + m 3) 2 a tr1,...,m−1 S Φ0 a 1 ... m−1 a=1 m−1 − (m−1) F F + m K m +1 tr1,...,m−1 S Φ0 a 1 ... m−1 . a=1 Nowweneedonemorelemma. Lemma 8.1.8. For any 1 a

(m) F [1]0 tr1,...,m S F1 ...Fm − m−1 N +2m 2 (m−1) = K + N − 2 tr S Φ F ...F − N +2m − 4 1,...,m−1 0 a 1 m 1 a=1 which is zero since K + N − 2 = 0 at the critical level. This completes the proof of the theorem. 128 8. GENERATORS IN TYPES B, C AND D

We will now prove the Feigin–Frenkel theorem (Theorem 6.3.1) for the orthogonal Lie algebras. Theorem . 8.1.9 The family φ22,φ44,...,φ2n 2n is a complete set of Segal– − Sugawara vectors for o2n+1 and the family φ22,φ44,...,φ2n−22n−2, Pf F [ 1] is a complete set of Segal–Sugawara vectors for o2n. Proof. The symbols of the elements of each family coincide with the images of certain elements of the symmetric algebra S(oN ) under the embedding S(oN ) → −1 −1 S t oN [t ] taking X ∈ oN to X[−1]. Due to Theorems 6.3.3 and 8.1.6 and Proposition 8.1.4, we only need to verify that the corresponding elements of S(oN ) oN are algebraically independent generators of the algebra of invariants S(oN ) ;see the argument after the proof of Theorem 6.3.3. However, this was already pointed out in Section 2.2; see Corollary 2.2.5. We conclude this section by providing equivalent expressions for the Segal– Sugawara vectors φma in the context of Howe duality. In the notation of Section 2.3, (m) m introduce the element F ∈ End P ⊗ U oN ⊕ Cτ by setting N F (m) : z ...z → z ...z ⊗Fi1... im , j1 jm i1 im j1... jm i1···im where 1 F i1... im = F ...F j1... jm iσ(1)jπ(1) iσ(m)jπ(m) α1! ...αN ! m! σ,π∈Sm and αi is the multiplicity of i in the multiset {i1,...,im}. Recall that the extremal projector p deﬁned by the equivalent formulas (2.64) and (2.66) is regarded as an P m element of End N via the action (2.63). Corollary . 8.1.10 The Segal–Sugawara vectors φma can be found from the expansion F (m) m m−1 ··· γm(N)trp = φm 0 τ + φm 1 τ + + φmm P m with the trace taken over the subspace of sl2-singular vectors in N . Proof. Recalling the formula (2.43) for the symmetrizer and using the cyclic property of trace, we can write the left hand side of (8.15) in the form (m) (m) (m) (m) γm(N)tr1,...,m S F1 ...Fm = γm(N)tr1,...,m S H F1 ...Fm H . Taking into account the isomorphism (2.61), we may regard the product H(m)F ...F H(m) 1 m P m ⊕ C as an operator on the vector space N with coeﬃcients in U oN τ .This operator coincides with F (m). It remains to note that the operator S(m) acts on P m N as the extremal projector p by Proposition 2.3.1.

8.2. Low degree invariants in trace form As we showed in Theorem 7.1.3, all coefficients of the polynomial tr E m defined in (7.9) are Segal–Sugawara vectors for glN . Their analogues for the orthogonal and symplectic Lie algebras are not known. In particular, the coefficients of the m polynomial tr F are not, in general, Segal–Sugawara vectors for oN . In this section we aim to construct examples of such vectors in trace form; cf. Example 7.1.6. This 8.2. LOW DEGREE INVARIANTS IN TRACE FORM 129 can be done by writing the Segal–Sugawara vectors φma provided by Theorem 8.1.6 in terms of the trace-type invariants

tr F [r1] ...F[rk],r1,...,rk < 0.

We will produce such expressions for φ22 and φ44. Using (1.46) and (1.69) we get N 1 P Q φ = tr + 12 − 12 F [−1] F [−1] + F [−2] . 22 N +2 1,2 2 2 N 1 2 2 Furthermore, by (1.67) and (1.68), − − − − − − − 2 tr1,2 P12F [ 1]1F [ 1]2 =tr1,2 F [ 1]2P12F [ 1]2 =tr2 F [ 1]2F [ 1]2 =trF [ 1] . Similarly, since F [−1] + F [−1] = 0, by (1.76) and (2.28) we have − − − − − tr1,2 Q12F [ 1]1F [ 1]2 = tr1,2 Q12F [ 1]2F [ 1]2 − − − − − 2 = tr2 F [ 1]2F [ 1]2 = tr F [ 1] . Taking into account that tr F [r] = 0 for any r, we conclude that

1 1 N φ = tr F [−1]2 = F [−1]F [−1]. 22 2 2 ij ji i,j=1

Thus, φ22 is proportional to the canonical Segal–Sugawara vector (6.11). To calculate φ44, note that the constant term of the polynomial (τ + F [−1]1 (τ + F [−1]2 (τ + F [−1]3 τ + F [−1]4 in τ is found by

F [ − 1]1F [−1]2F [−1]3F [−1]4 + F [−1]1F [−1]2F [−2]4 + F [−1]1F [−2]3F [−1]4

+ F [−1]1F [−1]3F [−2]4 + F [−2]2F [−1]3F [−1]4 + F [−1]2F [−2]3F [−1]4

+ F [−1]2F [−1]3F [−2]4 +2F [−1]1F [−3]4 +2F [−1]2F [−3]4 +2F [−1]3F [−3]4

+2F [−3]3F [−1]4 +2F [−2]3F [−2]4 + F [−2]2F [−2]4 +6F [−4]4. Using the properties (1.31) of the symmetrizer together with the cyclic property of trace and applying conjugations by appropriate permutation operators we can write φ44 in the form N +2 φ = tr S(4) F [−1] F [−1] F [−1] F [−1] +3F [−1] F [−1] F [−2] 44 N +6 1,2,3,4 1 2 3 4 1 2 3 +2F [−1] F [−2] F [−1] + F [−2] F [−1] F [−1] +6F [−1] F [−3] 1 2 3 1 2 3 1 2

+3F [−2]1F [−2]2 +2F [−3]1F [−1]2 +6F [−4]1 .

Applying Lemma 8.1.5 we get N +2 (4) − − − − − − − φ44 = tr1,2,3,4 S F [ 1]1F [ 1]2F [ 1]3F [ 1]4 +6F [ 1]1F [ 1]2F [ 2]3 N +6

+8F [−1]1F [−3]2 +3F [−2]1F [−2]2 +6F [−4]1 .

As a next step, calculate the partial trace tr4 by using the following general lemma. 130 8. GENERATORS IN TYPES B, C AND D

Lemma 8.2.1. Suppose that X is a certain linear combination of products of elements F [s]b of the algebra (8.3) with s<0 and 1 b m − 1. Then for any r<0 we have − N +2m − 2 m1 tr S(m)XF[r] = tr S(m−1)X F [r] . 1,...,m m m(N +2m − 4) 1,...,m−1 a a=1

Proof. This is essentially a version of Lemma 8.1.7, where Φ0 m should be replaced with F [r]m. It is veriﬁed by the same argument.

Lemmas 1.3.2 and 8.2.1 with m = 4 allow us to bring the expression for the Segal–Sugawara vector φ44 to the form N +2 φ = tr S(3) F [−1] F [−1] F [−1] F [−1] + F [−1] + F [−1] 44 4(N +4) 1,2,3 1 2 3 1 2 3

+6(N +1)F [−1]1F [−1]2F [−2]3 +8(N +1)F [−1]1F [−3]2

+3(N +1)F [−2]1F [−2]2 +6(N +1)F [−4]1 .

For the expression in the ﬁrst line we can write (3) − − − − − − tr1,2,3 S F [ 1]1F [ 1]2F [ 1]3 F [ 1]1 + F [ 1]2 + F [ 1]3 (3) − − − − − − =tr1,2,3 S F [ 1]1F [ 1]2F [ 1]3 + F [ 1]2F [ 1]1F [ 1]3

+ F [−1]2F [−1]3F [−1]1 F [−1]1.

Now use (8.16) repeatedly to move the factor F [−1]1 occurring in the second and third terms within the brackets to the leftmost position. After that, use the cyclic property of trace to write S(3) as the last factor to be able to use (8.17) to move the second factor F [−1]1 to the left. This brings the expression to the form (3) − 2 − − − − − tr1,2,3 S 3F [ 1]1F [ 1]2F [ 1]3 +3F [ 1]1F [ 1]2F [ 2]3 − 6F [−1] F [−2] F [−1] +2F [−2] F [−1] F [−1] +2F [−2] F [−2] 1 1 2 1 1 2 1 2 − − 2 − − − − − − − − 2F [ 2]1 + F [ 1]1F [ 2]2F [ 1]2 3F [ 1]1F [ 3]2 +3F [ 1]1F [ 3]1 .

Applying again Lemmas 1.3.2 and 8.2.1, this time with m = 3, and writing the symmetrizer S(2) explicitly, we get 1 1 P12 − Q12 − 2 − − − φ44 = tr1,2 + 3F [ 1]1F [ 1]2 F [ 1]1 + F [ 1]2 12 2 2 N 2 +(6N +9)F [−1]1F [−1]2 F [−2]1 + F [−2]2 +(8N +5N)F [−1]1F [−3]2

2 +(3N +5N)F [−2]1F [−2]2 −6NF[−1]1F [−2]1F [−1]2 +2NF[−2]1F [−1]1F [−1]2 − − 2 − − − − − − 2NF[ 2]1 +NF[ 1]1F [ 2]2F [ 1]2 +3NF[ 1]1F [ 3]1 +6N (N +1)F [ 4]1 .

Finally, calculating the traces in a way similar to the case of φ22 above, using (8.5) and taking into account the identity F [r]1 + F [r]2 Q12 =0,wecometothe 8.2. LOW DEGREE INVARIANTS IN TRACE FORM 131 formula 1 2 φ = 6trF [−1]4 +3 tr F [−1]2 +(3N + 15) tr F [−1]2F [−2] 44 24 +(N − 1) tr F [−1]F [−2]F [−1] + (2N +4)trF [−2]F [−1]2 +(5N 2 +27N + 22) tr F [−1]F [−3] + (N 2 +9N + 14) tr F [−2]2 .

Observe that the following relations hold for the traces occurring in the formula,

tr F [−1]2F [−2] = tr F [−2]F [−1]2 =(N/2 − 1) tr F [−2]2 and

tr F [−1]F [−2]F [−1] = −(N/2 − 1) tr F [−2]2 +(N − 2) tr F [−1]F [−3], which are easy to verify. For instance, using (8.5), for the ﬁrst element we can write

− 2 − − − − − tr F [ 1] F [ 2] = tr12 Q12F [ 1]1F [ 1]2F [ 2]2 − − − − − − − − = tr12 Q12F [ 1]2F [ 1]1F [ 2]2 tr12 Q12(P12 Q12)F [ 2]2F [ 2]2 − − − +tr12 Q12F [ 2]2(P12 Q12)F [ 2]2.

Observing that Q12F [−2]2Q12 = 0 and using the cyclic property of trace together with (1.71) we get

tr F [−1]2F [−2] = −tr F [−1]2F [−2] + (N − 2) tr F [−2]2 as required. Therefore, the formula for φ44 simpliﬁes to 1 − 4 − 2 2 φ44 = 6trF [ 1] +3 tr F [ 1] 24 +6(N +2)2 tr F [−1]F [−3] + (3N 2 +15N − 6) tr F [−2]2 .

Recall that the Feigin–Frenkel center z(oN ) is invariant with respect to the transla- 2 tion operator T ; see (6.7). In particular, the elements Tφ22 and T φ22 are Segal– Sugawara vectors. We have

− − 2 − 2 − − Tφ22 =trF [ 1]F [ 2] and T φ22 =trF [ 2] +2trF [ 1]F [ 3]. Therefore, 1 4φ − 2φ2 − (N +2)2 T 2φ =trF [−1]4 +(N/2 − 3) tr F [−2]2. 44 22 2 22

Hence, replacing φ44 by the element (8.25) tr F [−1]4 +(N/2 − 3) tr F [−2]2 in the complete sets of Segal–Sugawara vectors provided by Theorem 8.1.9 we get new complete sets of Segal–Sugawara vectors. 132 8. GENERATORS IN TYPES B, C AND D

Example 8.2.2. With the conventions of Remark 8.1.1, the following are complete sets of Segal–Sugawara vectors:

2 for o3 :trF [−1]

2 for o4 :trF [−1] , Pf F [−1] 1 for o :trF [−1]2, tr F [−1]4 − tr F [−2]2 5 2 − 2 − 4 − for o6 :trF [ 1] , tr F [ 1] , Pf F [ 1].

We will also give a direct proof that the element (8.25) is a Segal–Sugawara vector. Consider the algebra (8.20), where U now denotes the universal enveloping algebra U(oN ). Lemma 8.2.3. For any m 1 we have the relations − m (8.26) F [0]0 tr1 F [ 1]1 =0 and

− m (8.27) F [1]0 tr1 F [ 1]1 m − i−1 − m−i − − i−1 − m−i = tr1 F [ 1]1 Φ01F [ 1]1 Φ01 Φ01F [ 1]1 Φ01F [ 1]1 i=1 modulo the left ideal of U generated by oN [t] and K + N − 2,whereΦ=P − Q. Proof. Note the relation − m − − m − m − − m (8.28) F [0]0 F [ 1]1 F [ 1]1 F [0]0 =Φ01 F [ 1]1 F [ 1]1 Φ01 which is implied by (8.5). By taking tr1 and using the cyclic property of trace we get (8.26). Furthermore, calculating modulo the left ideal, we ﬁnd m − m − i−1 − − m−i F [1]0 tr1 F [ 1]1 = tr1 F [ 1]1 Φ01 F [0]0 + F [0]0 Φ01 +Φ01 K F [ 1]1 . i=1 By (8.28) we have − m−i − m−i − − m−i (8.29) F [0]0 F [ 1]1 =Φ01 F [ 1]1 F [ 1]1 Φ01. Now use the following general property of the transposition (2.27) which is applied to the copy of End CN labelled by 1:

(8.30) tr1 XY =tr1 X Y . − i−1 − m−i Taking X = F [ 1]1 F [0]0 and Y =Φ01 F [ 1]1 we obtain − i−1 − m−i − − i−1 − m−i tr1 F [ 1]1 F [0]0 Φ01 F [ 1]1 = tr1 F [ 1] 1 F [0]0 F [ 1] 1 Φ01, − where we also used Lemma 2.2.1 and the property Φ01 = Φ01. Applying the 8.2. LOW DEGREE INVARIANTS IN TRACE FORM 133 transposition to both sides of (8.29) we get − m−i − m−i − − m−i F [0]0 F [ 1] 1 =Φ01 F [ 1] 1 F [ 1] 1 Φ01. Thus, m − m − i−1 − m−i − − i−1 2 − m−i F [1]0 tr1 F [ 1]1 = tr1 F [ 1]1 Φ01 F [ 1]1 Φ01 F [ 1]1 Φ01 F [ 1]1 i=1 − − i−1 − m−i − i−1 − m−i 2 F [ 1] 1 Φ01 F [ 1] 1 Φ01 + F [ 1] 1 F [ 1] 1 Φ01 − i−1 − m−i + KF[ 1]1 Φ01 F [ 1]1 . Finally, the application of (8.30) to the terms in the middle line yields the desired expression as in the right hand side of (8.27) together with the sum m − i−1 2 − 2 − m−i tr1 F [ 1]1 (Φ01) Φ01 + K Φ01 F [ 1]1 . i=1 2 2 − − However, this sum is zero since (Φ01) =Φ01 +(N 2)Φ01 and K + N 2=0. Note that tr F [−1] = 0, and it is immediate from Lemma 8.2.3 that both 2 3 tr F [−1] and tr F [−1] are Segal–Sugawara vectors for oN . Furthermore, by (8.27), − 4 − − 2 − 2 − F [1]0 tr1 F [ 1]1 =tr1 F [ 1]1 Φ01 F [ 1]1 Φ01 + F [ 1]1 Φ01 F [ 1]1 Φ01 − − − 2 − − 2 − Φ01 F [ 1]1 Φ01 F [ 1]1 Φ01 F [ 1]1 Φ01 F [ 1]1 . By applying (8.30), we can write this as − 2 − 2 − − − − 2 − 2 tr1 F [ 1]1 +(F [ 1] )1 Φ01 F [ 1]1 Φ01 Φ01 F [ 1]1 Φ01 F [ 1]1 +(F [ 1] )1 . On the other hand,

Φ01 F [−1]1 Φ01 = F [−1]0 + Q01 F [−1]1 + F [−1]1 Q01 so that another application of (8.30) brings the expression to the form − 2 − 2 − − − tr1 F [ 1]1 +(F [ 1] )1 F [ 1]0 F [ 1]1 Φ01 − − − − − 2 − 2 F [ 1]0 Φ01 F [ 1]1 F [ 1]1 +(F [ 1] )1 . Now observe that − 2 − 2 − − − (F [ 1] )1 =tr2 F [ 1]1 Q12 = tr2 F [ 1]1 F [ 1]2 Q12 − − − − − − = tr2 F [ 1]2 F [ 1]1 Q12 +tr2 Φ12 F [ 2]1 F [ 2]1 Φ12 Q12 which gives (F [−1]2) = F [−1]2 − (N − 2)F [−2]. Using this observation and the commutation relations (8.5), we come to the sim- pliﬁed expression − 4 − − − − − − F [1]0 tr1 F [ 1]1 =(N 6) tr1 F [ 2]1 F [ 1]1 Φ01 Φ01 F [ 1]1 F [ 2]1 . 134 8. GENERATORS IN TYPES B, C AND D

Furthermore, we also ﬁnd with the use of (8.30) that − 2 − − − − − − tr1 F [ 2]1 = 2tr1 F [ 2]1 F [ 1]1 Φ01 Φ01 F [ 1]1 F [ 2]1 thus concluding that − 4 − − 2 F [1]0 tr1 F [ 1]1 +(N/2 3) tr1 F [ 2]1 =0 and so (8.25) is a Segal–Sugawara vector.

8.3. Segal–Sugawara vectors in type C We will keep the notation for the generators of the symplectic Lie algebra sp2n introduced in Chapter 5. That is, we can view sp2n is the subalgebra of gl2n spanned by the elements Fij defined in (5.3). Consider the affine Kac–Moody −1 ⊕ C r algebra sp2n = sp2n [t, t ] K as defined in (6.5) and set Fij[r]=Fijt for any ∈ Z r . The dual Coxeter number for the Lie algebra sp2n is given by (8.31) h∨ = n +1 and the normalized Killing form (6.4) can be written as ∈ X, Y =trXY, X,Y sp2n. By (5.4), the affine Kac–Moody algebra sp2n has the commutation relations (8.32) Fij[r],Fkl[s] = δkj Fil[r + s] − δil Fkj[r + s] − εi εj δki Fjl[r + s]+δjl Fki [r + s] +2rδr,−s K δkj δil − εi εj δki δjl , where the element K is central and the symmetry property Fij[r]+εi εj Fji [r]=0 holds. As with the orthogonal case, for any r ∈ Z regard the matrix F [r]= Fij[r] as the element 2n ⊗ ∈ C2n ⊗ F [r]= eij Fij[r] End U(sp2n). i,j=1 It has the symmetry property F [r]+F [r] = 0; see (2.24). For each a ∈{1,...,m} introduce the element F [r]a of the algebra C2n ⊗ ⊗ C2n ⊗ (8.33) End ... End U(sp2n) m by 2n ⊗(a−1) ⊗(m−a) (8.34) F [r]a = 1 ⊗ eij ⊗ 1 ⊗ Fij[r]. i,j=1 As in (5.6), the defining relations of the algebra U(sp2n) can be written in the matrix form

(8.35) F [r]1 F [s]2 − F [s]2 F [r]1

=(P − Q) F [r + s]2 − F [r + s]2 (P − Q)+2rδr,−s (P − Q)K, where both sides are elements of the algebra (8.33) with m =2. (m) The symmetrizer s in the Brauer algebra Bm(ω) was defined in Section 1.2. We keep the notation S(m) for the image of the symmetrizer s(m) under the action of the Brauer algebra Bm(−2n) in the tensor space defined by (1.73). We also 8.3. SEGAL–SUGAWARA VECTORS IN TYPE C 135 regard S(m) as an element of the algebra (8.33), identifying it with S(m) ⊗ 1. We will keep using the constant γm(−2n) defined in (2.40) so that 2n − m +2 γ (−2n)= . m 2(n − m +1)

The critical level of the vacuum module Vκ(sp2n) corresponds to the value − ∨ κ = h . We will construct Segal–Sugawara vectors for sp2n in a way similar to the orthogonal case. However, an expression analogous to (8.15) will only make senseforthevaluesofm bound by the condition m n + 1. The reason is that the formulas for the symmetrizer s(m) given in Section 1.2 will contain zero denominators under the specialization ω = −2n for the values m n +2. These formulas do define the operator S(m) for m = n+1 but it turns out to be identically zero; see Proposition 2.3.2. We will go around this problem in a way similar to the construction of Casimir elements in Section 5.5. Namely, we will let the parameter m be fixed and let n vary. Then we will show that the symplectic analogue of (8.15) still defines Segal–Sugawara vectors, and the values of n can be ‘analytically continued’ to the region n (m − 1)/2. Suppose first that n m and introduce the elements φma of the vacuum ∼ −1 −1 module V−h∨ (sp2n) = U t sp2n[t ] by the expansion (8.36) γ (−2n)tr S(m)F ...F = φ τ m + φ τ m−1 + ···+ φ , m 1,...,m 1 m m 0 m 1 mm where F = τ + F [−1] = δijτ + Fij[−1] is the 2n × 2n matrix with entries in the ⊕ C universal enveloping algebra of sp2n τ, and the commutation relations for τ are defined in (7.1). Recall the Manin matrices of type C as introduced in Definition 5.6.1. Lemma 8.3.1. The matrix F is a Manin matrix of type C. Moreover, if n m then for any 1 a

F1 F2 −F2 F1 = F [−1]1 F [−1]2 − F [−1]2 F [−1]1 − F [−2]1 + F [−2]2. Hence, applying (8.37) with m =2weget (2) S F1F2 −F2F1 =0 so that F satisﬁes (5.51), as required. Theorem . 8.3.2 Suppose that n m. Then all elements φma belong to the Feigin–Frenkel center z(sp2n). Proof. We will repeat the arguments used for the proof of Theorem 8.1.6 and indicate a few changes to be made. Consider the tensor product algebra CN ⊗ ⊗ CN ⊗ (8.39) End ... End U m+1 136 8. GENERATORS IN TYPES B, C AND D with m+1 copies of End CN labeled by 0, 1,...,m, where U stands for the universal ⊕ C enveloping algebra of the Lie algebra sp2n τ. We need to verify the relations (m) (m) (8.40) F [0]0 tr1,...,m S F1 ...Fm =0 and F [1]0 tr1,...,m S F1 ...Fm =0 ∨ modulo the left ideal of U generated by sp2n[t]andK + h . The ﬁrst relation follows by the same calculation as for oN which relies on a consequence of (8.35) given by

(8.41) [F [0]0, Fa]=Φ0 a Fa −Fa Φ0 a, wherewesetΦ=P − Q. To prove the second relation in (8.40) use another consequence of (8.35),

[F [1]0, Fa]=F [0]0 +Φ0 a F [0]a − F [0]a Φ0 a +2Φ0 a K, so that m (m)F F (m)F F F [1]0 tr1,...,m S 1 ... m = tr1,...,m S 1 ... a−1 a=1

× F [0]0 +Φ0 a F [0]a − F [0]a Φ0 a +2Φ0 a K Fa+1 ...Fm. This expression is then transformed in the same way as for the orthogonal case, taking into account that now (m) (m) (m) S Pab = PabS = −S for a

m−1 (m)F F − (m)F F (8.42) F [1]0 tr1,...,m S 1 ... m = 2 a tr1,...,m Φ0 a S 1 ... m−1 a=1 − (m) F F + 2mK + m(m 1) tr1,...,m S Φ0 m 1 ... m−1. The following is a symplectic analogue of Lemma 8.1.7 Lemma 8.3.3. If m n then for the m-th partial trace we have − n − m +1 m1 tr S(m)Φ = − S(m−1) Φ . m 0 m m(n − m +2) 0 a a=1

Proof. The argument is the same as for Lemma 2.2.7, where the role of Fm is taken by Φ0 m.

By calculating the partial trace trm on the right hand side of (8.42) with the use of (5.43) and Lemma 8.3.3, we get n − m +1 F [1] tr S(m)F ...F = − 0 1,...,m 1 m m (n − m +2) m−1 × − (m−1)F F (2n m +3) 2 a tr1,...,m−1 Φ0 a S 1 ... m−1 a=1 m−1 − (m−1)F F + m 2K + m 1 tr1,...,m−1 Φ0 a S 1 ... m−1 . a=1 Now prove a counterpart of Lemma 8.1.8. 8.3. SEGAL–SUGAWARA VECTORS IN TYPE C 137

Lemma 8.3.4. If m n then for any 1 a

The proof of the theorem is now completed by the application of Lemma 8.3.4 with m replaced by m − 1. We have

(m) F [1]0 tr1,...,m S F1 ...Fm − m−1 n m +1 (m−1) = −2 K + n +1 tr S Φ F ...F − n − m +2 1,...,m−1 0 a 1 m 1 a=1 which is zero since K + n + 1 = 0 at the critical level.

Our next goal is to extend the definition of the elements φma to the values − −1 −1 n (m 1)/2. Consider the symmetric algebra S t sp2n[t ] as a module over sp2n defined in (6.18) with r =0,sothatsp2n is identified with a subalgebra of → sp2n[t] via the embedding Y Y [0]. It follows from the results of Section 2.2 that −1 −1 for any positive integers r1,...,rk the elements of S t sp2n[t ] defined by

(8.43) tr F [−r1] ...F[−rk] are sp2n-invariants. Furthermore, by a standard application of the ﬁrst fundamental theorem of the invariant theory for the symplectic group, the algebra of invariants −1 −1 sp2n S t sp2n[t ] is generated by elements of the form (8.43). We have the graded decomposition of this algebra into ﬁnite-dimensional subspaces, −1 −1 sp2n d (8.44) S t sp2n[t ] = Sn, d0

d where Sn denotes the subspace of homogeneous elements of degree d with respect to the grading on the symmetric algebra deﬁned by deg Fij[−r]=r. For all n 2 identify sp2n−2 with the subalgebra of sp2n spanned by all generators Fij with i, j ∈{1,...,n− 1, (n − 1) ,...,1 }. Introduce the projections → ∈ πn : sp2n sp2n−2 such that πn(Fij) = 0 for all Fij / sp2n−2. The projection πn −1 −1 extends to the vector space t sp2n[t ] where it acts on the coeﬃcients of polynomials in t−1. We will keep the same notation for the associated homomorphism of symmetric algebras −1 −1 → −1 −1 πn :S t sp2n[t ] S t sp2n−2[t ] .

Note that the images of the elements (8.43) underπn take the same trace form and −1 −1 so belong to the algebra of sp2n−2-invariants in S t sp2n−2[t ] . 138 8. GENERATORS IN TYPES B, C AND D

Lemma 8.3.5. For each ﬁxed value d 0 there exists a nonnegative integer d n0 such that the dimension of the subspace Sn is independent of n for all n n0. d These subspaces possess respective bases Bn formed by monomials in the elements d d d (8.43) such that πn(Bn)=Bn−1 for all n>n0. Moreover, all subspaces Sn with ◦ ◦···◦ d 1 n

sp2n the algebra of invariants S(sp2n) ; see also the argument following the proof of Theorem 6.3.3. However, this was proved in Section 2.2; see Corollary 2.2.10.

The special role of the parameters satisfying the conditions n +1 m 2n +1 can be understood from the properties of the homomorphism B − → C2n ⊗m (8.47) m( 2n) End sp2n deﬁned by (1.73). The Brauer algebra Bm(−2n) is known to be semisimple for the values m 2n +1;see[2], [138]. The homomorphism (8.47) is an isomorphism for m n but it has a nonzero kernel for the values n +1 m 2n +1. Inparticular, if m = n + 1 then the kernel is one-dimensional and spanned by the symmetrizer s(m). To get explicit formulas for Segal–Sugawara vectors in this case, note the recurrence formula for the symmetrizer S(m); cf. the proof of Lemma 2.2.7:

1 S(m) = 2m(n − m +2) m−1 m−1 (m−1) × 2n − 2m +4− (2n − m +2) Pam − m Qam − ΦamΦbm S , a=1 a=1 a= b where Φam = Pam − Qam for a =1,...,m− 1. Hence, applying the transposition (2.27) with respect to the m-th copy of End C2n,weget n − m +1 S(m) − S(m) = − S(m−1) Y (m), m(n − m +2) where m−1 (m) Y = − Φam a=1 is the image of the Jucys–Murphy element ym − (ω − 1)/2 under the map (1.73); see (1.26). Therefore, assuming n m and using (8.30) for the m-th partial transposition, we obtain that the trace

(m)F F tr1,...,m S 1 ... m equals (m)F F − − tr1,...,m S 1 ... m−1 τ F [ 1]m − n m +1 (m−1) (m) − tr S Y F ...F − τ − F [−1] . m(n − m +2) 1,...,m 1 m 1 m

Since Y (m) = −Y (m), this implies

(m)F F (m)F F tr1,...,m S 1 ... m =tr1,...,m S 1 ... m−1 τ n − m +1 + tr S(m−1) Y (m)F ...F . 2m(n − m +2) 1,...,m 1 m 8.3. SEGAL–SUGAWARA VECTORS IN TYPE C 141

Together with (5.43) this gives 1 tr S(m)F ...F n − m +1 1,...,m 1 m − 2n m +3 (m−1) = tr S F ...F − τ m(n − m +2) 1,...,m 1 m 1 1 + tr S(m−1) Y (m)F ...F 2m(n − m +2) 1,...,m 1 m which can be taken as a definition of the left hand side for n = m−1. This definition is essentially equivalent to the one obtained in the proof of Proposition 8.3.7. We can now conclude that all coefficients of the polynomial in τ (m−1) (m)F F tr1,...,m S Y 1 ... m − are Segal–Sugawara vectors for sp2n for all n m 1. The branching properties (1.30) of the primitive idempotents imply the relation

(m−1) (m) S = S + EU + EV , where EU and EV denote the images of the idempotents eU and eV under the map (1.73), which are associated with the respective updown tableaux U = (1), (2),...,(m − 1), (m − 1, 1) and V = (1), (2),...,(m − 1), (m − 2) . Furthermore, by (1.27) we have

(m) (m) (m) (m) (m) S Y =(m − 1)S , EU Y = −EU and EV Y =(2n − m +3)EV and so (m−1) (m) (m) S (1 + Y )=mS +(2n − m +4)EV and (m−1) (m) (m) S (2n − m +3− Y )=2(n − m +2)S +(2n − m +4)EU . This implies that all coeﬃcients of the polynomials E F F E F F tr1,...,m U 1 ... m and tr1,...,m V 1 ... m − are Segal–Sugawara vectors for sp2n for all n m 1. It would be interesting to ﬁnd analogous formulas for Segal–Sugawara vectors for the values n (m − 1)/2 together with their direct proofs. To conclude this section we give equivalent expressions for the Segal–Sugawara vectors φma in the context of skew Howe duality. In the notation of Section 2.3, F (m) ∈ m ⊗ ⊕ C introduce the element End Λ2n U sp2n τ by F (m) : ζ ∧···∧ζ → ζ ∧···∧ζ ⊗Fi1... im , j1 jm i1 im j1... jm i1<···

Corollary . 8.3.9 The Segal–Sugawara vectors φma with m n can be found from the expansion − F (m) m m−1 ··· γm( 2n)trp = φm 0 τ + φm 1 τ + + φmm m with the trace taken over the subspace of sl2-singular vectors in Λ2n. Proof. Using the cyclic property of trace and applying the formula (2.52) for the symmetrizer, write the left hand side of (8.36) in the form (m) (m) (m) (m) γm(−2n)tr1,...,m S F1 ...Fm = γm(−2n)tr1,...,m S H F1 ...Fm H . (m) (m) The product H F1 ...Fm H canberegardedasanoperatoronthevector m ⊕ C space Λ2n with coeﬃcients in U sp2n τ , due to the isomorphism (2.67). This operator coincides with F (m). Finally, by Proposition 2.3.2, the operator S(m) acts m on Λ2n as the extremal projector p.

8.4. Low degree invariants in trace form As we pointed out in Section 8.2, the coeﬃcients of the polynomials tr F m are not, in general, Segal–Sugawara vectors for sp2n. In this section we construct examples of such vectors in trace form by using Theorem 8.3.8. We will calculate the Segal–Sugawara vectors φ22 and φ44 in a way similar to the orthogonal case; see Section 8.2. By (1.46) and (1.73) for n 2weget n 1 P Q φ = tr − 12 − 12 F [−1] F [−1] + F [−2] . 22 n − 1 1,2 2 2 2n 1 2 2 The same argument as in the orthogonal case yields 1 1 N φ = − tr F [−1]2 = − F [−1]F [−1] 22 2 2 ij ji i,j=1 which is proportional to the canonical Segal–Sugawara vector (6.11) for all n 1. For n 4 the Segal–Sugawara vector φ44 is found as the constant term of the polynomial in τ deﬁned by n − 1 tr S(4) (τ + F [−1] (τ + F [−1] (τ + F [−1] F [−1] . n − 3 1,2,3,4 1 2 3 4 Applying Lemma 8.3.6 we can write this polynomial in the form − − n 1 (3) − − − (8.48) − tr1,2,3 S (τ + F [ 1]1 (τ + F [ 1]2 (τ + F [ 1]3 4(n 2) × F [−1]1 + F [−1]2 + F [−1]3 . Due to Lemma 8.3.1, τ + F [−1] is a Manin matrix of type C so that using relation (5.55) we get (3) − − − − tr1,2,3 S (τ + F [ 1]1 (τ + F [ 1]2 (τ + F [ 1]3 F [ 1]2 (3) − − − − =tr1,2,3 S (τ + F [ 1]2 (τ + F [ 1]1 (τ + F [ 1]3 F [ 1]2.

Applying now the conjugation by P12 and using the cyclic property of trace we can write this expression as (3) − − − − tr1,2,3 S (τ + F [ 1]1 (τ + F [ 1]2 (τ + F [ 1]3 F [ 1]1. 8.4. LOW DEGREE INVARIANTS IN TRACE FORM 143

A similar calculation applied to the third summand in (8.48) shows that φ44 equals the constant term of the polynomial 3(n − 1) − tr S(3) (τ + F [−1] (τ + F [−1] (τ + F [−1] F [−1] . 4(n − 2) 1,2,3 1 2 3 1 Expanding and using Lemma 8.3.1 we get 3(n − 1) φ = − tr S(3) F [−1]2F [−1] F [−1] + F [−1] F [−1] F [−2] 44 4(n − 2) 1,2,3 1 2 3 1 2 3 − − − − 2 − − − +4F [ 1]1F [ 2]1F [ 1]2 + F [ 1]1F [ 2]2 +3F [ 2]1F [ 2]2

+2F [−1]1F [−3]2 +6F [−1]1F [−3]1 +6F [−4]1 . Now use Lemma 8.3.6 and relation (5.43) for m = 3 to bring this to the form 1 Q φ = tr 1 − P − 12 F [−1]2F [−1] F [−1] + F [−1] 44 8 1,2 12 n 1 2 1 2 − − − − − − − − − − 2 − + F [ 1]1F [ 1]2 F [ 1]1 + F [ 1]2 8nF[ 1]1F [ 2]1F [ 1]2 2nF[ 1]1F [ 2]2

− 6nF[−2]1F [−2]2 − 4nF[−1]1F [−3]2 − 12nF[−1]1F [−3]1 − 12nF[−4]1 . Calculating the traces with the use of (8.35) and the identity F [r]1+F [r]2 Q12 =0, we obtain 1 2 φ = tr F [−1]2 −2trF [−1]4 +(4n−1) tr F [−1]2F [−2]+(6n−6) tr F [−2]2 44 8 +(8n − 9) tr F [−1]F [−2]F [−1] + (−24n2 +18n + 10) tr F [−1]F [−3] . The following relations are analogous to the orthogonal case and veriﬁed in the same way: tr F [−1]2F [−2] = (n +1)trF [−2]2 and tr F [−1]F [−2]F [−1] = −(n +1)trF [−2]2 +(2n +2)trF [−1]F [−3].

Hence, the formula for φ44 simplifies to 1 − 2 2 − − 4 φ44 = tr F [ 1] 2trF [ 1] 8 − 8(n − 1)2 tr F [−1]F [−3] + (−4n2 +10n +2)trF [−2]2 . This is a Segal–Sugawara vector for all n 2. In fact, φ44 is well-defined for n =1 as well, but is equal to zero because S(2) =0. Since the Feigin–Frenkel center z(sp2n) is invariant with respect to the transla- 2 tion operator T defined in (6.7), the elements Tφ22 and T φ22 are Segal–Sugawara vectors. We have − − − 2 − − 2 − − − Tφ22 = tr F [ 1]F [ 2] and T φ22 = tr F [ 2] 2trF [ 1]F [ 3].

Therefore, the element φ44 in the complete set of Segal–Sugawara vectors of The- orem 8.3.8 can be replaced with the element − 2 − 2 2 4φ44 +2φ22 +2(n 1) T φ22 144 8. GENERATORS IN TYPES B, C AND D which coincides with

(8.49) tr F [−1]4 − (n +3)trF [−2]2.

Example 8.4.1. The following are complete sets of Segal–Sugawara vectors: − 2 for sp2 :trF [ 1] − 2 − 4 − − 2 for sp4 :trF [ 1] , tr F [ 1] 5trF [ 2] . We will also give a direct proof that the element (8.49) is a Segal–Sugawara vector. In the next lemma we use the tensor product algebra (8.39).

Lemma 8.4.2. For any m 1 we have the relations − m F [0]0 tr1 F [ 1]1 =0 and

− m F [1]0 tr1 F [ 1]1 m − i−1 − m−i − − i−1 − m−i = tr1 F [ 1]1 Φ01F [ 1]1 Φ01 Φ01F [ 1]1 Φ01F [ 1]1 i=1 − modulo the left ideal of U generated by sp2n[t] and K + n +1,whereΦ=P Q. Proof. The argument is essentially the same as for Lemma 8.2.3; it uses the 2 2 relation (Φ01) =Φ01 +2(n +1)Φ01 and the corresponding version of (8.30) which holds for the transposition (2.27) in the symplectic case.

We have tr F [−1] = 0, and Lemma 8.4.2 implies that both tr F [−1]2 and − 3 tr F [ 1] are Segal–Sugawara vectors for sp2n. Now we repeat the corresponding argument of Section 8.2 taking into account (1.75) and the respective symplectic analogues of the relations used in the argument:

Φ01 F [−1]1 Φ01 = F [−1]0 − Q01 F [−1]1 − F [−1]1 Q01 and (F [−1]2) = F [−1]2 − 2(n +1)F [−2]. This leads to the formulas − 4 − − − − − − F [1]0 tr1 F [ 1]1 = 2(n +3)tr1 F [ 2]1 F [ 1]1 Φ01 Φ01 F [ 1]1 F [ 2]1 and − 2 − − − − − − F [1]0 tr1 F [ 2]1 = 2tr1 F [ 2]1 F [ 1]1 Φ01 Φ01 F [ 1]1 F [ 2]1 which imply − 4 − − 2 F [1]0 tr1 F [ 1]1 (n +3)tr1 F [ 2]1 =0 so that (8.49) is a Segal–Sugawara vector. 8.5. SUGAWARA OPERATORS IN TYPES B, C AND D 145

8.5. Sugawara operators in types B, C and D Here we use the general results on Sugawara operators outlined in Section 6.6 to construct generators of the completed universal enveloping algebras at the critical level. As in Section 5.1, we will consider the three cases B, C and D simultaneously and use the notation gN for any of the Lie algebras oN or spN . The critical level κ = −h∨ is deﬁned in (8.1) and (8.31); see also Remark 8.1.1. Introduce the Laurent series in z with coeﬃcients in U−h∨ (gN )by −r−1 (8.50) Fij(z)= Fij[r] z ,i,j=1,...,N, r∈Z and combine them into the matrix F (z)sothat N F (z)= eij ⊗ Fij(z). i,j=1

As in (8.4) and (8.34), for a ∈{1,...,m} we use the notation F (z)a for the corresponding elements of the algebra CN ⊗ ⊗ CN ⊗ End ... End U, m where U stands for the algebra of polynomial differential operators over z with −1 coefficients in U−h∨ (gN )[[z,z ]]. We will keep using the rational function γm(ω) defined in (2.40). The variable ω takes value ω = N in the orthogonal case and ω = −N in the symplectic case. As suggested by (8.15) and (8.36) together with Proposition 8.3.7, define the Laurent series φma(z) by the following expansion with the usual reading of normal ordering from right to left: (m) : γm(ω)tr1,...,m S ∂z + F (z)1 ... ∂z + F (z)m : m m−1 ··· = φm 0(z) ∂z + φm 1(z) ∂z + + φmm(z).

Following (8.6), in the case gN = o2n we also introduce the (noncommutative) Pfaffian as the Laurent series 1 (8.51) Pf F (z)= sgn σ : F (z) ...F − (z):. 2nn! σ(1) σ(2) σ(2n 1) σ(2n) σ∈S2n The results of Section 8.1 in the orthogonal case and Section 8.3 in the symplectic case together with Proposition 6.6.1 lead to the following. Theorem . 8.5.1 All coefficients of the Laurent series φma(z), belong to the center of the algebra U−h∨ (gN ). Moreover, the coefficients of the Laurent series Pf F (z) belong to the center of U−h∨ (o2n). The coefficients of the Laurent series

φ22(z),φ44(z),...,φ2n 2n(z) are topological generators of the center of U−h∨ (gN ) in the cases gN = o2n+1 and gN = sp2n. Furthermore, the coeﬃcients of the Laurent series

φ22(z),φ44(z),...,φ2n−22n−2(z), Pf F (z) are topological generators of the center of U−h∨ (o2n). 146 8. GENERATORS IN TYPES B, C AND D

Example 8.5.2. By the calculations of Section 8.2 and Section 8.4 for the Segal–Sugawara vector φ22, the Laurent series φ22(z) is a scalar multiple of the series : tr F (z)2 :. We have

N N 2 :trF (z) :=: Fij(z)Fji(z):= Fij(z)+Fji(z)+Fji(z)Fij(z)− . i,j=1 i,j=1

Hence for each p ∈ Z the coeﬃcient of z−p−2 in this series is a central element of U−h∨ (gN )givenby

N Fij[r]Fji[p − r]+ Fji[p − r]Fij[r] . i,j=1 r<0 r0

Example 8.5.3. In the case gN = o4 we have 1 Pf F (z)= : F13(z)F31(z): − : F12(z)F21(z): + :F11(z)F22(z): 2

+:F31(z)F13(z): − : F21(z)F12(z): + :F22(z)F11(z): .

By Example 8.1.2, this can also be written as

Pf F (z)= :F11(z)F22(z): − : F12(z)F21(z): + :F13(z)F31(z): − ∂zF22(z).

Taking the coeﬃcient of z−p−2 in this series for any p ∈ Z, we get a central element ∨ of U−h∨ (o4) (assuming h =2)givenby F11[r]F22[p − r]+ F13[r]F31[p − r] − F12[r]F21[p − r] r∈Z r<0 + F31[p − r]F13[r] − F21[p − r]F12[r] +(p +1)F22[p]; r0 see Remark 8.1.1.

By the general construction of Section 6.5, some Casimir elements produced in Sections 5.4 and 5.5 can be obtained from the Segal–Sugawara vectors; cf. Sec- tion 7.1. Namely, apply the homomorphism deﬁned in (6.30) to the coeﬃcients of the polynomials (8.15) and (8.36). Multiplying the image by zm from the right we get (m) −1 −1 m γm(ω)tr1,...,m S −∂z + F1 z ... −∂z + Fm z z (m) = γm(ω)tr1,...,m S u + F1 ... u + Fm − m +1 , where u = −∂z z and the value of ω is N or −N, respectively. Using Lemma 5.3.4, and taking m =2k with the specializations u = k − 1andu = k we reproduce the respective Casimir elements (5.23) and (5.44). Equivalently, these Casimir elements can be obtained from the Sugawara operators constructed above in the same way as for type A as outlined at the end of Chapter 7. 8.6. BIBLIOGRAPHICAL NOTES 147

8.6. Bibliographical notes The exposition follows [110] with some modiﬁcations, especially in the symplectic case. If the Feigin–Frenkel theorem (Theorem 6.3.1) is assumed in this case, then the second part of Proposition 8.3.7 follows by a simpler argument as in the proof of Proposition 5.5.3. A noncommutative form of the property that the elements (8.43) generate the algebra of invariants was derived by Kumar [97].

CHAPTER 9

Commutative subalgebras of U(g)

We will now use the explicit generators of the algebra z(g) produced in Chap- ters 7 and 8 to construct commutative subalgebras Aμ of the universal enveloping algebra U(g) parameterized by elements μ ∈ g∗. The corresponding subalgebras of the symmetric algebra S(g) are Poisson commutative and known as the Mishchenko– Fomenko or shift of argument subalgebras. The construction thus provides an aﬃr- mative solution of Vinberg’s quantization problem in terms of explicit generators. Furthermore, we obtain a direct proof of a theorem of Feigin, Frenkel and Toledano Laredo [42] describing the structure of Aμ for regular elements μ in the case where g is a classical Lie algebra. In a more general context, a connection of the center at the critical level with a commutative family of Gaudin Hamiltonians will be discussed in Chapter 14.

9.1. Mishchenko–Fomenko subalgebras A Poisson algebra A is a commutative associative algebra equipped with a Poisson bracket which is a bilinear map (9.1) { , } : A × A → A satisfying the following properties: A is a Lie algebra with respect to this bracket, and the Leibniz rule (9.2) {x, yz} = {x, y}z + y {x, z} holds for any three elements x, y, z ∈ A. In particular, the map (9.1) is skew- symmetric, {x, y} = −{y, x}, and satisfies the Jacobi identity {x, {y, z}} + {y, {z,x}} + {z,{x, y}} =0 for all x, y, z ∈ A.ThePoisson center of A is defined by (9.3) Z(A)={P ∈ A |{x, P } =0 forall x ∈ A}. Clearly, Z(A) is a subalgebra of A. Now let g be an arbitrary finite-dimensional Lie algebra over C. The symmetric algebra S(g) is a Poisson algebra with the Lie–Poisson bracket such that for any two elements x, y ∈ g the value {x, y} is the commutator [x, y]. The bracket extends to all elements of S(g) via the Leibniz rule (9.2). If J1,...,Jd is a basis of g such that d k (9.4) [Ji,Jj ]= cij Jk k=1

149 150 9. COMMUTATIVE SUBALGEBRAS OF U(g)

k for structure constants cij,then d { } k (9.5) Ji,Jj = cij Jk. k=1 By the Leibniz rule, the Poisson bracket of two monomials is found explicitly by r s (9.6) {x1 ...xr,y1 ...ys} = x1 ...xa ...xr y1 ...yb ...ys {xa,yb} a=1 b=1 with the hats indicating that the corresponding factors should be skipped. By the deﬁnition (9.3), the Poisson center of S(g) coincides with the subalgebra S(g)g of g-invariants in S(g). Fix μ ∈ g∗ and t ∈ C. Consider another set of generators of the algebra S(g) by ‘shifting the arguments’. Namely, set Ji = Ji + tμi for i =1,...,d with μi = μ(Ji). Equip S(g) with another Poisson bracket { , }t which is deﬁned by the same formulas as in (9.5) but for the shifted generators: d { } k (9.7) Ji ,Jj t = cij Jk. k=1 In terms of the basis elements of g it takes the form d { } k (9.8) Ji,Jj t = cij (Jk + tμk). k=1

It is clear from (9.7), that the Poisson center Zt of S(g) with respect to the bracket g { , }t coincides with the subalgebra obtained from S(g) by shifting the arguments of the invariants J → J + tμ .Thatis, i i i g (9.9) Zt = P (J + tμ) | P ∈ S(g) , where we regard elements of the symmetric algebra S(g)aspolynomialsP = P (J) in the variables J =(J ,...,J ) and write 1 d P (J + tμ)=P J1 + tμ1,...,Jd + tμd .

Definition 9.1.1. Denote by Aμ the subalgebra of S(g) generated by the Pois- son centers Zt for all t ∈ C. This subalgebra Aμ of S(g)iscalledtheMishchenko– Fomenko subalgebra or shift of argument subalgebra.

Proposition 9.1.2. The subalgebra Aμ is Poisson commutative with respect to any bracket { , }t. That is, {x, y}t =0for all x, y ∈ Aμ and any t ∈ C. Proof. ∈ C Suppose that t1,t2 . Take an arbitrary element of Zt1 and an arbitrary element of Zt2 . By (9.9) such elements should have the respective forms P (J +t μ)andQ(J +t μ)forsomeP, Q ∈ S(g)g. By (9.6) and (9.8), the expression 1 2 (9.10) P (J + t1 μ),Q(J + t2 μ) t is linear in t.Ift1 = t2 then this expression vanishes at t = t1 and t = t2 because the elements belong to the Poisson centers of the respective brackets. This implies that (9.10) is zero for all t. On the other hand, for a ﬁxed value of t, the expression (9.10) depends polynomially on t1 and t2. Since this polynomial vanishes for all t1 = t2 it must be identically zero. 9.1. MISHCHENKO–FOMENKO SUBALGEBRAS 151

Taking a slightly different point of view, let P = P (J1,...,Jd)beanelement g of S(g) of a certain degree m as a polynomial in the Ji. Let us now regard t as a variable. Make the substitution J → J + tμ and expand as a polynomial in t, i i i ··· m (9.11) P J1 + tμ1,...,Jd + tμd = P(0) + P(1) t + + P(m) t , ∈ A to define elements P(i) S(g) associated with P and μ. The subalgebra μ is ∈ g generated by all elements P(i) associated with all g-invariants P S(g) .By Proposition 9.1.2, Aμ is Poisson commutative with respect to the initial bracket { , } (which corresponds to t =0). When P is a homogeneous polynomial, we will also use an equivalent form −1 of (9.11), where we make the substitution Ji → Ji z + μi for a variable z,and expand as a polynomial in z−1, −1 −1 −m ··· −1 (9.12) P J1 z + μ1,...,Jd z + μd = P(0) z + + P(m−1) z + P(m). Now suppose that g is a simple Lie algebra. The bilinear form (6.4) allows us to identify g∗ with g, as in the proof of Theorem 6.3.3. It will also be convenient to 1 d use the basis J ,...,J of g which is dual to J1,...,Jd with respect to (6.4). For ∗ i i any element μ ∈ g set μi = μ(Ji)andμ = μ(J ). Then μ is identified with the element of g given by 1 d 1 d (9.13) μ = μ1J + ···+ μdJ = μ J1 + ···+ μ Jd i i and we have μ, Ji = μi and μ, J = μ .Weletn denote the rank of g.An ∗ ∼ element μ ∈ g = g is called regular, if the centralizer gμ of μ in g has minimal possible dimension; this minimal dimension coincides with n. As we recalled in Chapter 2, the subalgebra of g-invariants in the symmetric algebra S(g) admits a family P1,...,Pn of algebraically independent generators; see (2.2).

Theorem 9.1.3. Let μ be regular. If P1,...,Pn are algebraically independent g generators of S(g) of the respective degrees d1,...,dn, then the elements − (9.14) Pk(i),k=1,...,n, i=0, 1,...,dk 1, are algebraically independent generators of Aμ. Proof. A It is clear that the Pk(i) generate the algebra μ. We only need to show that these polynomials are algebraically independent. It is enough to demonstrate that the differentials dPk (i) evaluated at a certain point are linearly independent. We will regard the differential dP of any polynomial P = P (J) ∈ S(g)asthe element of S(g) ⊗ g defined by ∂P ∂P dP = ⊗ J1 + ···+ ⊗ Jd. ∂J1 ∂Jd It is straightforward to verify that it does not depend on the chosen basis. The vector space S(g) ⊗ g will be understood as a Lie algebra with the bracket [P ⊗ X, Q ⊗ Y ]=PQ⊗ [X, Y ],P,Q∈ S(g)andX, Y ∈ g. Introduce the element X of this Lie algebra by d i X = J ⊗ Ji. i=1 152 9. COMMUTATIVE SUBALGEBRAS OF U(g)

Lemma 9.1.4. If P ∈ S(g)g then in the Lie algebra S(g) ⊗ g we have X ,dP =0. Proof. By using the structure coeﬃcients deﬁned by (9.4) we can write d d d i ⊗ i ∂P ⊗ k i ∂P ⊗ (9.15) J Ji,dP = J [Ji,Jj]= cij J Jk. ∂Jj ∂Jj i=1 i,j=1 i,j,k=1 k k k k On the other hand, we have cij = [Ji,Jj ],J ,andsocij = Ji, [Jj ,J ] since the form (6.4) is invariant. This implies the expansion d k − k i [J ,Jj ]= cij J . i=1 The condition P ∈ S(g)g means that for any k =1,...,d,

d ∂P d ∂P J k · P = [J k,J ]=− c k J i =0 ∂J j ij ∂J j=1 j i,j=1 j and so (9.15) is equal to zero.

The identity of Lemma 9.1.4 will hold if we replace the generators of S(g)by Ji → Ji + tμi for all i =1,...,d,wheret is a variable. In terms of the dual generators this replacement is written as J i → J i + tμi.Thatis, d i i (J + tμ ) ⊗ Ji,dP =0. Ji→Ji+tμi i=1 Therefore, expanding P (J +tμ) as in (9.11) and taking the coeﬃcients of all powers of t we get the relations X ,dP(0) =0, (9.16) X ,dP(i+1) + 1 ⊗ μ, dP(i) =0,i=0, 1,...,m− 2, 1 ⊗ μ, dP(m−1) =0. As a next step, we complete the proof of the theorem for the particular case where μ is regular nilpotent. Consider an sl2-subalgebra of g with the standard basis elements e, f, h satisfying the commutation relations (6.12) where e = μ.Set x = h/2 and introduce the evaluation map

ev : S(g) → C,Ji → xi for i =1,...,d, 1 d where x is written as x = x1J + ···+ xdJ . We will use the same notation for its extension to the Lie algebra (9.17) ev : S(g) ⊗ g → g which is the identity map on g. Clearly, ev(X )=x. Consider the respective evaluations of the diﬀerentials of the polynomials Pk(i) and set P − (9.18) k(i) =ev dPk(i) ,k=1,...,n, i=0, 1,...,dk 1. 9.1. MISHCHENKO–FOMENKO SUBALGEBRAS 153

The maximum of all integers dk is the Coxeter number hg of the Lie algebra g.For each j =0, 1,...,hg − 1 introduce the subspace of g by

Dj =spanof {Pk(i) | k =1,...,n, i=0, 1,...,j}, assuming that Pk(i) =0fori dk. By the results of Kostant [93], we have the decomposition of g regarded as an adjoint sl2-module, into irreducible components,

g = m1 ⊕···⊕mn, where dim mk =2dk − 1. The highest vector of the sl2-module mk is annihilated by e and has the x-weight equal to dk − 1. Hence its weight decomposition with respect to the adjoint action of x has the form − − − (dk 1) ⊕ (dk 2) ⊕···⊕ ( dk+1) mk = mk mk mk .

For each integer −hg +1 i hg − 1 introduce the subspace of g by (i) (i) ⊕···⊕ (i) (9.19) g = m1 mn .

Lemma 9.1.5. For each j =0, 1,...,hg − 1 we have

(0) (1) (j) (9.20) Dj = g ⊕ g ⊕···⊕g . Proof. By applying the evaluation map (9.17) to the relations (9.16) we get x, Pk(0) =0, (9.21) x, Pk(i+1) + e, Pk(i) =0,i=0, 1,...,dk − 2, P e, k(dk−1) =0 for each k =1,...,n. We will be proving (9.20) by the induction on j.Ifj =0then (0) the ﬁrst relation in (9.21) shows that D0 ⊂ g . However, x is a regular element of g as its centralizer coincides with the n-dimensional subspace g(0). Hence, by another theorem of Kostant [94], the diﬀerentials dP1,...,dPn evaluated at the (0) regular point x are linearly independent and so D0 = g . Suppose that (9.20) folds for j =0, 1,...,m. By (9.21) we have

(9.22) ad x ·Dm+1 =ade ·Dm. (i) → (i+1) For each k the map ad e : mk mk is surjective for all i 0andsoisthemap ad e : g(i) → g(i+1). Hence by the induction hypothesis,

(1) (m+1) (9.23) ad e ·Dm = g ⊕···⊕g . Observe that for each i = 0 the restriction of the operator ad x on the subspace g(i) is invertible. Moreover, the kernel of this operator on g coincides with g(0). Therefore, (9.22) and (9.23) imply that

(0) (1) (m+1) Dm+1 ⊂ g ⊕ g ⊕···⊕g

(0) and that the projection of Dm+1 on the direct sum in (9.23) along g is surjective. (0) However, Dm+1 contains D0 = g and thus (9.20) holds for j = m + 1, completing the proof of the lemma. 154 9. COMMUTATIVE SUBALGEBRAS OF U(g)

Taking j = hg − 1 in Lemma 9.1.5 we find that the span of all differentials (9.18) coincides with the direct sum − (9.24) g(0) ⊕ g(1) ⊕···⊕g(hg 1) which is a Borel subalgebra of g. Therefore the differentials (9.18) are linearly independent since their total number d1 + ···+ dn equals the dimension of the Borel subalgebra. Thus we showed that the polynomials (9.14) are algebraically independent in the case where μ is regular nilpotent. To complete the proof of the theorem in the general case, introduce the subset S ⊂ g which consists of all elements μ satisfying the property that the corresponding polynomials (9.14) are algebraically independent. Then S is a nonempty Zariski open subset of g. It follows from the definition of the algebra Aμ via the expansions A A ∈ C A A (9.11) that μ = cμ for any nonzero c .Moreover,Adg( μ)= Adg (μ) for any element g ∈ G of the adjoint group G of g. Hence, S is a conic subset of g which is invariant under the adjoint action of G. We will keep the sl2-subalgebra of g introduced in the previous argument. It is well-known by Kostant [94], that the adjoint orbit of any regular element of g intersects the affine subspace e + gf (Kostant’s slice)atonepoint.Heregf denotes the centralizer of f in g which is a subspace of the Borel subalgebra opposite to (9.24), − − gf ⊂ g(0) ⊕ g( 1) ⊕···⊕g( hg+1). Thus, for the rest of the proof we may assume that μ is regular and μ ∈ e + gf . Suppose that μ/∈ S and for any a ∈ C set

χa =expa · Adexp(−ax)(μ). All these elements do not belong to S.Notethatify ∈ g(p) then exp ad(−ax) (y)=exp(−ap)y. (1) Since e ∈ g ,thisimpliesthatifexpa → 0thenχa → e. This makes a contradiction as the element e of the open set S is approximated by elements not belonging to S.So,μ ∈ S and the proof is complete.

The total number of the algebraically independent generators of the algebra Aμ provided by Theorem 9.1.3 is d1 +···+dn which equals (dim g+n)/2 and coincides with the dimension of a Borel subalgebra of g. The transcendence degree of any Poisson commutative subalgebra of S(g) is known not to exceed this number.

Remark 9.1.6. The Mishchenko–Fomenko subalgebra Aμ of S(g) associated with a regular element μ ∈ g∗ is maximal Poisson commutative; see [132]and [147]. Example . 9.1.7 Theorem 9.1.3 extends to the reductive Lie algebra g = glN in an obvious way; we just need to include an extra invariant of degree one, as ∈ ∗ compared to the case of the simple Lie algebra slN . Any element μ glN can be regarded as the N ×N matrix μ =[μij], where μij = μ(Eij). Equip the Lie algebra 1 glN with the nondegenerate invariant bilinear form ∈ X, Y =trXY, X,Y glN .

In accordance with (9.13), we can identify μ with the element of glN represented t ∗ by the transposed matrix μ .Thenμ is a regular element of glN if and only if

1Although we use the same notation, this form is diﬀerent from the Killing form (7.2). 9.2. VINBERG’S QUANTIZATION PROBLEM 155 the Jordan canonical form of the matrix [μij] (or, equivalently, its transpose) does not contain two Jordan cells with the same eigenvalue. Furthermore, μ is regular semisimple if this matrix is diagonalizable and all eigenvalues are distinct. In the case where the canonical form is a single Jordan cell with the zero eigenvalue, μ is called regular nilpotent. Using the invariants C1,...,CN of S(glN ) deﬁned in (2.6), introduce elements Cm (i) of the subalgebra Aμ by the expansion N −1 N N−m (9.25) det u + μ + Ez = u + Cm(z)u m=1 with −m ··· −1 Cm(z)=Cm (0) z + + Cm (m−1) z + Cm (m).

If μ is regular, then the elements Cm (i) with m =1,...,N and i =0, 1,...,m− 1 are algebraically independent generators of Aμ. Similarly, taking the invariants T deﬁned in (2.11), set m −1 m −m ··· −1 tr μ + Ez = Tm (0) z + + Tm (m−1) z + Tm (m). A All elements Tm (i) belong to the subalgebra μ of S(glN ). Furthermore, if μ is regular, then the elements Tm (i) with m =1,...,N and i =0, 1,...,m− 1are algebraically independent generators of Aμ. The invariants in types B, C and D constructed in Section 2.2 can be used to produce analogous families of algebraically independent generators of the subalgebra Aμ of S(gN ) for regular elements μ.

9.2. Vinberg’s quantization problem The universal enveloping algebra U(g) can be regarded as a ‘quantization’ of the Poisson algebra S(g). Namely, we can re-scale the basis elements of the Lie algebra g by introducing the ‘deformation parameter’ h (taking nonzero complex values) so that the new basis elements are Ji = hJi with i =1,...,d. In the new basis the commutation relations (9.4) take the form d k [Ji, Jj ]=h cij Jk. k=1 In the limit h → 0 we get the symmetric algebra S(g), and the Poisson bracket (9.5)isrecoveredby [Ji, Jj ] {Ji, Jj } = lim . h→0 h An equivalent viewpoint which we took in Chapter 4 is to equip the universal enveloping algebra U(g) with the canonical ﬁltration so that the associated graded algebra gr U(g) is isomorphic to the symmetric algebra S(g). Given that the subalgebra Aμ of S(g) is Poisson commutative, one could wonder whether it is possible to construct a commutative subalgebra Aμ of U(g)which ‘quantizes’ Aμ in the sense that gr Aμ = Aμ.Thisquantization problem was raised by Vinberg in [149], where, in particular, some commuting families of elements of U(g) were produced. Theorem 9.2.4 below provides a positive solution of Vinberg’s problem which uses the Feigin–Frenkel center z(g). 156 9. COMMUTATIVE SUBALGEBRAS OF U(g)

Suppose that z is a variable. For any μ ∈ g∗ consider the following generalization of the evaluation homomorphism (6.30), −1 −1 → ⊗ C −1 → r (9.26) μ :U t g[t ] U(g) [z ],X[r] Xz + δr,−1 μ(X), for any X ∈ g and r<0. If S ∈ z(g) is a homogeneous element of degree d with respect to the grading defined by the derivation (6.14), define the elements ∈ S(a) U(g) (depending on μ) by the expansion −d ··· −1 (9.27) μ(S)=S(0) z + + S(d−1) z + S(d). Since z(g) is invariant with respect to the translation operator T defined in (6.7), the property (6.31) extends to the homomorphism (9.26) so that − (9.28) μ(TS)= ∂z μ(S).

Note that by Proposition 6.5.1 the element S(0) is central in U(g). Suppose that the variable z takes a particular nonzero value in C. Then the formula (9.26) deﬁnes a homomorphism −1 −1 → → r (9.29) μ,z :U t g[t ] U(g),X[r] Xz + δr,−1 μ(X), for any X ∈ g and r< 0. The Feigin–Frenkel center z(g) is a commutative subal- −1 −1 gebra of U t g[t ] , and so its image under μ,z is a commutative subalgebra of U(g). We will see below in Lemma 9.2.2 that the image does not depend on the value of z.

Definition 9.2.1. The commutative subalgebra Aμ of U(g) is deﬁned as the image of z(g) under the homomorphism (9.29) with a nonzero z ∈ C.

Lemma 9.2.2. The subalgebra Aμ does not depend on z.

Proof. Suppose that the algebra Aμ is defined for a certain fixed nonzero value z ∈ C. It will be sufficient to verify that for any homogeneous element S ∈ z(g) A of degree d all coefficients S(a) of the polynomial (9.27) belong to μ. Introduce another polynomial ··· d ∈ ⊗ C P (z)=S(0) + S(1)z + + S(d)z U(g) [z]. d Then P (z)=z μ(S) and by (9.28) all derivatives of P (z)overz can be expressed C r as [z]-linear combinations of the images μ(T S). The claim now follows by evaluating z at the chosen fixed value.

Suppose S1,...,Sn is a complete set of Segal–Sugawara vectors of the respective degrees d1,...,dn as provided by Theorem 6.3.1. Introduce the corresponding polynomials (9.27) by − − (9.30) (S )=S z dk + ···+ S z 1 + S . μ k k (0) k (dk−1) k (dk)

Corollary 9.2.3. The algebra Aμ is generated by the elements − Sk (i),k=1,...,n, i=0, 1,...,dk 1.

Proof. As the proof of Lemma 9.2.2 shows, the algebra Aμ canbedeﬁnedas the subalgebra of U(g) generated by all coeﬃcients S(a) of the polynomials (9.27) associated to all homogeneous elements S ∈ z(g). On the other hand, the algebra r z(g) is generated by the elements T Sk; see (6.15). Hence the required property follows from (9.28). 9.3. GENERATORS OF COMMUTATIVE SUBALGEBRAS OF U(glN ) 157

Due to Proposition 6.5.2 the algebra Aμ contains the center Z(g)ofU(g). In the particular case μ = 0 it coincides with the center. The following theorem provides a positive solution of Vinberg’s quantization problem for an arbitrary simple Lie algebra g and a regular element μ ∈ g∗.We use the notation (9.30). ∗ Theorem 9.2.4. Let μ ∈ g be regular. If S1,...,Sn is a complete set of Segal–Sugawara vectors of the respective degrees d1,...,dn then the elements − Sk (i),k=1,...,n, i=0, 1,...,dk 1, are algebraically independent generators of Aμ. Moreover, gr Aμ = Aμ. A general proof can be found in [42]. In the next sections we give a direct proof for the Lie algebras g of classical types relying on the explicit constructions of complete sets of Segal–Sugawara vectors provided in Chapters 7 and 8.

Remark 9.2.5. Under the assumptions of Theorem 9.2.4, the subalgebra Aμ of U(g) is maximal commutative. This follows from the corresponding property of the Poisson commutative subalgebra Aμ of S(g); see Remark 9.1.6.

9.3. Generators of commutative subalgebras of U(glN )

We let g = glN and use the tensor notation as in Section 7.1. Consider the tensor product algebra N N (9.31) End C ⊗ ... ⊗ End C ⊗ U, m ⊗ C −1 where U = U(glN ) [z ,∂z] is the algebra of differential operators whose elements are finite sums of the form −k l ∈ xkl z ∂z ,xkl U(glN ). k,l0 ∈ ∗ × As in Example 9.1.7, take an element μ glN which will be regarded as the N N matrix μ =[μ ], where μ = μ(E ). Introduce the matrix ij ij ij −1 −1 (9.32) M = −∂z + μ + Ez = −δij ∂z + μij + Eijz with entries in U. It follows from Lemma 4.5.3 that M is a Manin matrix. −1 Define polynomials φma(z), ψma(z)andθma(z)inz (depending on μ) with coefficients in U(glN ) by the expansions (m) − m − m−1 ··· tr1,...,m A M1 ...Mm = φm 0(z)( ∂z) + φm 1(z)( ∂z) + + φmm(z), (m) − m − m−1 ··· tr1,...,m H M1 ...Mm = ψm 0(z)( ∂z) + ψm 1(z)( ∂z) + + ψmm(z), and m − m − m−1 ··· tr M = θm 0(z)( ∂z) + θm 1(z)( ∂z) + + θmm(z); cf. (7.7), (7.8) and (7.9). Furthermore, following (7.11) define the polynomials φa(z)by − N − N−1 ··· (9.33) cdet M =( ∂z) + φ1(z)( ∂z) + + φN (z). By the same argument as for the proof of (7.12) we get N − a (9.34) φ (z)= φ (z), 0 a m N, ma m − a a 158 9. COMMUTATIVE SUBALGEBRAS OF U(g) and so φmm(z)=φm(z) for all m. Introduce the coefficients of the polynomials by −m ··· −1 φm(z)=φm (0)z + + φm (m−1)z + φm (m), −m ··· −1 ψmm(z)=ψmm(0)z + + ψmm(m−1)z + ψmm(m), and −m ··· −1 θmm(z)=θmm(0)z + + θmm(m−1)z + θmm(m). The following theorem will imply Theorem 9.2.4 in type A. Theorem . ∈ ∗ 9.3.1 (i) Given any μ glN , all coefficients of the polynomials A φm(z), ψma(z) and θma(z) belong to the commutative subalgebra μ of U(glN ). Moreover, the elements of each of the families

(9.35) φm (k),ψmm(k) and θmm(k) with m =1,...,N and k =0, 1,...,m− 1, are generators of the algebra Aμ. (ii) If μ is regular, then each of the three families (9.35) is algebraically independent. Moreover, gr Aμ = Aμ. Proof. Under the homomorphism μ defined in (9.26) we have − → −1 μ : Eij[ 1] Eij z + μij − → −1 so that E[ 1] μ+Ez .Extend μ to the homomorphism of the tensor product algebras −1 −1 ⊗ C → ⊗ C −1 (9.36) μ :U t glN [t ] [τ] U(glN ) [z ,∂z] →− by setting τ ∂z. The coefficients of the polynomials φm(z), ψma(z)andθma(z) are the images under the homomorphism μ of the respective elements of the Feigin– A Frenkel center z(glN ) provided by Theorem 7.1.3. Hence they belong to μ.To- gether with Theorem 7.1.4 and Corollary 9.2.3 this completed the proof of part (i) of the theorem. Now suppose that μ is regular and consider the symbols of the elements of the ∼ families (9.35) in the associated graded algebra S(glN ) = gr U(glN ). These symbols coincide with the respective coefficients of the polynomials in z−1 given by (m) −1 −1 tr1,...,m A μ1 + E1z ... μm + Emz , (m) −1 −1 tr1,...,m H μ1 + E1z ... μm + Emz , and m tr μ + Ez−1 for m =1,...,N; cf. Example 9.1.7. However, these coefficients belong to the A Mishchenko–Fomenko subalgebra μ of S(glN ) since they arise from generators of glN S(glN ) by the rule (9.12); see Corollary 2.1.6 and (2.12). Moreover, by Theo- rem 9.1.3 the coefficients are algebraically independent generators of Aμ. Hence part (ii) of the theorem follows.

Note that, in particular, the coeﬃcients of the column-determinant cdet M A are respective quantizations of the generators of the algebra μ provided by the ‘shifted’ characteristic polynomial det u + μ + Ez−1 deﬁned in (9.25). 9.3. GENERATORS OF COMMUTATIVE SUBALGEBRAS OF U(glN ) 159

Example 9.3.2. Using the complete sets of Segal–Sugawara vectors given in Example 7.1.6, we get the following algebraically independent generators of the algebra Aμ for regular μ: 2 for gl2 :trE, tr μE, tr E 2 2 2 3 for gl3 :trE, tr μE, tr μ E, tr E , tr μE , tr E 2 3 2 2 for gl4 :trE, tr μE, tr μ E, tr μ E, tr E , tr μE , 2trμ2E2 +tr(μE)2, tr E3, tr μE3, tr E4. This follows by calculating the coefficients of the powers of z−1 in the polynomials of the form tr (μ + Ez−1)m. There are some relations between these coefficients which we have used to simplify the set of generators. One easily verifies that tr EμE =trμE2 +trμ tr E − N tr μE and tr EμE2 =trE2μE =trμE3 +trμ tr E2 − N tr μE2. Indeed, using (4.10), for the first relation we have

tr EμE =tr12 P12 E1μ1E2 =tr μ P E E =tr μ P (E E + P E − P E ) 12 1 12 1 2 12 1 12 2 1 12 2 12 1 − 2 − =tr12 μ1 E1P12E1 + μ1 E2 μ1 E1 =trμE +trμ tr E N tr μE, as claimed. The second relation is proved in a similar way. ∈ ∗ The condition that the element μ glN is regular turns out to be necessary A for the subalgebra μ of U(glN ) to have the maximum possible transcendence degree. This can be seen directly from the next theorem providing an algebraically A ∈ ∗ independent family of generators of μ for an arbitrary element μ glN . It also shows that the subalgebra Aμ is a quantization of the shift of argument subalgebra Aμ for any μ thus proving a conjecture from [42, Conjecture 1]. Consider the matrix M introduced in (9.32). We will work with the generators A φm (k) of μ obtained as the coeﬃcients of the column-determinant N m−1 − N −m+k − N−m cdet M =( ∂z) + φm (k) z ( ∂z) m=1 k=0 as deﬁned in (9.33). To state the theorem, suppose that the distinct eigenvalues of μ are λ1,...,λr and the Jordan canonical form of μ is the direct sum of the (i) (i) ··· (i) respective Jordan blocks J (i) (λi) of sizes α1 α2 αsi 1, where we αj use the notation ⎡ ⎤ λ 10... 0 ⎢ ⎥ ⎢ 0 λ 1 ... 0 ⎥ ⎢ ⎥ ⎢ ...... ⎥ Jp(λ)=⎢ . . . . . ⎥ ⎣ 000... 1 ⎦ 000... λ for the Jordan block of size p.Weletα(i) denote the corresponding Young diagram (i) | (i)| (i) whose j-th row is αj and let α be the number of boxes of α . Given these 160 9. COMMUTATIVE SUBALGEBRAS OF U(g) data, introduce another Young diagram γ =(γ1,γ2,...) by setting r (i) (9.37) γl = αj , i=1 jl+1 so that γl is the total number of boxes which are strictly below the l-throwsinall (i) diagrams α . Furthermore, associate the elements of the family φm (k) with the boxes of the diagram Γ = (N,N − 1,...,1), as illustrated:

φN (N−1) φN−1(N−2) ... φ2(1) φ1(0)

φN (N−2) φN−1(N−3) ... φ2(0) (9.38) Γ = ......

φN (1) φN−1(0)

φN (0)

It is clear that the diagram γ iscontainedinΓ.Theskew diagram Γ/γ is obtained by deleting from Γ all boxes which belong to γ. Theorem . 9.3.3 The elements φm (k) corresponding to the boxes of the skew diagram Γ/γ are algebraically independent generators of the subalgebra Aμ.More- over, the subalgebra Aμ is a quantization of Aμ so that gr Aμ = Aμ. Proof. As we pointed out above, the matrix M deﬁned in (9.32) is a Manin matrix. Therefore, applying relation (3.17) in Proposition 3.2.2 we get (m) I (9.39) tr1,...,m A M1 ...Mm = MI , I,|I|=m summed over the subsets I = {i1,...,im} with i1 < ···

For each l =1,...,N introduce the polynomial in a variable t with coeﬃcients in Aμ by N−l N−l−1 ··· Φl(t, μ)=φl (0)(μ)t + φl+1 (1)(μ)t + + φN (N−l)(μ), where we indicated the dependence of μ of the elements φm (k). Note that the coeﬃcients of Φl(t, μ) are the entries of the l-th row of the diagram Γ; see (9.38). Lemma 9.3.4. For any a ∈ C we have the relation

Φl(t, μ + a1) = Φl(t + a, μ). Proof. We have m (m) p (m) tr1,...,m A (a + M1) ...(a + Mm)= a tr1,...,m A Mi1 ...Mim−p .

p=0 i1<···

(m) (m) Furthermore, A =sgnσ · A Pσ for any σ ∈ Sm,wherePσ denotes the image of σ ∈ Sm in the algebra (9.31) under the action of Sm deﬁned in (1.65). Hence, applying conjugations by appropriate elements Pσ and using the cyclic property of trace, we can write the expression as m m p (m) a tr A M ...M − . p 1,...,m 1 m p p=0 The partial trace of the anti-symmetrizer over the m-th copy of End Cn is found by (3.26) which implies − − (m) (N m + p)! (m p)! (m−p) tr − A = A . m p+1,...,m (N − m)! m! Hence,

(m) tr1,...,m A (a + M1) ...(a + Mm) m − N m + p p (m−p) = a tr − A M ...M − . p 1,...,m p 1 m p p=0 Now equate the constant terms of the diﬀerential operators on both sides and take the coeﬃcients of z−m+k to get the relation k − N m + p p φ (μ + a1) = a φ − − (μ). m (k) p m p (k p) p=0

Therefore, for the polynomial Φl(t, μ + a1) we ﬁnd N−l N−l−k Φl(t, μ + a1) = φl+k (k)(μ + a1)t k=0 N−l k − − N l k + p p N−l−k = a φ − − (μ)t p l+k p (k p) k=0 p=0 162 9. COMMUTATIVE SUBALGEBRAS OF U(g) which equals

− N−l−p Nl N − l − r ap φ (μ)t N−l−p−r p l+r (r) p=0 r=0 − Nl ap d p = Φ (t, μ)=Φ(t + a, μ), p! dt l l p=0 as claimed.

Lemma 9.3.5. Suppose that μ has the form of a block-diagonal matrix J (0) O (9.41) μ = α , O μ where Jα(0) is the nilpotent Jordan matrix associated with a certain Young diagram α =(α1,α2,...) and μ is an arbitrary square matrix of size q such that |α|+q = N. Then for any l 1 we have

φl+k (k) =0 for all N − l − δl +1 k N − l, where δl = αl+1 + αl+2 + ... is the number of boxes of α below its row l.

Proof. The generator φl+k (k) is found by (9.40) for m = l + k.Theinternal sum is a linear combination of k × k minors of the matrix μ satisfying the condition that the union B ∪ C of the row and column indices of each minor is a set of size not exceeding k + l. Hence the lemma will be implied by the claim that under the B given condition on k, the minor μ C can be nonzero only if the union of row and column indices is of the size at least k + l + 1. Indeed, observe that if p is a positive integer, then any nonzero p × p minor of a nilpotent Jordan block has the property that the minimal possible size of the union of its row and column indices is p +1. However, the condition k N − l − δl +1 meansthatk α1 + ···+ αl − l +1+q. Therefore, any nonzero k × k minor must involve at least l + 1 Jordan blocks and the claim follows.

(i) (i) In the notation of the theorem, for each diagram α denote by δl the corresponding parameter δl introduced in Lemma 9.3.5, so that for the number γl deﬁned in (9.37) we have r (i) γl = δl . i=1

Lemma 9.3.6. The polynomial Φl(t, μ) admits the factorization

δ(1) δ(r) Φl(t, μ)=(t + λ1) l ...(t + λr) l Φl(t, μ) for a certain polynomial Φl(t, μ) in t.

Proof. Observe that by formulas (9.39), the elements φm (k) are unchanged under the simultaneous replacements μ → gμg−1 and E → gEg−1 for any element g ∈ GLN . ThisimpliesthatAgμg−1 can be identiﬁed with the algebra Aμ associated −1 with the image of U(glN ) under the automorphism sending E to gEg . Therefore, the algebra Aμ depends only on the coadjoint orbit of μ. This is a particular case of the general property; cf. the proof of Theorem 9.1.3. 9.3. GENERATORS OF COMMUTATIVE SUBALGEBRAS OF U(glN ) 163

For any i ∈{1,...,r} the Jordan canonical form of μ − λi1 is a matrix of the (i) form (9.41), where α = α . By Lemma 9.3.5, the polynomial Φl(t, μ − λi1) is δ(i) divisible by t l . By Lemma 9.3.4, the polynomial Φl(t, μ)=Φl(t + λi,μ− λi1) is δ(i) then divisible by (t + λi) l . Returning to the proof of the theorem, note that by Lemma 9.3.6 for any l =1,...,N the generators φl+k (k) with N − l − γl +1 k N − l are linear combinations of those generators with k =0, 1,...,N − l − γl. Therefore, the elements φl+k (k) corresponding to the boxes of the skew diagram Γ/γ generate the algebra Aμ. It remains to verify that these generators are algebraically independent. ∈ Consider the elements φm (k) S(glN ) which are deﬁned by · B I\B φm (k) = sgn σ μ C E I\C , I,|I|=m B,C⊂I |B|=|C|=k with the notation as in (9.40), where the entries of the matrix E are now regarded as elements of the symmetric algebra S(glN ). Equivalently, the elements φm (k) are found by (m) −1 −1 tr1,...,m A μ1 + E1z ... μm + Emz −m ··· −1 = φm (0)z + + φm (m−1)z + φm (m).

By Theorem 9.3.1(i), they generate the subalgebra Aμ. The above arguments applied to these generators instead of the φm (k) show that the algebra Aμ is generated by the subset of the elements φm (k) corresponding to the boxes of the skew diagram Γ/γ. Furthermore, we have the following. Lemma . A 9.3.7 The generators φm (k) of the subalgebra μ corresponding to the boxes of the skew diagram Γ/γ are algebraically independent. Proof. Regarding the elements φm (k) as polynomials in the variables Eij,we will see that their differentials d φm (k) are linearly independent at a certain point. A Since these elements generate μ, the linear span of the differentials d φm (k) at any point coincides with the linear span of all differentials

d Aμ =spanof{dφ | φ ∈ Aμ}. The dimension of the space of diﬀerentials evaluated at a certain regular point can be found from Bolsinov’s criterion [11, Theorem 3.2] . To state it for the Lie algebra glN , recall that the index ind g of an arbitrary Lie algebra g is the minimal dimension ind g =mindim Ann ξ ξ∈g∗ of the annihilators Ann ξ for the coadjoint representation, where { ∈ | ∗ } Ann ξ = X g adX ξ =0 . By Bolsinov’s criterion, the relation 1 dim d A =rankgl + dim gl − dim Ann μ μ N 2 N for the diﬀerentials evaluated at a certain regular point holds if and only if

(9.42) ind Ann μ =rankglN . 164 9. COMMUTATIVE SUBALGEBRAS OF U(g)

This equality does hold for any μ [154], and so, to show that the differentials d φm (k) of the generators are linearly independent at a certain point, we only need to verify that the number of boxes of the skew diagram Γ/γ coincides with 1 1 rank gl + dim gl − dim Ann μ = N + N 2 − dim Ann μ . N 2 N 2 Since |Γ| = N(N +1)/2, the desired formula is equivalent to the relation (9.43) dim Ann μ =2|γ| + N. The dimension of the annihilator Ann μ coincides with that of the centralizer of μt in dim glN ; see Example 9.1.7. We have r (i) dim Ann μ = dim Ann gl μ , ni i=1 where μ(i) denotes the direct sum of all Jordan blocks of μ with the eigenvalue (i) λi,andni is the size of μ . Hence, by the definition of γ, the verification of (9.43) reduces to the case where μ has only one eigenvalue. Let α1 ··· αs be the respective sizes of the Jordan blocks of such a matrix μ. The dimension of the annihilator (or, equivalently, the centralizer of μt) is found by the well-known formula [80, Section 3.1],

dim Ann μ = α1 +3α2 + ···+(2s − 1)αs, while

|γ| = α2 +2α3 + ···+(s − 1)αs and N = α1 + ···+ αs, thus implying (9.43).

Now consider the generators φm (k) of the algebra Aμ associated with the boxes of the diagram Γ/γ. By Lemma 9.3.7, the corresponding elements φm (k) are nonzero, so that the image of φm (k) in the (m − k)-th component of the graded ∼ algebra gr U(glN ) = S(glN )coincideswithφm (k). Moreover, the generators φm (k) corresponding to the boxes of the diagram Γ/γ are algebraically independent. This completes the proof of the theorem. Remark 9.3.8. Theorem 9.3.3 is valid in the same form for some other families A of generators of the algebra μ in place of the elements φm (k);see[53]. Example 9.3.9. Take N =6andletμ be a nilpotent matrix with the Jordan blocks of sizes (2, 2, 1, 1). Then γ =(4, 2, 1) and the skew diagram Γ/γ is

so that Aμ has 14 algebraically independent generators. If the element μ is regular, then all Jordan blocks correspond to distinct eigenvalues so that each α(i) is a singe row diagram. Therefore, γ = ∅ and all generators φm (k) associated with the boxes of Γ are algebraically independent. On the other hand, if μ is a scalar matrix (in particular, μ = 0), then γ =(N − 1,N − 2,...,1). 9.4. GENERATORS OF COMMUTATIVE SUBALGEBRAS OF U(oN ) AND U(spN ) 165

A In this case, μ is generated by φ1(0),...,φN (0) and it coincides with the center of U(glN ); cf. Section 6.5.

9.4. Generators of commutative subalgebras of U(oN ) and U(spN )

Now let g = gN be the orthogonal Lie algebra oN or the symplectic Lie algebra spN . We keep using the tensor product notation and consider the algebra N N End C ⊗ ... ⊗ End C ⊗ U, m −1 where U = U(gN )⊗ C[z ,∂z] is the algebra of diﬀerential operators whose elements are ﬁnite sums of the form −k l ∈ xkl z ∂z ,xkl U(gN ). k,l0

We will also use the rational function γm(ω) deﬁned in (2.40) with the specializations ω = N and ω = −N in the orthogonal and symplectic case, respectively. ∈ ∗ × An arbitrary element μ gN will be regarded as the N N matrix μ =[μij], where μij = μ(Fij). The matrix μ has the property μ + μ = 0; see (2.24). Deﬁne −1 the polynomials φma(z)inz (depending on μ) by the expansion (m) − −1 − −1 (9.44) γm(ω)tr1,...,m S ∂z + μ1 + F1 z ... ∂z + μm + Fm z − m − m−1 ··· = φm 0(z)( ∂z) + φm 1(z)( ∂z) + + φmm(z).

In the symplectic case gN = sp2n we adopt the same convention for (9.44) as for the elements (8.36); cf. the remark just before Theorem 8.3.8. That is, the left hand side is assumed to be written in an equivalent form which is well-defined for all values 1 m 2n + 1 as provided by Proposition 8.3.7. In fact, in both the orthogonal and symplectic case the expression (9.44) coincides with the respective image of (8.15) or (8.36) under the homomorphism (9.26) with the extension to the differential operators by τ →−∂z as in (9.36). In the case gN = o2n we also define the (noncommutative) Pfaffian of the matrix μ + Fz−1 by Pf μ + Fz−1 1 −1 −1 = sgn σ · μ + Fz ... μ + Fz . 2nn! σ(1) σ(2) σ(2n−1) σ(2n) σ∈S2n Introduce the coefficients of the polynomials by −m ··· −1 φmm(z)=φmm(0)z + + φmm(m−1)z + φmm(m) and −1 −n ··· −1 Pf μ + Fz = π(0)z + + π(n−1)z + π(n).

In particular, π(0) coincides with Pf F . The next theorem will imply Theorem 9.2.4 in types B,C and D. Theorem 9.4.1. ∈ ∗ (i) Given any μ gN , all coeﬃcients of the polynomial φma(z) belong to the A commutative subalgebra μ of U(gN ). Moreover, all coeﬃcients of the polynomial −1 Pf μ + Fz belong to the subalgebra Aμ of U(o2n). 166 9. COMMUTATIVE SUBALGEBRAS OF U(g)

(ii) In the case where μ is regular, the elements − φ2k 2k (i) with k =1,...,n and i =0, 1,...,2k 1 are algebraically independent generators of the algebra Aμ in the cases B and C, while the elements − − φ2k 2k (i) with k =1,...,n 1 and i =0, 1,...,2k 1 together with π(0),...,π(n−1) are algebraically independent generators of the algebra Aμ in the case D. Moreover, in all three cases, gr Aμ = Aμ. Proof. −1 The coefficients of the polynomials φma(z)andPf μ+Fz are the images under the homomorphism μ of the respective elements of the Feigin–Frenkel center z(gN ) provided by Proposition 8.1.4, Theorem 8.1.6 and Proposition 8.3.7. Therefore, they belong to Aμ. To prove part (ii), suppose that μ is regular and consider the symbols of the ∼ elements φ2k 2k (i) and π(j) in the associated graded algebra S(gN ) = gr U(gN ). These symbols coincide with the respective coefficients of the polynomials in z−1 given by (2k) −1 −1 γ2k(ω)tr1,...,2k S μ1 + F1 z ... μ2k + F2k z and Pf μ+Fz−1 . The coefficients belong to the Mishchenko–Fomenko subalgebra gN Aμ of S(gN ) since they arise from generators of S(gN ) by the rule (9.12); see Section 2.2. Furthermore, taking into account Corollaries 2.2.5 and 2.2.10, we derive from Theorem 9.1.3 that the respective coefficients are algebraically independent generators of Aμ. Hence part (ii) of the theorem follows.

Example 9.4.2. By making use of the Segal–Sugawara vectors provided in Example 8.2.2, we get algebraically independent generators of the algebra Aμ for regular μ (cf. Example 9.3.2):

2 for o3 :trμF, tr F

2 for o4 :trμF, tr F ,π(0),π(1) 2 3 2 2 2 3 4 for o5 :trμF, tr F , tr μ F, 2trμ F +tr(μF) , tr μF , tr F

2 3 2 2 2 3 4 for o6 :trμF, tr F , tr μ F, 2trμ F +tr(μF) , tr μF , tr F ,

π(0),π(1),π(2). As in Example 9.3.2, we calculate the coeﬃcients of the powers of z−1 in the −1 k polynomials of the form tr (μ+Fz ) . We also use some relations (for oN ) between these coeﬃcients which can be derived from (5.2) or (5.6):

tr FμF2 =trF 2μF =trμF3 − (N − 2) tr μF2, 2trμF2 =(N − 2) tr μF and tr Fμ2F =trμ2F 2, tr μ2F =0. 9.5. BIBLIOGRAPHICAL NOTES 167

Example 9.4.3. Similarly, by Example 8.4.1 we get algebraically independent generators of the algebra Aμ for regular μ in the symplectic case: 2 for sp2 :trμF, tr F 2 3 2 2 2 3 4 for sp4 :trμF, tr F , tr μ F, 2trμ F +tr(μF) , tr μF , tr F . 9.5. Bibliographical notes

The Poisson commutative subalgebras Aμ of S(g) were first introduced by Mishchenko and Fomenko [108] in connection with the Euler equations; see also Manakov [105]. Theorem 9.1.3 is contained in the original paper [108]inthecase where μ is regular semisimple. The general regular case is due to Bolsinov [11], [12]. We followed the proof given by Feigin, Frenkel and Toledano Laredo [42]; see also Kostant [95]. The paper [42] also contains a general proof of Theorem 9.2.4. In the case of regular semisimple μ itwaspreviouslyprovedbyRybnikov[139]. Explicit formulas of Theorem 9.3.1 originate in the paper by Talalaev [145]; see also Chervov and Talalaev [24]. These families of generators are related to each other via the identities satisfied by Manin matrices; see the proof of Theorem 7.1.3. A different construction of a commutative subalgebra of U(glN )wasprovidedby Tarasov [146] via a symmetrization map. In the case where μ ∈ g∗ is regular semisimple, explicit constructions of maximal commutative subalgebras of U(g) were given in an earlier work of Nazarov and Olshanski [126] with the use of the Yangian for glN and the twisted Yangians associated with the orthogonal and symplectic Lie algebras. In the case of glN their elements are found by an expansion of the quantum determinant which is essentially equivalent to (9.33). In types B, C and D their elements appear to be quite different from those of Theorem 9.4.1 whichwereprovidedin[110]. For an arbitrary simple Lie algebra g, the spectra of the subalgebra Aμ of U(g) in finite-dimensional irreducible representations were described by Feigin, Frenkel and Rybnikov [41]. In the proof of Theorem 9.3.3 we followed the arguments of [53]whichrely on Bolsinov’s work [11]; see also Bolsinov and Zhang [13]. Relation (9.42) is a particular case of the Bolsinov–Elashvili conjecture whichoriginatesin[12]and applies to any reductive Lie algebra; see also Panyushev [130]. The first published proof of (9.42) is due to Yakimova [154], which extends to all classical Lie algebras; see also Charbonnel and Moreau [17] for its proof covering all simple Lie algebras and more references.

CHAPTER 10

Yangian characters in type A

As we recalled in the Preface, the Harish–Chandra isomorphism (0.1) leads to a description of the structure of the center Z(g) as an algebra of polynomials. We will now aim to prove an aﬃne analogue of the isomorphism theorem for the classical Lie algebras. In its basic form, this analogue will involve the Feigin–Frenkel center z(g) which will play the role of Z(g). The algebra of W -invariants in U(h) will now be replaced by a classical W-algebra; we will review these algebras in Chapter 12. Our arguments will rely on the explicit constructions of generators of z(g)given in Chapters 7 and 8. The calculation of the Harish–Chandra images as elements of the respective classical W-algebra will be completed in Chapter 13 and will be basedontheYangian characters. We will derive the necessary character formulas in the current chapter for type A and in Chapter 11 for types B, C and D. The Yangian Y(g) is a Hopf algebra which is a deformation of the universal enveloping algebra U g[t] in the class of Hopf algebras. We will work with the RT T -presentations of the Yangians to calculate the characters of certain classes of representations. Essentially the same calculation will give the character formulas for + representations of thedual Yangian Y (g) which is a deformation of the universal −1 −1 enveloping algebra U t g[t ] . By taking appropriate classical limits we will recover the Harish-Chandra images of certain elements of U(g[t]) and U t−1g[t−1] which will then be translated into the desired formulas for the corresponding Harish- Chandra images of the Segal–Sugawara vectors. The Yangian approach can also be used to obtain an alternative construction of the Segal–Sugawara vectors of Section 7.1 which we discuss below in Section 10.5. Moreover, this approach can be extended to any simple Lie algebra g provided that an appropriate version of the fusion procedure is developed. We begin by reviewing some well-known properties of the Yangian for glN following [109].

10.1. Yangian for glN (r) The Yangian Y(glN ) is a unital associative algebra with generators tij ,where 1 i, j N and r =1, 2,... and the deﬁning relations (r+1) (s) − (r) (s+1) (r) (s) − (s) (r) (10.1) [tij ,tkl ] [tij ,tkl ]=tkj til tkj til , (0) where r, s =0, 1,... and tij = δij with the usual notation for the commutator [a, b]=ab − ba. In terms of the formal series ∞ (r) −r ∈ −1 tij(u)=δij + tij u Y(glN )[[u ]] r=1 the deﬁning relations are written in the form

(10.2) (u − v)[tij(u),tkl(v)] = tkj(u) til(v) − tkj(v) til(u).

169 170 10. YANGIAN CHARACTERS IN TYPE A

By dividing both sides by u − v we get an equivalent form of (10.1), min{r,s} (r) (s) (a−1) (r+s−a) − (r+s−a) (a−1) (10.3) [tij ,tkl ]= tkj til tkj til . a=1 Set N ⊗ ∈ CN ⊗ −1 (10.4) T (u)= eij tij(u) End Y(glN )[[u ]] i,j=1 and adopt the matrix notation of Sections 1.4 and 1.5, so that Ta(u)fora =1,...,m will be understood as formal series in u−1 with coeﬃcients in the tensor product algebra CN ⊗ ⊗ CN ⊗ End ... End Y(glN ). m

The Yang R-matrix R12(u) is a rational function in a complex parameter u with values in the tensor product algebra End CN ⊗ End CN deﬁned by −1 (10.5) R12(u)=1− P12 u , where P is the permutation operator (1.64). This function satisﬁes the Yang–Baxter equation

(10.6) R12(u) R13(u + v) R23(v)=R23(v) R13(u + v) R12(u).

The deﬁning relations for the algebra Y(glN ) can be written in the matrix form as

(10.7) R12(u − v) T1(u) T2(v)=T2(v) T1(u) R12(u − v).

The algebra Y(glN ) possesses two natural ascending filtrations defined by (r) (r) − deg tij = r and deg tij = r 1 for all r 1. Denote by gr Y(glN )andgr Y(glN ) the corresponding graded algebras. We have the isomorphisms ∼ ∼ gr Y(glN ) = S glN [t] and gr Y(glN ) = U glN [t] ¯(r) so that, in particular, the algebra gr Y(glN ) is commutative. The image tij of (r) − the generator tij in the (r 1)-th component of the graded algebra gr Y(glN ) − corresponds to the element Eij[r 1] of U glN [t] . It follows easily from Proposition 4.2.2 that the mapping (10.8) ev : T (u) → 1+Eu−1, → defines a surjective homomorphism Y(glN ) U(glN )knownastheevaluation homomorphism. Every representation V of glN can thus be extended to a Yangian evaluation module via (10.8). The universal enveloping algebra U(glN ) will be identified with a subalgebra of → (1) the Yangian Y(glN ) via the embedding Eij tij .ThenY(glN ) can be regarded as a glN -module with the adjoint action, and the defining relations (10.3) give (s) (s) − (s) (10.9) [Eij,tkl ]=δkj til δil tkj . We have the following generalization of Theorem 4.5.1, where we keep the notation used in that theorem. 10.1. YANGIAN FOR glN 171

Proposition 10.1.1. For any s ∈ C[Sm] and variables u1,...,um,allcoeﬃ- cients of the series

tr1,...,m ST1(u1) ...Tm(um) commute with glN . Proof. Consider the tensor product CN ⊗ CN ⊗m ⊗ End End ( ) Y(glN ) with the additional copy of the endomorphism algebra End CN labeled by 0. Using (10.7) or writing (10.9) in a matrix form, we obtain the relations

[E0,Ta(ua)] = P0 aTa(ua) − Ta(ua)P0 a for a =1,...,m. Now the relation E0, tr1,...,m ST1(u1) ...Tm(um) =0, which is a counterpart of (4.32), is veriﬁed by the same argument as in the proof of Theorem 4.5.1.

h Denote by Y(glN ) the subalgebra of h-invariants under the action (10.9) so that h { ∈ | } Y(glN ) = y Y(glN ) [Eii,y]=0 for i =1,...,N . (r) Consider the left ideal I of the algebra Y(glN ) generated by all elements tij with the conditions 1 i

As before, by ca = ca(U) we denote the content c(α)=j−i of the box α =(i, j) in U occupied by a ∈{1,...,m}. Theorem 10.1.2. The series T E (10.11) μ(u)=tr1,...,m U T1(u + c1) ...Tm(u + cm) does not depend on the standard tableau U of shape μ. Moreover, the images of its coefficients under the Harish-Chandra homomorphism (10.10) are found by T μ(u) → λT (α) u + c(α) , sh(T )=μ α∈μ summed over semistandard tableau T of shape μ with entries in {1,...,N}. Proof. To prove the first claim it suffices to show that the right hand side of (10.11) remains unchanged if U is replaced by the standard tableau U = sa U for some a ∈{1,...,m− 1} such that the entries a and a +1 of U do not belong to the same row or the same column. By (1.67) and the Yangian defining relations (10.7), we have

Paa+1Raa+1(ca − ca+1) Ta(u + ca) Ta+1(u + ca+1)

= Ta(u + ca+1) Ta+1(u + ca)Paa+1Raa+1(ca − ca+1). Hence, representing the right hand side of (10.11) in the form E − −1 − tr1,...,m U Raa+1(ca ca+1) Paa+1Paa+1Raa+1(ca ca+1)

× T1(u + c1) ...Tm(u + cm) and using the cyclic property of trace, we can write it as − E − −1 tr1,...,m Paa+1Raa+1(ca ca+1) U Raa+1(ca ca+1) Paa+1

× T1(u + c1) ...Ta(u + ca+1) Ta+1(u + ca) ...Tm(u + cm). −1 Using the notation d =(ca+1 − ca) as in Lemma 1.1.1, we ﬁnd

−1 (10.12) EU Raa+1(ca − ca+1) Paa+1 1 1 = E (P − d) = E P E , U aa+1 1 − d 2 U aa+1 U 1 − d 2 where the last equality holds by (1.6). Lemma 10.1.3. In the notation of the theorem, we have

EU T1(u + c1) ...Tm(u + cm)=EU T1(u + c1) ...Tm(u + cm) EU .

Proof. Denote by R(u1,...,um) the image of the expression φ(u1,...,um) deﬁned in (1.23), under the action of the symmetric group in (CN )⊗ m.Then (10.13) R(u1,...,um)= Rab(ua − ub), 1a

R(u1,...,um)T1(u1) ...Tm(um)=Tm(um) ...T1(u1)R(u1,...,um). 10.1. YANGIAN FOR glN 173

Due to the fusion procedure of Proposition 1.1.7, the consecutive evaluations of the variables ua = u + ca for a =1,...,m imply the relation

(10.14) EU T1(u + c1) ...Tm(u + cm)=Tm(u + cm) ...T1(u + c1) EU .

2 This completes the proof of the lemma since EU = EU .

Applying (10.12), the cyclic property of trace and Lemma 10.1.3 with U replaced by U , we bring the right hand side of (10.11) to the form

1 tr E P E P E 1 − d 2 1,...,m U aa+1 U aa+1 U

× T1(u + c1) ...Ta(u + ca+1)Ta+1(u + ca) ...Tm(u + cm).

By (1.7), this coincides with the right hand side of (10.11), where U is replaced by U , as required. To prove the second part of the theorem, set Eμ(u)= EV T1(u + c1) ...Tm(u + cm), sh(V)=μ summed over the standard tableaux V of shape μ, where it is understood that the contents ca = ca(V) depend on V.

Lemma 10.1.4. For any s ∈ Sm we have

Ps Eμ(u)=Eμ(u) Ps.

Proof. We will use the isomorphism (1.2) between the group algebra C[Sm] and the direct sum of matrix algebras. We will show that the element Eμ(u)commutes with the image EUU of an arbitrary matrix unit eUU under the action of the symmetric group in (CN )⊗ m,whereU and U are standard tableaux of the same shape. By Lemma 10.1.3 we have Eμ(u)= EV T1(u + c1) ...Tm(u + cm) EV . sh(V)=μ

Hence, if U and U are of the shape ν and ν = μ,then

EUU Eμ(u)=Eμ(u) EUU =0.

Now suppose that U and U are of the shape μ. In the case where U = U we have

EU Eμ(u)=Eμ(u) EU = EU T1(u + c1) ...Tm(u + cm) EU ,

where ca = ca(U) for all a. Finally, suppose U and U are diﬀerent tableaux of the shape μ. It suﬃces to consider the case, where U is obtained from U by swapping the entries a and a+1 for some a ∈{1,...,m−1}, which do not belong to the same row or the same column. That is, U = sa U. In this case, relation (1.8) implies 1 1 EUU = √ EU (Paa+1 − d) EU = √ EU Paa+1 Raa+1(ca+1 − ca) EU . 1 − d 2 1 − d 2 174 10. YANGIAN CHARACTERS IN TYPE A

Now, using (1.67), (10.7) and Lemma 10.1.3 we ﬁnd

Eμ(u) EU Paa+1 Raa+1(ca+1 − ca) EU

= EU T1(u + c1) ...Tm(u + cm) EU Paa+1 Raa+1(ca+1 − ca) EU

= EU T1(u + c1) ...Tm(u + cm) Paa+1 Raa+1(ca+1 − ca) EU E − E = U Paa+1 Raa+1(ca+1 ca) T1(u + c1) ...Tm(u + cm) U , U where we denoted ca = ca( ) for all a. Transform this expression further by using relation (1.6) which gives

EU Paa+1 Raa+1(ca+1 − ca)=EU Paa+1 Raa+1(ca+1 − ca) EU . So, by Lemma 10.1.3, applied to U instead of U, the expression equals E − E E U Paa+1 Raa+1(ca+1 ca) U T1(u + c1) ...Tm(u + cm) U

= EU Paa+1 Raa+1(ca+1 − ca) EU Eμ(u), thus completing the proof of the lemma. Write the series (10.11) in the form T 1 E μ(u)= tr1,...,m μ(u), fμ where fμ is the number of standard tableaux of shape μ. Introduce the matrix elements of Eμ(u)by E (u)= e ⊗ ...⊗ e ⊗E (u) i1... im , μ i1j1 imjm μ j1... jm I,J summed over all m-tuples of indices I =(i1,...,im)andJ =(j1,...,jm)from { } 1,...,N .Then tr E (u)= E (u) i1... im . 1,...,m μ μ i1... im I By Lemma 10.1.4, for any σ ∈ Sm the summands corresponding to the m-tuples (i1,...,im)and(iσ(1),...,iσ(m)) are equal. Hence we may restrict the summation to the weakly increasing sets of m-tuples. Taking into account the number of equal summands, we come to the expression m! (10.15) tr E (u)= E (u) i1... im , 1,...,m μ μ i1... im α1! ...αN ! i1···im where αi denotes the multiplicity of the index i among the m-tuple (i1,...,im). Let ΦU = ΦU (s) Ps−1

s∈Sm be the image of the diagonal matrix element φU deﬁned in (1.5), under the action of the symmetric group in (CN )⊗ m. Note that by the invariance of the inner product, −1 −1 ΦU (s)=(s · vU ,vU )=(vU ,s · vU )=ΦU (s ). By (1.4), f (10.16) E (u)= μ Φ T (u + c ) ...T (u + c ), μ m! U 1 1 m m sh(U)=μ 10.1. YANGIAN FOR glN 175 where ca = ca(U)fora =1,...,m, as before. Therefore, i1... im fμ Eμ(u) = ΦU (s) ti i (u + c1) ...ti i (u + cm). i1... im m! s(1) 1 s(m) m sh(U)=μ s∈Sm The series (10.11) can now be written in the form T 1 μ(u)= ΦU (s) tis(1) i1 (u + c1) ...tis(m) im (u + cm). α1! ...αN ! i1···im sh(U)=μ s∈Sm

This form of T μ(u) allows us to get a formula for its Harish-Chandra image which canbewrittenas 1 (10.17) ΦU (s) λi1 (u + c1) ...λim (u + cm), α1! ...αN ! i1···im sh(U)=μ s∈S(i) ∼ × ··· × where S(i) = Sα1 SαN denotes the subgroup of Sm which consists of α1 α2 αN the permutations stabilizing the m-tuple (i1,...,im)=(1 2 ...N ). Indeed, tis(m) im (u + cm) is zero modulo the left ideal I of Y(glN ) unless is(m) = im.Inthis case the permutation s preserves the subset of indices {r | ir = im}. The image of timim (u + cm) under the homomorphism (10.10) is λim (u + cm), and the claim follows by applying the same argument to tis(m−1) im−1 (u + cm−1)etc. Given an m-tuple (i1,...,im)with1 i1 ··· im N, for each standard tableau U of shape μ denote by T = i(U) the tableau obtained from U by replacing the entry r with the number ir for r =1,...,m. The entries of T then weakly increase along the rows and down the columns. Changing the order of summation in the expression (10.17), we can write it as 1 (10.18) ΨT λT (α) u + c(α) , α1! ...αN ! ∈ i1···im sh(T )=μ α μ summed over the tableaux T with the entries i1,...,im, where we have set (10.19) ΨT = ΦU (s).

U,i(U)=T s∈S(i) For a μ-tableau T with entries in {1,...,N} such that the entries of T weakly increase along the rows and down the columns, consider the skew diagrams ω1,...,ωN , where ωr is the union of the cells of T occupied by r. Then (10.19) can be rewritten as N

ΨT = χωr (sr), ∈ r=1 sr Sαr where χωr denotes the skew character of Sαr associated with ωr. However, the expression 1 χωr (s) αr! ∈ s Sαr coincides with the standard inner product of χωr with the trivial character of Sαr . This equals the multiplicity of the trivial representation in the skew representation of Sαr associated with ωr. By the Pieri rule, the multiplicity is nonzero only if ωr does not contain two cells in the same column, in which case the multiplicity is 1. 176 10. YANGIAN CHARACTERS IN TYPE A

This implies that the summation in (10.18) can be restricted to those tableaux T whose columns strictly increase and so expression (10.18) equals λT (α) u + c(α) , sh(T )=μ α∈μ completing the proof of the theorem.

Remark 10.1.5. (i) Applying the evaluation homomorphism (10.8) to the series T μ(u) deﬁned in (10.11) and multiplying by a polynomial in u we recover the Casimir element S μ(u)deﬁnedin(4.44): S μ(u)=(u + c1) ...(u + cm)ev T μ(u) .

Consider the Harish-Chandra homomorphism (4.6) for g = glN with the standard Cartan subalgebra h spanned by the elements λi = Eii with i =1,...,N and the subalgebra n+ spanned by the elements Eij with 1 i

h −−−−→ C U(glN ) [λ1,...,λN ], where the horizontal arrows are the respective Harish-Chandra homomorphisms (10.10) and (4.6), while the right vertical arrow is the homomorphism deﬁned by (1) → (r) → λi λi and λi 0,r 2.

Using the diagram, we get the Harish-Chandra images of the polynomials S μ(u) from Theorem 10.1.2 and hence prove Proposition 4.9.1. Note also that apart from the calculation of the central characters χ(S μ), The- orem 10.1.2 also implies the formula for the character of the irreducible representation L(λ)ofGLN . Namely, the character formula (4.27) is recovered by the specialization T (u) → h. (ii) The Yangian characters can be defined for all finite-dimensional representations of the Yangian as the Harish-Chandra images of the associated transfer matrices by analogy with the quantum affine algebras; see [51]. However, the proof that T μ(u) coincides with the transfer matrix associated with L(μ)requiresad- ditional arguments which use some properties of the universal R-matrix; cf. [48] and [124]. By the properties of the transfer matrices, the coefficients of all series T μ(u) belong to a commutative subalgebra of the Yangian. This fact can also be established directly with the use of the fusion formula of Proposition 1.1.7 for the idempotents eU involved in the definition of the series. Another proof of commutativity of the subalgebra generated by the coefficients can be obtained by using a quantum vertex algebra structure associated with the double Yangian [37]; see Remark 10.4.2 below. We will point out two particular cases of Theorem 10.1.2 corresponding to the respective constructions of the Casimir elements for glN in Propositions 4.6.1 and 4.7.1. They provide the characters for the Yangian representations in the exterior powers Λm(CN ) and symmetric powers Sm(CN ) of the vector representation. Here μ is a column or row diagram with m boxes and U is the unique standard 10.2. DUAL YANGIAN FOR glN 177 tableau of shape μ. The primitive idempotent EU in (10.11) coincides with the anti-symmetrizer A(m) or symmetrizer H(m), respectively. Corollary 10.1.6. For the images under the Harish-Chandra homomorphism (10.10) we have (m) − → ··· − tr1,...,m A T1(u) ...Tm(u m +1) λi1 (u) λim (u m +1)

1i1<···

1i1···imN

10.2. Dual Yangian for glN + By analogy with (10.3), define the dual Yangian Y (glN ) as the unital asso- (−r) ciative algebra with generators tij ,where1 i, j N and r =1, 2,... subject to the defining relations (−r) (−s) (−r−s) − (−r−s) [tij ,tkl ]=δkj til δil tkj min{r,s} (−r−s+a−1) (−a) − (−a) (−r−s+a−1) + tkj til tkj til . a=1 Combining the generators into the formal power series ∞ + − (−r) r−1 ∈ + tij(u)=δij tij u Y (glN )[[u]] r=1 we can write the defining relations as − + + + + − + + (10.20) (u v)[tij(u),tkl(v)] = tkj(u) til (v) tkj(v) til (u) which thus take the same form as (10.2). Hence, they can also be written in the matrix form − + + + + − (10.21) R12(u v) T1 (u) T2 (v)=T2 (v) T1 (u) R12(u v) as in (10.7), where we use the Yang R-matrix (10.5) and N + ⊗ + ∈ CN ⊗ + (10.22) T (u)= eij tij(u) End Y (glN )[[u]]. i,j=1 + Similar to the matrix notation used above, Ta (u)fora =1,...,m denote formal series in u defined as in (1.61) with coefficients in the tensor product algebra CN ⊗ ⊗ CN ⊗ + (10.23) End ... End Y (glN ). m + The algebra Y (glN ) possesses a natural ascending filtration defined by setting (−r) − deg tij = r for all r 1. We have the isomorphism for the associated graded algebra + ∼ −1 −1 gr Y (glN ) = U t glN [t ] . ¯(−r) (−r) − The image tij of the generator tij in the ( r)-th component of the graded + − −1 −1 algebra gr Y (glN ) corresponds to the element Eij[ r]ofU t glN [t ] . 178 10. YANGIAN CHARACTERS IN TYPE A

+ We will regard Y (glN ) as a module over the Cartan subalgebra h of glN ,where each basis element Eii of h acts as a derivation and the action is deﬁned on the generators by

· + + − + Eii tkl(u)=δki til (u) δil tki(u).

+ h Denote by Y (glN ) the subalgebra of h-invariants under this action. Consider + (−r) the left ideal J of the algebra Y (glN ) generated by all elements tij with the conditions N i>j 1andr 1. As with the Yangian, the quotient of + h + h ∩ Y (glN ) by the two-sided ideal Y (glN ) J is isomorphic to the commutative (−r) algebra freely generated by the images of the elements tii with i =1,...,N and (−r) (−r) r 1 in the quotient. We will use the notation λi for this image of tii .The corresponding analogue of the Harish-Chandra homomorphism (10.10) now takes the form

+ h → C (−r) | (10.24) Y (glN ) [λi i =1,...,N, r 1].

(−r) We combine the elements λi into the formal series

∞ + − (−r) r−1 (10.25) λi (u)=1 λi u ,i=1,...,N, r=1

+ which will be understood as the images of the series tii (u) under the homomorphism (10.24). Our next goal is to prove a dual Yangian version of Theorem 10.1.2. Note, however, that the shift automorphism of the Yangian taking T (u)toT (u + c)fora given constant c, which is used in the definition of the series T μ(u) is not defined for the dual Yangian. To introduce counterparts of such series we need to extend + + the algebra Y (glN ) by embedding it into the completed dual Yangian Y (glN ) defined as follows. Consider the descending filtration

+ + ⊃ + ⊃ + ⊃ (10.26) Y (glN )=Y0 Y1 Y2 ...

+ (−r) on Y (glN ) defined by setting the degree of the generator tij with r 1tobe + equal to r. That is, the subspace Yk is spanned by all monomials in the generators + + of the total degree at least k.ThenY (glN )isthecompletionofY (glN ) with respect to this filtration. It is clear that for any c ∈ C the coefficients of u in the + + power series tij(u+c)canberegardedaselementsofY (glN ). The Harish-Chandra + h homomorphism (10.24) extends by continuity to the homomorphism from Y (glN ) C (−r) | to the corresponding completed polynomial algebra [λi i =1,...,N, r 1]. + Working with the tensor product algebra (10.23) with Y (glN ) replaced with + T + Y (glN ), we can now define the series μ (u) by setting

T + E + + (10.27) μ (u)=tr1,...,m U T1 (u + c1) ...Tm (u + cm), 10.2. DUAL YANGIAN FOR glN 179 using the same idempotents EU as in Theorem 10.1.2. Given a diagram μ,areverse tableau of shape μ is obtained by filling in the boxes of the diagram with the numbers in a given set in such a way that the entries weakly decrease along each row from left to right and strictly decrease in each column from top to bottom. Theorem . T + 10.2.1 The series μ (u) does not depend on the standard tableau U of shape μ. Moreover, the images of its coefficients under the Harish-Chandra homomorphism (10.24) are found by T + → + μ (u) λT (α) u + c(α) , sh(T )=μ α∈μ summed over reverse tableau T of shape μ with entries in {1,...,N}. Proof. The first part follows from the respective argument of the proof of Theorem 10.1.2 because it only relies on the matrix form of the defining relations which is the same for both the Yangian and dual Yangian. In particular, the relation (10.14) holds in the same form, E + + + + E (10.28) U T1 (u + c1) ...Tm (u + cm)=Tm (u + cm) ...T1 (u + c1) U . A slight change is needed for the proof of the second part. Namely, using the matrices T +(u)inplaceofT (u) throughout the arguments, we need the following counterpart of (10.15), m! tr E +(u)= E +(u) i1... im , 1,...,m μ μ i1... im α1! ...αN ! i1···im where αi denotes the multiplicity of the index i among the m-tuple (i1,...,im)of weakly decreasing indices. By applying (10.28), we get the following counterpart of (10.16), f E +(u)= μ T +(u + c ) ...T+(u + c )Φ , μ m! m m 1 1 U sh(U)=μ where ca = ca(U)fora =1,...,m, as before. Hence, + i1... im fμ + + E (u) = ΦU (s) t (u + cm) ...t (u + c1) μ i1... im m! im is(m) i1 is(1) sh(U)=μ s∈Sm and so + 1 + + T (u)= ΦU (s) t (u + c ) ...t (u + c ). μ im is(m) m i1 is(1) 1 α1! ...αN ! i1···im sh(U)=μ s∈Sm T + Using the notation as in (10.17), for the Harish-Chandra image of μ (u)weget 1 + + ΦU (s) λ (u + c ) ...λ (u + c ). im m i1 1 α1! ...αN ! i1···im sh(U)=μ s∈S(i) Repeating the concluding part of the argument with the reverse ordering of the indices we come to the desired expression for the Harish-Chandra image where the role of semistandard tableaux is played by the reverse tableaux.

The following is the counterpart of Corollary 10.1.6 for the dual Yangian. 180 10. YANGIAN CHARACTERS IN TYPE A

Corollary 10.2.2. For the images under the Harish-Chandra homomorphism (10.24) we have tr A(m) T +(u) ...T+(u − m +1) → λ+ (u) ···λ+ (u − m +1) 1,...,m 1 m i1 im Ni1>···>im1 and tr H(m) T +(u) ...T+(u + m − 1) → λ+ (u) ···λ+ (u + m − 1). 1,...,m 1 m i1 im Ni1···im1

10.3. Double Yangian for glN

The double Yangian DY(glN )forglN is defined as the associative algebra gen- (r) (−r) erated by the central element C and elements tij and tij ,where1 i, j N and r =1, 2,..., subject to the defining relations written in terms of the generator matrices (10.4) and (10.22) as follows. They are given by (10.7), (10.21) together with the relation − + + − − (10.29) R12 u v + C/2 T1(u) T2 (v)=T2 (v) T1(u) R12 u v C/2 , where −1 R12(u)=g(u) R12(u)=g(u) 1 − P12 u and ∞ −i (10.30) g(u)=1+ gi u ,gi ∈ C, i=1 is a formal power series in u−1 whose coefficients are uniquely determined by the relation (10.31) g(u + N)=g(u)(1− u−2). Its first few terms are 1 N 2 +1 g(u)=1+ u−1 + u−2 + .... N 2N 2

The relation (10.31) ensures that the R-matrix R12(u) possesses the crossing symmetry properties −1 t1 t1 −1 t2 t2 (10.32) R12(u) R12 u + N =1 and R12(u) R12 u + N =1, where t1 and t2 are the standard matrix transpositions t : eij → eji acting on the ﬁrst and second copy of End CN , respectively. Moreover, the following unitarity property holds

(10.33) R12(u)R12(−u)=1. Indeed, replacing u with −u − N in (10.31) we get g(−u)=g(−u − N) 1 − (u + N)−2 and so g(u)g(−u)(1− u−2)=g(u + N)g(−u − N) 1 − (u + N)−2 . So the series on the left hand side is invariant under the shift u → u + N which is only possible when g(u)g(−u)(1− u−2)=1 10.3. DOUBLE YANGIAN FOR glN 181 thus implying (10.33). The series g(u) can be defined equivalently as the unique formal power series of the form (10.30) satisfying the relation −1 (10.34) g(u) g(u +1)...g(u + N − 1) = 1 − u−1 . To see the equivalence of the definitions, observe that by (10.31), the series G(u) defined by the left hand side of (10.34) satisfies G(u+1) = G(u)(1−u−2). However, G(u) is uniquely determined by this relation and so coincides with the right hand side of (10.34). Given any c ∈ C we will introduce the double Yangian at the level c as the − quotient DYc(glN )ofDY(glN ) by the ideal generated by C c. As a vector space, this quotient is isomorphic to the tensor product ∼ + ⊗ (10.35) DYc(glN ) = Y (glN ) Y(glN ) of the Yangian and dual Yangian. This property relies on the Poincaré–Birkhoff– Witt theorem for the algebra DY(glN ). For the level zero case c =0thetheorem canbeprovedbyusingtheargumentsof[35]and[125] and then extended to an arbitrary level with the use of level 1 representations constructed in [72]; see [82]. Consider an ascending filtration on the double Yangian DY(glN ) defined by (r) − (−r) − deg tij = r 1 and deg tij = r for all r 1; the degree of the central element C is defined to be equal to zero. Denote by gr DY(glN ) the corresponding graded algebra. We will use the notation ¯(r) ¯(−r) tij and tij for the images of the generators in the respective components of the graded algebra and let C be the image of C in the zeroth component. −1 ⊕ C Recall the affine Kac–Moody algebra glN = glN [t, t ] K defined by the commutation relations (7.3). Proposition 10.3.1. The assignments − → ¯(r) − → ¯(−r) → (10.36) Eij[r 1] tij ,Eij[ r] tij and K C with r 1 define an algebra isomorphism → U(glN ) gr DY(glN ). Proof. As we pointed out in the previous sections, there are isomorphisms ∼ −1 −1 ∼ + U glN [t] = gr Y(glN )andUt glN [t ] = gr Y (glN ). They are defined by the assignments (10.36). We will now use the defining relations ¯(r) ¯(−s) (10.29) to verify that the images of the generators tij and tkl with r, s 1 satisfy the desired relations in U(glN ). Introduce re-scaled generators of DY(glN ) by setting (r) r−1 (r) (−r) −r (−r) tij = h tij and tij = h tij ¯(r) for r 1, where h is a complex-valued parameter. The relations satisfied by tij ¯(−s) and tkl in the graded algebra gr DY(glN ) will be recovered by calculating the (r) (−s) → relations between tij and tkl and then taking the limit as h 0. Set ∞ 1 u t (u)= t (r) u−r = t − δ ij ij h ij h ij r=1 182 10. YANGIAN CHARACTERS IN TYPE A and ∞ − 1 v t+ (v)= t ( s) vs−1 = δ − t+ . kl kl h kl kl h s=1 Write (10.29) in terms of the generating series: 1 (10.37) g u − v + C/2 t (u) t+ (v) − t (u) t+(v) ij kl u − v + C/2 kj il 1 = g u − v − C/2 t+ (v) t (u) − t+ (v) t (u) . kl ij u − v − C/2 kj il Note the expansion into a power series in (u − v)−1: g u − v − C/2 C (10.38) =1+ + ... g u − v + C/2 N (u − v)2 Now replace u by u/h and v by v/h in (10.37) to get the corresponding relations + between the series tij(u)andtkl(v). We have h δ + ht (u) δ − ht+ (v) − δ + ht (u) δ − ht+(v) ij ij kl kl u − v + hC/2 kj kj il il h − δ − ht+ (v) δ + ht (u) − δ − ht+ (v) δ + ht (u) kl kl ij ij u − v − hC/2 kj kj il il h2 C × 1+ + ... =0. N (u − v)2 As a power series in h, the left hand side is divisible by h2. Hence, dividing by h2 we get the relation modulo h, 1 t (u), t+ (v) ≡ δ t (u)+t+(v) − δ t (u)+t+ (v) ij kl u − v kj il il il kj kj C + Nδ δ − δ δ . N (u − v)2 kj il ij kl Thus, taking the coefficients of u−rvs−1 with r, s 1 on both sides, in the limit h → 0 in the graded algebra we get ⎧ ⎪ − − ⎪ ¯(r s) − ¯(r s) − − δij δkl ⎨δkj til δil tkj +(r 1) δr,s+1 C δkj δil ¯(r) ¯(−s) N tij , tkl = ⎪ − − − − δ δ ⎩δ t¯(r s 1) − δ t¯(r s 1) +(r − 1) δ C δ δ − ij kl kj il il kj r,s+1 kj il N with the top line chosen for r>sand the bottom line for r s. Comparing with (7.3), we may conclude that the assignments (10.36) define a homomorphism → U(glN ) gr DY(glN ). It is clear that the homomorphism is surjective whereas its injectivity is implied by the Poincaré–Birkhoff–Witt theorem for the double Yangian. The critical level c = −N will play a particular role in what follows. Under this specialization the fraction (10.38) admits a closed form, g u − v + N/2 1 =1− . g u − v − N/2 (u − v − N/2)2 10.4. INVARIANTS OF THE VACUUM MODULE OVER THE DOUBLE YANGIAN 183

This implies the following form of the deﬁning relations (10.37) at the critical level: 1 1 t (u),t+ (v) = − t+ (v) t (u)+ t+(v) t (u) ij kl u − v + N/2 kj il u − v − N/2 il kj 1 − t+(v) t (u). (u − v)2 − N 2/4 ij kl

10.4. Invariants of the vacuum module over the double Yangian By analogy with the vacuum module over glN defined in (6.6) (with g = glN ), V ∈ C introduce the vacuum module c(glN ) at the level c over DY(glN )asthe (r) quotient of the algebra DYc(glN ) by the left ideal generated by all elements tij with r 1: V (r) | c(glN )=DYc(glN )/DYc(glN ) tij r 1 . V We let 1 denote the image of 1 in the quotient. In particular, c(glN ) has a structure of a module over the Yangian Y(glN ) obtained by restriction. As a vector space, the + vacuum module is isomorphic to the dual Yangian Y (glN ) due to the decomposi- V tion (10.35). Therefore, c(glN ) is equipped with the descending filtration (10.26). V The completed vector space c(glN )isalsoaY(glN )-module. Now assume that the level is critical, c = −N and introduce the subspace of Y(glN )-invariants by V { ∈ V | } z( cri)= v −N (glN ) tij(u)v = δij v , V (r) so that any element of z( cri) is annihilated by all operators tij with r 1. As V with the Feigin–Frenkel center z(glN ), the subspace z( cri) can be regarded as a + subalgebra of the completed dual Yangian Y (glN ). Moreover, this subalgebra turns out to be commutative; see [82]. Suppose that μ is a diagram with m boxes whose length does not exceed N. T + Consider the power series μ (u)inu defined in (10.27). Theorem . T + 10.4.1 All coefficients of the series μ (u)1 belong to the subalgebra of invariants z(Vcri) of the vacuum module. In particular, the coefficients are + pairwise commuting elements of the completed dual Yangian Y (glN ). Proof. Similar to the proof of Theorem 7.1.3, consider the tensor product space CN ⊗ ⊗ CN ⊗ V End ... End −N (glN ) m+1 with the m + 1 copies of End CN labeled by 0, 1,...,m. We need to verify the identity T + T + T0(z) μ (u)1 = μ (u)1. By the defining relations (10.29), for all a =1,...,m we can write

+ T0(z) Ta (u + ca) − − − −1 + − − = R0 a(z u ca N/2) Ta (u + ca) T0(z) R0 a(z u ca + N/2). 184 10. YANGIAN CHARACTERS IN TYPE A

Hence, suppressing the arguments of the R-matrices we get E + + T0(z)tr1,...,m U T1 (u + c1) ...Tm (u + cm)1 − − E 1 1 + + =tr1,...,m U R01 ...R0 m T1 (u + c1) ...Tm (u + cm) T0(z) R0 m ...R011 − − E 1 1 + + =tr1,...,m U R01 ...R0 m T1 (u + c1) ...Tm (u + cm) R0 m ...R011, where the second equality holds since T0(z) acts as the identity operator on the subspace End (CN )⊗ (m+1) ⊗ 1. Note the following consequence of the Yang–Baxter equation (10.6),

(10.39) R(u1,...,um)R0 m(u0 − um) ...R01(u0 − u1)

= R01(u0 − u1) ...R0 m(u0 − um)R(u1,...,um), where u0 is another variable and we use the notation (10.13). Relation (10.39) will remain valid if each factor R0 a(u0 − ua) is replaced with R0 a(u0 − ua). Hence, by the fusion procedure of Proposition 1.1.7, the consecutive evaluations ua = ca for a =1,...,m imply

EU R0 m(u0 − cm) ...R01(u0 − c1)=R01(u0 − c1) ...R0 m(u0 − cm) EU . By inverting the R-matrices we also get

−1 −1 EU R01(u0 − c1) ...R0 m(u0 − cm)

−1 −1 = R0 m(u0 − cm) ...R01(u0 − c1) EU . T + E Returning now to the calculation of T0(z) μ (u)1, recall that U in an idempotent and use the cyclic property of trace to write E ◦ E ◦ E 2 tr1,...,m U XY =tr1,...,m X U Y =tr1,...,m X U Y E ◦ E ◦ E 2 ◦ E E =tr1,...,m U XY U =tr1,...,m XY U =tr1,...,m XY U =tr1,...,m X U Y, whereweset − − 1 1 + + X = R01 ...R0 m,Y= T1 (u + c1) ...Tm (u + cm) R0 m ...R01 andusedthenotationX◦ and Y ◦ for the same products written in the opposite order. Thus, we can write t ...t T + E t1...tm E 1 m T0(z) μ (u)1 =tr1,...,m X U Y 1 =tr1,...,m X U Y 1. We have E t1...tm tm t1 E + + t1...tm U Y = R0 m ...R01 U T1 (u + c1) ...Tm (u + cm) and −1 t1 −1 tm t1...tm X = R01 ... R0 m . By the crossing symmetry (10.32), we have ta −1 ta R0 a R0 a =1 for all a =1,...,m and so T + E + + t1...tm T + T0(z) μ (u)1 =tr1,...,m U T1 (u + c1) ...Tm (u + cm) 1 = μ (u)1 as required. 10.5. FROM YANGIAN INVARIANTS TO SEGAL–SUGAWARA VECTORS 185

Remark 10.4.2. As pointed out above, it was derived in [82] that the co- T + efficients of all series μ (u) pairwise commute as they belong to a commutative V + subalgebra of z( cri) of the completed dual Yangian Y (glN ). Since the defining relations of both the Yangian and dual Yangian written in terms of the generating series are given by the identical formulas (10.2) and (10.20), we thus recover a well- known property that the coefficients of all series T μ(u)belongtoacommutative subalgebra of the Yangian Y(glN ).

10.5. From Yangian invariants to Segal–Sugawara vectors In this section we give another proof of Theorem 7.1.3. By Proposition 10.3.1, ∈ V + for any element S z( cri) its image S in the graded algebra gr Y (glN ) belongs to the Feigin–Frenkel center z(glN ). We will use Theorem 10.4.1 to construct an appropriate linear combination of elements of z(Vcri) whose graded image will coincide with the elements given by (7.7). As explained in Section 7.1, the claim for the elements (7.8), (7.9) and (7.10) will then follow. (−r) − Extend the ascending ﬁltration on the dual Yangian deﬁned by deg tij = r + − to the algebra of formal series Y (glN )[[u, ∂u]] by setting deg u =1anddeg ∂u = 1 −1 −1 so that the associated graded algebra is isomorphic to U t glN [t ] [[u, ∂u]]. Then the element − − (m) − + ∂u − + ∂u (10.40) tr1,...,m A 1 T1 (u)e ... 1 Tm (u)e has degree −m and its image in the graded algebra coincides with (m) (10.41) tr1,...,m A ∂u + E(u)+1 ... ∂u + E(u)+ m , where, as in Section 7.2, ∞ r−1 E(u)+ = E[−r]u . r=1 On the other hand, the element (10.40) equals m − tr A(m) (−1)k T +(u) ...T+(u − k +1)e k∂u . 1,...,m i1 ik k=0 1i1<···

Transform the expression by applying conjugations by elements of Sm and using the cyclic property of trace to bring it to the form m m − tr A(m) (−1)k T +(u) ...T+(u − k +1)e k∂u . 1,...,m k 1 k k=0 Calculating partial traces of the anti-symmetrizer with the use of (3.26) we can write this as m N − k − (10.42) (−1)k tr A(k) T +(u) ...T+(u − k +1)e k∂u . m − k 1,...,k 1 k k=0 Apply Theorem 10.4.1 to the case of the one-column diagram μ =(1k). There is (k) a unique standard tableau U of shape μ so that EU = A and the contents are ci = −i +1fori =1,...,k. We find that all elements (k) + + − tr1,...,k A T1 (u) ...Tk (u k +1)1 186 10. YANGIAN CHARACTERS IN TYPE A belong to z(Vcri). This proves that all coefficients of (10.41) are Segal–Sugawara 0 vectors for glN . In particular, this holds for the coefficient of u which coincides with (7.7) after the replacement of ∂u by τ. The last step is essentially an application of the vacuum axiom for the affine vertex algebra. This completes the proof of Theorem 7.1.3.

The above construction of Segal–Sugawara vectors for glN based on the use of the double Yangian can be extended, in principle, to any simple Lie algebra g. In accordance to Drinfeld [32], the corresponding Yangian Y(g) admits an RT T presentation of the form (10.7). The double Yangian DY(g) can then be deﬁned by additional relations analogous to (10.21) and (10.29). Its associated graded algebra at the critical level should reproduce U−h∨ (g). Hence, Segal–Sugawara vectors for g could be obtained by taking the graded images of the invariants of the vacuum module over the double Yangian as in the above argument for glN . The proof of Theorem 10.4.1 providing invariants of the vacuum module at the critical level relies on the existence of a fusion procedure. Namely, the primitive idempotents E = EU are represented as consecutive evaluations of the product of the R-matrices 1 (10.43) E = R(u1,...,um) ... h(μ) u1=c1 u2=c2 um=cm as stated in Proposition 1.1.7, where R(u1,...,um)= Rab(ua − ub), 1a

10.6. Screening operators The Harish-Chandra images provided by Theorem 10.1.2 can be regarded as polynomials in the formal variables λi(u + a), where i =1,...,N and a ∈ C.It was conjectured in [51] and proved in [48], that the polynomials which arise as the characters of Yangian representations are precisely those which belong to the intersection of the kernels of the screening operators. Although the Harish-Chandra image of T μ(u) manifestly coincides with the q-character of the evaluation module 10.6. SCREENING OPERATORS 187 under an appropriate identiﬁcation of the parameters, we will show it belongs to the kernels of all screening operators thus illustrating the use of the result of [48]. We will suppress the variable u from the notation and introduce the algebra of polynomials

L = C[λi(a) | i =1,...,N,a∈ C] in the variables λi(a). For every i ∈{1,...,N − 1} consider the free left L-module Li with the generators σi(a), where a runs over C and denote by Li its quotient by the relations

(10.44) λi(a) σi(a)=λi+1(a) σi(a +1),a∈ C. Deﬁne the linear operator S : L→L by the formula i ⎧ i ⎨⎪λi(a) σi(a)forj = i (10.45) Si : λj (a) → −λ (a) σ (a +1) for j = i +1 ⎩⎪ i+1 i 0forj = i, i +1 and the Leibniz rule (10.46) Si(AB)=B Si(A)+ASi(B). Now the i-th screening operator

Si : L→Li is defined as the composition of Si and the projection Li →Li. L The subalgebra Rep Y(glN ) of Yangian characters in can be defined as the intersection of the kernels of the screening operators: N−1 Rep Y(glN )= ker Si. i=1 ∈L For any diagram μ of length (μ) N define the polynomial χL(μ) by (10.47) χL(μ) = λT (α) c(α) , sh(T )=μ α∈μ summed over all semistandard tableaux T of shape μ with entries in the set {1,...,N}; see Theorem 10.1.2. Proposition . ∈ 10.6.1 We have χL(μ) Rep Y(glN ). Proof. We will verify that χL(μ) is annihilated by all screening operators Si for i =1,...,N − 1. By the definition (10.45), the operator Si annihilates all variables except for λi(a)andλi+1(a). Since each tableau U is semistandard, its entries i and i + 1 occupy a skew subdiagram which can have at most two boxes in the same column, as illustrated:

Therefore, changing the order of summation in (10.47), we can reduce the calculation to the case where the diagram μ is replaced by such a skew diagram and the corresponding tableaux take entries in the set {i, i+1}. Two such entries occurring 188 10. YANGIAN CHARACTERS IN TYPE A in the same two-box column contribute the factor of the form λi(c)λi+1(c−1) where c is the content of the upper box. However,

Si : λi(c)λi+1(c − 1) → λi(c)λi+1(c − 1)σi(c) − λi(c)λi+1(c − 1)σi(c)=0. Excluding the two-box columns from the diagram, we get a disjoint union of rows of boxes. Such a row of length k will contribute the sum

k (10.48) λi(c)λi(c +1)...λi(c + p − 1) p=0

× λi+1(c + p)λi+1(c + p +1)...λi+1(c + k), where c is now the content of the leftmost box of the row. By applying the screening operator Si we get k λi(c)λi(c +1)...λi(c + p − 1)λi+1(c + p)λi+1(c + p +1)...λi+1(c + k) p=0 × σi(c)+σi(c +1)+···+ σi(c + p − 1) − σi(c + p +1)−···−σi(c + k +1) . By (10.44) we can transform the products by using the relations

λi+1(c + p) ...λi+1(c + p + r)σi(c + p + r +1)

= λi(c + p) ...λi(c + p + r)σi(c + p).

This implies that the result of the application of Si to the polynomial (10.48) will contain each monomial of the form

λi(c) ...λi(c + q − 1)λi+1(c + q) ...λi+1(c + k) σi(c + s − 1) with s =1,...,q exactly twice with the opposite signs. The proof is completed by taking the Leibniz rule (10.46) into account. Remark 10.6.2. By analogy with (4.34), introduce the shift operator e∂ such that ∂ ∂ ∂ ∂ e λi(a)=λi(a +1)e and e σi(a)=σi(a +1)e . ∂ Each screening operator Si commutes with e . The polynomials (10.48) can be combined into the generating series ∂ −1 ∂ −1 1 − λi(c)e 1 − λi+1(c)e . Therefore, as an alternative to the last step of the proof of Proposition 10.6.1, we can verify that the inverse series ∂ ∂ 1 − λi+1(c)e 1 − λi(c)e is annihilated by S . By applying S we get i i ∂ ∂ ∂ ∂ λi+1(c)σi(c +1)e 1 − λi(c)e + 1 − λi+1(c)e −λi(c)σi(c)e

∂ ∂ = λi+1(c)σi(c +1)e − λi(c)σi(c)e =0 by (10.44). Note also that the proof of Proposition 10.6.1 obviously extends to arbitrary skew diagrams μ which do not contain columns of length exceeding N.The corresponding polynomials (10.47) are the characters of the skew representations of the Yangian Y(glN )[127]; see also [109, Section 8.5]. 10.6. SCREENING OPERATORS 189

We will now state the counterpart of Proposition 10.6.1 for the Harish-Chandra T + + image of the series μ (u) associated with the dual Yangian Y (glN )asprovided by Theorem 10.2.1. Observe that the only formal diﬀerence with the corresponding polynomial of Theorem 10.1.2 is the use of reverse tableaux. However, there is an obvious bijection between the semistandard tableaux U and reverse tableaux U of the same shape μ with the entries in the set {1,...,N}. The bijection replaces each entry i of U by i = N − i + 1. Therefore the following deﬁnitions and results are essentially reformulations of those associated with the Yangian Y(glN ). Introduce the algebra of polynomials

L+ C + | ∈ C (10.49) = [λi (a) i =1,...,N,a ] + ∈{ − } L+ in the variables λi (a). For every i 1,...,N 1 consider the free left -module L+ + C L+ i with the generators σi (a), where a runs over and denote by i its quotient by the relations + + + + ∈ C (10.50) λi (a) σi (a +1)=λi+1(a) σi (a),a.

+ L+ → L+ Deﬁne the linear operator Si : i by the formula ⎧ ⎪− + + ⎨ λi (a) σi (a +1) for j = i + + (10.51) S : λ (a) → λ+ (a) σ+(a)forj = i +1 i j ⎩⎪ i+1 i 0forj = i, i +1 and the Leibniz rule

+ + + (10.52) Si (AB)=B Si (A)+ASi (B). Now the i-th screening operator

+ L+ →L+ Si : i + L+ →L+ is deﬁned as the composition of Si and the projection i i . + L+ The subalgebra Rep Y (glN )of is deﬁned as the intersection of kernels of the screening operators:

N−1 + + (10.53) Rep Y (glN )= ker Si . i=1

+ ∈L+ For any diagram μ of length (μ) N deﬁne the polynomial χL(μ) by + + χL(μ) = λT (α) c(α) , sh(T )=μ α∈μ summed over all reverse tableaux T of shape μ with entries in the set {1,...,N}; see Theorem 10.2.1. Proposition . + ∈ + 10.6.3 We have χL(μ) Rep Y (glN ). Proof. This follows by the same argument as for Proposition 10.6.1. 190 10. YANGIAN CHARACTERS IN TYPE A

10.7. Bibliographical notes

A detailed account of the properties of the Yangian Y(glN )anditsrepresen- tations together with references to original papers can be found in the books by Chari and Pressley [18] and the author [109]. The proof of Theorem 10.1.2 follows Okounkov’s paper [128]. Although it deals with U(glN ), the arguments extend to the Yangian without significant changes. Some other calculations of the character of the evaluation module can be found in Frenkel and Mukhin [49, Section 4.5] and Brundan and Kleshchev [16, Section 7.4]. Our definition of the double Yangian follows Iohara [72] (we put h = −1in his notation); see also references therein. These algebras were used by Etingof and Kazhdan [36], [37] in the construction of quantum vertex algebras. Dual Yangians were studied in their previous work [35]andalsobyNazarov[125] who introduced them for the queer Lie superalgebras. The results of Section 10.4 are based on [45] and follow [82]. The theory of characters originates in the work of Knight [91] in the Yangian context and Frenkel and Reshetikhin [51] in the context of quantum affine algebras (the latter are commonly known as the q-characters). The theory was further developed by Frenkel and Mukhin [48] where an algorithm for the calculation of the q-characters was proposed while conjectures for functional relations satisfied by the q-characters were proved by Hernandez [66]andNakajima[122]. An extensive review of the role of the q-characters in classical and quantum integrable systems is given by Kuniba, Nakanishi and Suzuki [99]. CHAPTER 11

Yangian characters in types B, C and D

Our goal in this chapter is to prove some analogues of Theorem 10.1.2 for the Yangian characters in types B, C and D. We will do this for one particular idempotent in the Brauer algebra, the symmetrizer s(m) associated with the trivial representation; see Section 1.2. These particular Yangian characters will be used in Chapter 13 for the computation of the Harish-Chandra images of the generators of the Feigin–Frenkel center constructed in Chapter 8.

11.1. Yangian for gN

Recall that the Lie subalgebra of glN spanned by the elements Fij deﬁned in (2.23) is isomorphic to the orthogonal Lie algebra oN or the symplectic Lie algebra spN . We will keep the notation gN for oN (with N =2n or N =2n +1)orspN (with N =2n). We let h denote the Cartan subalgebra of gN spanned by the basis elements F ,...,F .Set 11 nn N/2 − 1 in the orthogonal case, κ = N/2 + 1 in the symplectic case.

In the notation of Section 1.5, the R-matrix R12(u) is a rational function in a complex parameter u with values in the tensor product algebra End CN ⊗ End CN deﬁned by P Q (11.1) R (u)=1− 12 + 12 . 12 u u − κ It is well known by [155] that this function satisﬁes the Yang–Baxter equation (10.6). (r) The extended Yangian X(gN ) is an associative algebra with generators tij , where 1 i, j N and r =1, 2,..., satisfying certain quadratic relations. Intro- duce the formal series ∞ (r) −r ∈ −1 (11.2) tij(u)=δij + tij u X(gN )[[u ]] r=1 and set N N −1 T (u)= eij ⊗ tij(u) ∈ End C ⊗ X(gN )[[u ]]. i,j=1

The deﬁning relations for the algebra X(gN ) are written in the form

(11.3) R12(u − v) T1(u) T2(v)=T2(v) T1(u) R12(u − v).

The Yangian Y(gN )isthequotientofthealgebraX(gN )bytherelation (11.4) T (u + κ) T (u)=1,

191 192 11. YANGIAN CHARACTERS IN TYPES B, C AND D where we use the matrix notation of Section 1.4 and the prime denotes the matrix transposition defined in (2.24). In terms of the series (11.2) the defining relations (11.3) can be written as 1 [t (u),t (v)] = t (u) t (v) − t (v) t (u) ij kl u − v kj il kj il 1 N N − δ θ t (u) t (v) − δ θ t (v) t (u) , u − v − κ ki ip pj p l lj jp kp ip p=1 p=1 wherewesetθij ≡ 1 in the orthogonal case, and θij = εi εj in the symplectic case. By taking the coefficients of u−rv−s on both sides one can get the defining relations (r) explicitly in terms of the tij as in (10.3), although their general form will not be used below. Similarly, relation (11.4) reads as N θki ti k (u + κ) til(u)=δkl. i=1

We will identify the universal enveloping algebra U(gN ) with a subalgebra of → (1) the Yangian Y(gN ) via the embedding Fij tij .ThenY(gN ) can be regarded as a gN -module with the adjoint action determined by the relations (s) (s) − (s) − (s) − (s) (11.5) [Fij,tkl ]=δkj til δil tkj θij δki tjl δjl tki . In the following proposition we use the notation of Theorem 5.3.1.

Proposition 11.1.1. For any s ∈Bm(ω) and variables u1,...,um,allcoeﬃ- cients of the series

tr1,...,m ST1(u1) ...Tm(um) commute with gN . Proof. Take the tensor product N N ⊗m End C ⊗ End (C ) ⊗ Y(gN ) with the additional copy of the endomorphism algebra End CN labeled by 0. A matrix form of the relations (11.5) reads

[F0,Ta(ua)] = (P0 a − Q0 a)Ta(ua) − Ta(ua)(P0 a − Q0 a) for a =1,...,m.Therelation E0, tr1,...,m ST1(u1) ...Tm(um) =0 is a counterpart of (5.20) and it follows by the same argument as in the proof of Theorem 5.3.1.

h Denote by Y(gN ) the subalgebra of h-invariants under the action (11.5), h Y(gN ) = {y ∈ Y(gN ) | [Fii,y]=0 for i =1,...,n}. (r) Consider the left ideal I of the algebra Y(gN ) generated by all elements tij with the conditions 1 i

(r) of tii and extend this notation to all values i =1,...,N. Thus, we get an analogue of the Harish-Chandra homomorphisms (4.6) and (10.10), h → C (r) | (11.6) Y(gN ) [λi i =1,...,n, r 1]. (r) We combine the elements λi into the formal series ∞ (r) −r (11.7) λi(u)=1+ λi u ,i=1,...,N, r=1 which can be understood as the image of the series tii(u) under the homomorphism (11.6). Evaluating the coeﬃcients of each series λi(u)inC,wegettheN-tuple of numerical series (11.8) λ(u)= λ1(u),...,λN (u) which can be regarded as the highest weight of the corresponding Verma module M(λ(u)) over the Yangian Y(gN ). By the Yangian representation theory, the Verma module is nonzero if and only if the N-tuple (11.8) satisﬁes certain conditions which are given in [7, Propositions 5.2 and 5.14]. These conditions imply that the Harish- Chandra images (11.7) must satisfy the relations

(11.9) λi(u + κ − i) λi (u)=λi+1(u + κ − i) λ(i+1) (u), − for i =0, 1,...,n 1ifgN = o2n or sp2n,andfori =0, 1,...,n if gN = o2n+1, where λ0(u)=λ0 (u):=1. Note that the Harish-Chandra homomorphism can be deﬁned in a similar way for the extended Yangian X(gN ). Then relations (11.9) will hold in the same form for all values i 1. The relations for i = 0 are excluded for the extended Yangian because they are consequences of (11.4). Recall the element S(m) introduced in Section 2.2. We will identify it with the element S(m) ⊗ 1 of the algebra CN ⊗ ⊗ CN ⊗ (11.10) End ... End Y(gN ). m By Proposition 1.2.8, it can be given by the following multiplicative formula in the orthogonal and symplectic case, respectively: 1 P Q (11.11) S(m) = 1+ ab − ab m! b − a N/2+b − a − 1 and 1a

We will now consider types B, C and D separately. In types B and D introduce (m) the formal series T (u) with coeﬃcients in the Yangian Y(oN ) by the formula

(m) (m) (11.14) T (u)=tr1,...,m S T1(u) T2(u +1)...Tm(u + m − 1)

N with the trace taken over all m copies of End C in (11.10), where gN = oN .By h Proposition 11.1.1, all coeﬃcients of this series belong to Y(oN ) .

Type Bn. Take N =2n + 1 in the deﬁnition (11.14). Theorem 11.1.2. The image of the series T (m)(u) under the homomorphism (11.6) is found by T (m) → − (u) λi1 (u) λi2 (u +1)...λim (u + m 1)

1i1···imN with the condition that n+1 occurs among the summation indices i1,...,im at most once.

Proof. By (1.31) the symmetrizer S(m) has the properties

(m) (m) (m) (m) (m) (11.15) S Qab = Qab S =0 and S Pab = Pab S = S . The subspace of harmonic tensors in (CN )⊗m is spanned by the tensors v with the property Qab v = 0 for all 1 a

By Proposition 2.3.1, Hm is then identiﬁed with the subspace of sl2-singular vec- P m tors in N under the action (2.63) whose elements are homogeneous harmonic polynomials of degree m. They belong to the kernel of the Laplace operator n 1 2 ∂ ∂ + ∂ , i i 2 n+1 i=1 where ∂i denotes the partial derivative over zi. The basis vectors of Hm will be parameterized by the N-tuples of nonnegative integers (k1,...,kn,δ,ln,...,l1), where the values of δ are restricted to δ ∈{0, 1} and the sum of all entries is m. Given such a tuple, the corresponding harmonic 11.1. YANGIAN FOR gN 195 polynomial is deﬁned by ··· a1+···+an 2a1+ +2an+δ (−2) (a1 + ···+ an)! z (11.18) n+1 a1! ...an!(2a1 + ···+2an + δ)! a1,...,an

n ki−ai li−ai z z × i i , (k − a )! (l − a )! i=1 i i i i summed over nonnegative integers ai satisfying ai min{ki,li}. Each polynomial contains a unique monomial (which we call the leading monomial)wherethevari- able zn+1 occurs with the power not exceeding 1. It is straightforward to see that these polynomials are indeed all harmonic and linearly independent. Furthermore, the number of the polynomials equals N + m − 2 N + m − 3 + m m − 1 which coincides with the dimension dim L(m, 0,...,0) in (11.16). Hence the polynomials (11.18) form a basis of the subspace Hm. Similar to the proof of (10.14), relations (11.3) and (11.13) imply that the product occurring in (11.14) can be written as (m) (m) (11.19) S T1(u) ...Tm(u + m − 1) = Tm(u + m − 1) ...T1(u) S . This relation shows that the product on each side can be regarded as an operator N ⊗m −1 on (C ) with coefficients in the algebra Y(oN )[[u ]] such that the subspace Hm is invariant under this operator. Now fix a basis vector v ∈Hm of the form (11.18). Denote the operator on the right hand side of (11.19) by A and consider the coefficient of v in the expansion of Av as a linear combination of the basis vectors. Use (11.17) to write the vector v as a linear combination of the tensors ⊗ ⊗ (m) ej1 ... ejm .WehaveS v = v, while the matrix elements of the remaining product are found from the expansion − ⊗ ⊗ Tm(u + m 1) ...T1(u)(ej1 ... ejm ) − ⊗ ⊗ = timjm (u + m 1) ...ti1j1 (u)(ei1 ... eim ).

i1,...,im The coeﬃcient of v in the expansion of Av is uniquely determined by the coeﬃcient ⊗ ⊗ ··· of the tensor ei1 ... eim with i1 im which corresponds to the leading monomial of v under the isomorphism (11.17) so that

(11.20) e ⊗ ...⊗ e = e ⊗ ...⊗ e ⊗ e ⊗ ...⊗ e ⊗ ...⊗ e ⊗ ...⊗ e . i1 im 1 1 2 2 1 1

k1 k2 l1 ⊗ ⊗ It follows from the formula (11.18) that if a tensor of the form ej1 ... ejm corresponds to a non-leading monomial occurring in the expansion of v, then the − matrix element timjm (u + m 1) ...ti1j1 (u) vanishes under the homomorphism (11.6). Indeed, the multiplicity of the index 1 in the multiset {j1,...,jm} equals − − k1 a1 for a nonnegative integer a1, whereas the coeﬃcients of t1jc (u+c 1) belong to the left ideal I if 1

Thus, a nonzero contribution to the image of the diagonal matrix element of the operator A corresponding to v under the homomorphism (11.6) only comes − from the term timim (u + m 1) ...ti1i1 (u). Taking the sum over all basis vectors (11.18) yields the desired formula for the image of the element (11.14). Remark . 11.1.3 As with the Yangian Y(glN ), we take the Harish-Chandra image provided by Theorem 11.1.2 as the definition of the Yangian character of the representation of Y(o2n+1)onL(m, 0,...,0) which extends the action of o2n+1.The existence of such an extension is explained by the fact that the projection S(m) is a product of evaluated R-matrices (11.13). This Yangian representation is known as a Kirillov–Reshetikhin module and the character provided by Theorem 11.1.2 coincides with the q-character under an appropriate identification of the parameters; see reviews covering all classical types in [99, Section 7] and [121, Section 2]. In particular, the relations (11.9) coincide with those for the q-characters, as the λi(u) correspond to the ‘single box variables’. Note also that all coefficients of all series T (m)(u) defined in (11.14) pairwise commute as it can be interpreted as a transfer matrix [98]; cf. Remark 10.1.5(ii). A direct proof of the commutativity property is given in [110]. (m) Type Dn. Consider the series T (u) given by (11.14), where the parameter N now takes the even value 2n. Theorem 11.1.4. The image of the series T (m)(u) under the homomorphism (11.6) is given by T (m) → − (u) λi1 (u) λi2 (u +1)...λim (u + m 1)

1i1···imN with the condition that n and n do not occur simultaneously among the summation indices i1,...,im. Proof. As in the proof of Theorem 11.1.2, we use the properties (11.15) of the symmetrizer S(m). Following the argument of that proof we identify the image (m) N ⊗m S (C ) with the space Hm of homogeneous harmonic polynomials of degree m in variables z1,...,zN by (11.17). This time the harmonic polynomials are annihilated by the Laplace operator of the form n ∂i ∂ i . i=1

The vector space Hm carries an irreducible representation of oN isomorphic to L(m, 0,...,0) whose dimension is found by the formula (11.16) with N =2n.The basis vectors of Hm will be parameterized by the N-tuples (k1,...,kn,ln,...,l1), where the ki and li are arbitrary nonnegative integers, the sum of all entries is m and at least one of kn and ln is zero. Given such a tuple, the corresponding harmonic polynomial is now defined by (11.21) ··· a +···+a − +k a +···+a − +l − a1+ +an−1 ··· 1 n 1 n 1 n 1 n ( 1) (a1 + + an−1)! zn zn a1! ...an−1!(a1 + ···+ an−1 + kn)! (a1 + ···+ an−1 + ln)! a1,...,an−1 − n1 ki−ai li−ai z z × i i , (k − a )! (l − a )! i=1 i i i i 11.1. YANGIAN FOR gN 197 summed over nonnegative integers a1,...,an−1 satisfying ai min{ki,li}.A unique leading monomial corresponds to the values a1 = ··· = an−1 =0.Itis easily verified that these polynomials are indeed all harmonic and linearly independent. The number of the polynomials equals N + m − 2 N + m − 3 2 − m m which coincides with the dimension dim L(m, 0,...,0) in (11.16). Hence the polynomials (11.21) form a basis of Hm. The argument is now completed in the same way as for Theorem 11.1.2 by considering the diagonal matrix elements of the operator A on right hand side of (11.19) for the basis vectors (11.21). These matrix elements are determined by the ⊗ ⊗ ··· coefficients of the tensors ei1 ... eim with i1 im corresponding to the leading monomials of v under the isomorphism (11.17). The tensors are given by the same formula (11.20) with N =2n and the condition that at least one of kn and ln is zero. A nonzero contribution to the image of the diagonal matrix element of the operator A corresponding to a given basis vector v under the homomorphism − (11.6) only comes from the term timim (u+m 1) ...ti1i1 (u) with the condition that the indices i1,...,im correspond to a leading monomial. This yields the desired formula for the image of the element (11.14).

(m) As with type Bn, the coeﬃcients of the series T (u)belongtoacommutative subalgebra of the Yangian Y(o2n); cf. Remark 11.1.3.

(m) Type Cn. Now let gN = spN with N =2n. The symmetrizer S is deﬁned in (11.12) with m n + 1. Introduce the formal series

(m) (m) (11.22) T (u)=tr1,...,m S T1(u) T2(u − 1) ...Tm(u − m +1), with the trace taken over all m copies of End C2n in (11.10). By Proposition 11.1.1, T (m) h all coeﬃcients of (u)belongtoY(sp2n) . Theorem 11.1.5. TheimageoftheseriesT (m)(u) with m n under the homomorphism (11.6) is given by T (m) → − − (11.23) (u) λi1 (u) λi2 (u 1) ...λim (u m +1)

1i1<···

(m) (m) (m) (m) (m) (11.24) S Qab = Qab S =0 and S Pab = Pab S = −S . As with the orthogonal case, the subspace of harmonic tensors in (CN )⊗m is spanned by the tensors v with the property Qab v = 0 for all 1 a

The subspace Hm is then identiﬁed with the subspace of homogeneous harmonic polynomials of degree m; they belong to the kernel of the skew Laplace operator n (11.26) ∂i ∧ ∂ i , i=1 where ∂i denotes the (left) partial derivative over ζi. The basis vectors of Hm will be parameterized by the subsets {i1,...,im} of the set {1,...,2n} satisfying the condition as stated in the theorem, assuming that the elements i1,...,im are written in the increasing order. We will call such subsets admissible and extend this deﬁnition to all values 1 m n +1.

Lemma 11.1.6. For m n +1 the number of admissible subsets {i1,...,im} of the set {1,...,2n} equals 2n 2n − . m m − 2 In particular, this number is zero for m = n+1 and it coincides with the dimension dim L(1,...,1, 0,...,0) in (11.25) for m n. Proof. We will construct a bijection π between the set of non-admissible subsets {i1,...,im} and the set of all subsets {j1,...,jm−2} of {1,...,2n} which will imply the desired formula. Our bijection will involve some multisets so it will be convenient to introduce some related notation. Let {i1,...,im} be a multiset with elements in {1,...,2n} with i1 i2 ··· im.If(q, q ) is a pair of indices such that q = ir and q = is for some 1 r

Now we will construct the desired bijection as a composition of certain transformations πq of multisets. The transformations πq are parameterized by elements q ∈{1,...,n− 1} and deﬁned on the multisets {i1,...,im} as follows. Consider all pairs (q, q ) occurring in a given multiset {i1,...,im} such that the distance between q and q is n − q. Note that such pairs are disjoint. The transformation πq replaces each of these pairs by the pair q +1, (q +1) so that q → q +1and q → (q +1).Ifnopair(q, q ) with that distance condition occurs, then πq is the identity map. Given a non-admissible subset {i1,...,im},let(p, p ) be its maximal breaking pair so that the distance between p and p is n−p. One veriﬁes that the application of the compositions of the transformations πq yields a multiset of the following form πn−1 ◦···◦πp+1 ◦ πp : {i1,...,im} →{j1,...,ja,n,n,ja+1,...,jm−2}, where (11.27) j1 < ···

π : {i1,...,im} →{j1,...,ja,ja+1,...,jm−2}. To show that π is a bijection, we construct the inverse map. Starting with an arbitrary subset {j1,...,jm−2} of {1,...,2n}, adjoin the entries n and n to make it into an m-multiset J satisfying (11.27). Deﬁne the map τ by setting τ {j1,...,jm−2} = τp+1 ◦ τp+2 ◦···◦τn (J), where the transformations τq and the parameter p are deﬁned as follows. Consider all pairs (q, q ) occurring in a given multiset {i1,...,im} such that the distance − between q and q is n q + 1. The transformation τq replaces each of these pairs by the pair q − 1, (q − 1) .Ifnopair(q, q) with this distance condition occurs then τq is the identity map. The parameter 1 p n is determined by the property that this is the maximal value of q ∈{1,...,n} such that τq ◦ τq+1 ◦···◦τn)(J)= τq+1 ◦ τq+2 ◦···◦τn)(J), where the right hand side is understood as equal to J if q = n. It is straightforward to verify that τ {j1,...,jm−2} is indeed a non-admissible subset of {1,...,2n} and that the compositions π ◦ τ and τ ◦ π are the identity maps. Example 11.1.7. To illustrate the bijection used in the proof of Lemma 11.1.6, take n = m = 7 and consider the non-admissible set {3, 4, 5, 6, 4, 3, 1}. Its unique breaking pair is (3, 3) which is therefore maximal. The transformations act by π3 : {3, 4, 5, 6 , 4 , 3 , 1 } →{4, 4, 5, 6 , 4 , 4 , 1 }, π4 : {4, 4, 5, 6 , 4 , 4 , 1 } →{5, 5, 5, 6 , 5 , 5 , 1 }, π5 : {5, 5, 5, 6 , 5 , 5 , 1 } →{5, 6, 6, 6 , 6 , 6 , 1 }, π6 : {5, 6, 6, 6 , 6 , 6 , 1 } →{5, 7, 7, 7 , 7 , 6 , 1 }. 200 11. YANGIAN CHARACTERS IN TYPES B, C AND D

~~Hence, for the image with respect to the map π we have π : {3, 4, 5, 6, 4, 3, 1} →{5, 7, 7, 6, 1}.~~

~~Clearly, the transformations τ7,τ6,τ5,τ4 are respectively inverse to π6,π5,π4,π3 and τ3 acts as the identity map on {3, 4, 5, 6 , 4 , 3 , 1 }.~~

~~To construct a basis of Hm, consider monomials of the form~~

(11.28) ζ ∧ ζ ∧···∧ζ ∧ ζ ∧ ζ ∧···∧ζ a1 a1 ak ak b1 bl with 1 a1 < ··· ai for all i. ∧···∧ Our condition on the parameters bi implies that the monomial ζb1 ζbl is annihilated by the skew Laplace operator (11.26). We denote this monomial by y and set xa = ζa ∧ ζa for a =1,...,n. The vector k (−1)p x ∧···∧x ∧···∧x ∧···∧x ∧ x ∧···∧x ∧ y, a1 ad1 adp ak cd1 cdp p=0 d1<···

~~κi(u)=λi(u), κ2n−i+3(u)=λ2n−i+1(u)fori =1,...,n, and~~

~~(11.30) κn+2(u)=−κn+1(u), −1 where κn+1(u) is the formal series in u with constant term 1 uniquely determined by the condition~~

(11.31) κn+1(u)κn+1(u − 1) = λn(u)λn (u − 1). Corollary 11.1.9. The image of the series (11.22) with m n under the homomorphism (11.6) can be written as T (m) → κ κ − κ − (11.32) (u) i1 (u) i2 (u 1) ... im (u m +1).

~~1i1<···~~

~~(11.30) and (11.31). Therefore, the sum in (11.32) is obtained by subtracting the expression~~

~~m−2 − (11.34) λj1 (u) ...λja (u a +1) a=0 1j1<···~~

× − − − − − − λn(u a)λn (u a 1)λja+1 (u a 2) ...λjm−2 (u m +1) from (11.33). Hence, the proof will be completed if we show that (11.34) coincides with the sum − − (11.35) λi1 (u) λi2 (u 1) ...λim (u m +1),

~~1i1<···~~

(11.36) λq(v) λq (v − n + q − 1) = λq+1(v) λ(q+1) (v − n + q − 1) for q =1,...,n− 1. So the products in (11.35) are preserved by the respective transformations πq used in the construction of the bijection π since the relation (11.36) is invoked precisely in the case where the distance between the entries q and q is n − q.

Remark 11.1.10. Since the relation (11.4) was not used in the proofs, the Harish-Chandra images provided by Theorems 11.1.2, 11.1.4 and 11.1.5 take the same form if the Yangian Y(gN ) is replaced by the extended Yangian X(gN ).

11.2. Dual Yangian for gN + Define the extended dual Yangian X (gN ) as the associative algebra with gen- (−r) erators tij ,where1 i, j N and r =1, 2,... subject to the defining relations written in a matrix form as follows. Combine the generators into the formal power series ∞ + − (−r) r−1 ∈ + tij(u)=δij tij u X (gN )[[u]] r=1 and introduce the matrix N + ⊗ + ∈ CN ⊗ + T (u)= eij tij(u) End X (gN )[[u]]. i,j=1 The defining relations are − + + + + − (11.37) R12(u v) T1 (u) T2 (v)=T2 (v) T1 (u) R12(u v), + whereweusetheR-matrix (11.1). As in (1.61), Ta (u)fora =1,...,m denotes the corresponding element of the tensor product algebra CN ⊗ ⊗ CN ⊗ + End ... End X (gN ). m 11.2. DUAL YANGIAN FOR gN 203

+ Similar to (10.26), define the descending filtrationonX (gN ) by setting the degree (−r) + of the generator tij with r 1tobeequaltor.WeletX (gN )denotethe + completion of X (gN ) with respect to this filtration. + + The dual Yangian Y (gN ) is now defined as the quotient of the algebra X (gN ) by the relations T + (u + κ) T +(u)=1, where the prime denotes the matrix transposition defined in (2.24). + (−r) − Now consider the ascending filtration on Y (gN ) defined by deg tij = r for all r 1. A version of the Poincaré–Birkhoff–Witt theorem can be proved + for Y (gN ) by using the approach of [7] which implies the isomorphism for the associated graded algebra + ∼ −1 −1 gr Y (gN ) = U t gN [t ] . ¯(−r) (−r) − The image tij of the generator tij in the ( r)-th component of the graded + −1 −1 algebra gr Y (gN ) corresponds to the element Fij[−r]ofU t gN [t ] . + We will regard Y (gN ) as a module over the Cartan subalgebra h of gN ,where each basis element Fii of h with i =1,...,n acts as a derivation and the action is defined on the generators by + + + + + · − − Fii tkl(u)=δki til (u) δil tki(u) δki til(u)+δil tki (u). + h Denote by Y (gN ) the subalgebra of h-invariants under this action. Consider + (−r) the left ideal J of the algebra Y (gN ) generated by all elements tij with the + h conditions N i>j 1andr 1. The quotient of Y (gN ) by the two-sided + h ideal Y (gN ) ∩J is isomorphic to the commutative algebra freely generated by the (−r) images of the elements tii with i =1,...,n and r 1 in the quotient. We will (−r) (−r) use the notation λi for this image of tii and extend this notation to all values i =1,...,N. An analogue of the Harish-Chandra homomorphism (11.6) now takes the form + h → C (−r) | (11.38) Y (gN ) [λi i =1,...,n, r 1], C (−r) where [λi ] is the completion of the algebra of polynomials with respect to the (−r) gradation defined by setting the degree of λi to be equal to r. We combine the (−r) elements λi into the formal series ∞ + − (−r) r−1 λi (u)=1 λi u ,i=1,...,N, r=1 + which will be understood as the images of the series tii (u) under the homomorphism (11.38). We will work with the tensor product algebra CN ⊗ ⊗ CN ⊗ + (11.39) End ... End Y (gN ) m to state the dual Yangian analogues of the results of Section 11.1. These analogues follow in uniform way in all three cases as we will point out below. 204 11. YANGIAN CHARACTERS IN TYPES B, C AND D

(m) Type Bn. Consider the element S deﬁned by (11.11) and introduce the + +(m) formal series T (u) with coeﬃcients in the dual Yangian Y (oN )withodd N =2n +1bytheformula T +(m) (m) + + + − (11.40) (u)=tr1,...,m S T1 (u) T2 (u +1)...Tm (u + m 1) with the trace taken over all m copies of End CN in (11.39). Theorem 11.2.1. The image of the series T +(m)(u) under the homomorphism (11.38) is found by T +(m)(u) → λ+ (u) λ+ (u +1)...λ+ (u + m − 1) i1 i2 im Ni1···im1 with the condition that n+1 occurs among the summation indices i1,...,im at most once.

+(m) Type Dn. Deﬁne the series T (u) by the same formula (11.40), where the parameter N now takes an even value 2n. Theorem 11.2.2. The image of the series T +(m)(u) under the homomorphism (11.38) is given by T +(m)(u) → λ+ (u) λ+ (u +1)...λ+ (u + m − 1) i1 i2 im Ni1···im1 with the condition that n and n do not occur simultaneously among the summation indices i1,...,im.

Type Cn. Set T +(m) (m) + + − + − (11.41) (u)=tr1,...,m S T1 (u) T2 (u 1) ...Tm (u m +1), CN with the trace taken over all m copies of End in (11.39) with gN = spN and S(m) deﬁned in (11.12) with m n. Theorem 11.2.3. TheimageoftheseriesT +(m)(u) with m n under the homomorphism (11.38) is given by T +(m)(u) → λ+ (u) λ+ (u − 1) ...λ+ (u − m +1) i1 i2 im 2ni1>···>im1 with the condition that if for any i both i and i occur among the summation indices as i = ir and i = is for some 1 r

Corollary 11.2.4. The image of the series (11.41) with m n under the homomorphism (11.38) can be written as (11.43) T +(m)(u) → κ+(u) κ+(u − 1) ...κ+ (u − m +1). i1 i2 im 2n+2i1>···>im1 Moreover, the sum in (11.43) is zero for m = n +1. κ+ κ+ Note that the formal parameters n+1(u)and n+2(u) are eliminated from the formula for the image in (11.43) as in the proof of Corollary 11.1.9. Namely, due to (11.42), the terms corresponding to two sets of summation indices which differ only at one index with ik = n +2andik = n +1forsomek ∈{1,...,m} cancel with each other, whereas the term corresponding to a set of indices with ik = n +2 + and ik+1 = n + 1 is expressed in terms of the λi (u + a). Theorems 11.2.1, 11.2.2 and 11.2.3 follow from their respective counterparts which we stated in Section 11.1. Namely, the mapping (11.44) θ : T (u) → T t(−u), where t : eij → eji is the standard transposition, defines an automorphism of the extended Yangian X(gN ). This follows by the application of the transposition t1t2 to both sides of (11.3) with the use of Lemma 2.2.1 and the following properties of the R-matrix (11.1): − Rt1t2 (u)=R(u)andR(u)R(−u)=1− u 2. Hence, applying the automorphism (11.44), we obtain the image of θ T (m)(u) under the Harish-Chandra-type homomorphism, where instead of I we use the left (r) ideal of the extended Yangian generated by all elements tij with the conditions N i>j 1andr 1; see Remark 11.1.10. Since the defining relations take the same matrix form for the extended Yangian and extended dual Yangian, we may conclude that in the orthogonal case the image of the series (m) + t − + t − − + t − − (11.45) tr1,...,m S T1 ( u) T2 ( u 1) ...Tm ( u m +1) under the homomorphism (11.38) equals λ+ (−u) λ+ (−u − 1) ...λ+ (−u − m +1) i1 i2 im 1i1···imN with the respective additional conditions on the summation as in Theorems 11.1.2 and 11.1.4. Now apply the transposition t1t2 ...tm to the expression under the trace in (11.45) and observe that it leaves S(m) unchanged. The latter property holds since the application of this transposition to the right hand sides of (11.11) and (11.12) will result in the products written in the opposite order. However, the factors are the values of the R-matrix (11.1) so that the products coincide with S(m) by the Yang–Baxter equation (10.6). Thus, (11.45) equals + − + − − (m) tr1,...,m T1 ( u) ...Tm ( u m +1)S + − + − − (m) =tr1,...,m Tm ( u) ...T1 ( u m +1)S (m) + − − + − T +(m) − − =tr1,...,m S T1 ( u m +1)...Tm ( u)= ( u m +1), 206 11. YANGIAN CHARACTERS IN TYPES B, C AND D where we used the conjugation by the permutation P1 m P2 m−1 ... and repeated application of (11.37). Theorems 11.2.1 and 11.2.2 now follow by the replacement of u with −u − m +1. Theorem 11.2.3 and Corollary 11.2.4 are derived by the same argument from Theorem 11.1.5 and Corollary 11.1.9 with the use of (11.12). Remark 11.2.5. The coefficients of all series T +(m)(u) pairwise commute as elements of the dual Yangian. This can be shown with the use of the quantum vertex algebra structure associated with the corresponding double Yangian; see [37]. Moreover, this also implies that the coefficients of all series T (m)(u) pairwise commute as elements of the Yangian Y(gN ); cf. Remark 10.4.2.

11.3. Screening operators We will show that the Harish-Chandra images of the series T (m)(u)belongto the intersection of the kernels of the screening operators and so they coincide with the characters of Yangian representations; cf. Section 10.6. Suppressing the variable u from the notation λi(u + a), consider the algebra of polynomials in variables λi(a) with i =1,...,N and a ∈ C and denote by L = L(gN ) its quotient by the relations

(11.46) λi(a + κ − i) λi (a)=λi+1(a + κ − i) λ(i+1) (a),a∈ C, − for i =0, 1,...,n 1ifgN = o2n or sp2n,andfori =0, 1,...,n if gN = o2n+1, where λ0(a)=λ0 (a) = 1; cf. (11.9). For i =1,...,n consider the free left L-module Li with the generators σi(a), where a runs over C and denote by Li its quotient by the relations

~~λi(a) σi(a)=λi+1(a) σi(a +1),i=1,...,n− 1,a∈ C, together with~~

~~λn(a) σn(a)=λn+1(a) σn(a +1/2) for gN = o2n+1,~~

~~(11.47) λn(a) σn(a)=λn+1(a) σn(a +2) for gN = sp2n,~~

λn−1(a) σn(a)=λn+1(a) σn(a +1) for gN = o2n. For every i ∈{1,...,n} deﬁne a linear operator Si : L→Li satisfying the Leibniz rule (10.46). For i =1,...,n− 1set ⎧ ⎪ ⎪λi(a) σi(a)forj = i ⎪ ⎨⎪−λi+1(a) σi(a +1) for j = i +1 → − − Si : λj(a) ⎪ λi (a) σi(a + κ i +1) for j = i ⎪ ⎪λ(i+1) (a) σi(a + κ − i)forj =(i +1) ⎩⎪ 0forj = i, i,i+1, (i +1). The action of Sn depends on the type and is given as follows. Case gN = o2n+1. Sn : λj (a) → 0ifjn and λn(a) → λn(a) σn(a)+σn(a − 1/2) λn+1(a) → λn+1(a) σn(a − 1/2) − σn(a +1/2) λn (a) →−λn (a) σn(a)+σn(a +1/2) . 11.3. SCREENING OPERATORS 207

~~→ Case gN = sp2n. Sn : λj (a) 0ifjn and~~

~~λn(a) → λn(a) σn(a)~~

~~λn (a) →−λn (a) σn(a +2). Case gN = o2n. Sn : λj (a) → 0ifj(n − 1) and~~

~~λn−1(a) → λn−1(a) σn(a)~~

~~λn(a) → λn(a) σn(a)~~

~~λn (a) →−λn (a) σn(a +1)~~

~~λ(n−1) (a) →−λ(n−1) (a) σn(a +1). Relations (11.46) are easily seen to be preserved by the action of the Si so that the operators on L are well-deﬁned. The i-th screening operator~~

Si : L→Li is now deﬁned as the composition of Si and the projection Li →Li. Deﬁne the subalgebra Rep Y(gN ) of Yangian characters in L as the intersection of the kernels of the screening operators: n Rep Y(gN )= ker Si. i=1

~~Following Theorem 11.1.2, for gN = o2n+1 deﬁne the polynomial χm ∈Lby − χm = λi1 (0) λi2 (1) ...λim (m 1)~~

1i1···imN with the condition that n + 1 occurs among the summation indices i1,...,im at most once. Similarly, taking Theorem 11.1.4 into account, for gN = o2n deﬁne the polynomial χm ∈Lby − χm = λi1 (0) λi2 (1) ...λim (m 1)

1i1···imN with the condition that n and n do not occur simultaneously among the summation indices i1,...,im. Following Theorem 11.1.5, for gN = sp2n deﬁne the polynomial ∈L χm by − − χm = λi1 (0) λi2 ( 1) ...λim ( m +1)

~~1i1<···~~

~~This formula deﬁnes χm for all values m =1, 2,...,2n +2.~~

~~Proposition 11.3.1. In all three cases we have χm ∈ Rep Y(gN ). 208 11. YANGIAN CHARACTERS IN TYPES B, C AND D~~

Proof. A calculation quite analogous to the one used in the proof of Propo- sition 10.6.1 (see also Remark 10.6.2) shows that χm belongs to the kernel of the screening operator Si for each i =1,...,n− 1. To verify this property for Sn we will consider the three cases separately.

~~Case gN = o2n+1. Using the notation of Remark 10.6.2, combine the polynomials χm into the generating series~~

~~∞ m∂ ∂ −1 ∂ −1 1+ χm e = 1 − λ1(0)e ... 1 − λn(0)e m=1 ∂ ∂ −1 ∂ −1 × 1+λn+1(0)e 1 − λn (0)e ... 1 − λ1 (0)e .~~

It suﬃces to verify the desired property for the inverse series so that we need to see that the product ∂ ∂ −1 ∂ (11.48) 1 − λn (0)e 1+λn+1(0)e 1 − λn(0)e is annihilated by Sn.Wehave ∂ −1 Sn : 1+λn+1(0)e ∂ −1 ∂ −1 → σn(−1/2) 1+λn+1(0)e − 1+λn+1(0)e σn(−1/2).

Hence the image of (11.48) under Sn is ∂ ∂ −1 ∂ λn (0) σn(0) + σn(1/2) e 1+λn+1(0)e 1 − λn(0)e ∂ ∂ −1 + 1 − λn (0)e σn(−1/2) 1+λn+1(0)e ∂ −1 ∂ − 1+λn+1(0)e σn(−1/2) 1 − λn(0)e ∂ ∂ −1 ∂ − 1 − λn (0)e 1+λn+1(0)e λn(0) σn(0) + σn(−1/2) e which simpliﬁes to ∂ ∂ −1 ∂ λn (0)σn(0)e + σn(−1/2) 1+λn+1(0)e 1 − λn(0)e ∂ ∂ −1 ∂ − 1 − λn (0)e 1+λn+1(0)e λn(0)σn(0)e + σn(−1/2) .

~~Using (11.46) for i = n and (11.47) write~~

~~(11.49) λn (0)σn(0) = λn+1(0)σn(−1/2) and λn(0)σn(0) = λn+1(0)σn(1/2) to simplify the image further to ∂ ∂ σn(−1/2) 1 − λn(0)e − 1 − λn (0)e σn(−1/2).~~

~~This is zero due to the relation λn(0)σn(−1/2) = λn (0)σn(1/2) implied by (11.49). 11.3. SCREENING OPERATORS 209~~

Case gN = o2n. We have the generating series ∞ m∂ ∂ −1 ∂ −1 1+ χm e = 1 − λ1(0)e ... 1 − λn−1(0)e m=1 ∂ −1 ∂ −1 ∂ −1 ∂ −1 × 1−λn(0)e + 1−λn (0)e −1 1−λ(n−1) (0)e ... 1−λ1 (0)e which can also be written as ∂ −1 ∂ −1 1 − λ1(0)e ... 1 − λn(0)e ∂ ∂ ∂ −1 ∂ −1 × 1 − λn(0)e λn (0)e 1 − λn (0)e ... 1 − λ1 (0)e . By taking the inverse series, we come to verifying that the product ∂ ∂ 1 − λ(n−1) (0)e 1 − λn (0)e ∂ ∂ −1 ∂ ∂ × 1 − λn(0)e λn (0)e 1 − λn(0)e 1 − λn−1(0)e is annihilated by Sn. This follows easily by using ∂ ∂ −1 Sn : 1 − λn(0)e λn (0)e ∂ ∂ −1 ∂ ∂ −1 → σn(0) 1 − λn(0)e λn (0)e − 1 − λn(0)e λn (0)e σn(0) and the relations

~~λ(n−1) (0)σn(1) = λn(0)σn(0) and λn−1(0)σn(0) = λn (0)σn(1).~~

Case gN = sp2n. We have 2n+2 −m∂ −∂ −∂ 1+ χm e = 1+κ1(0)e ... 1+κ2n+2(0)e . m=1 It suﬃces to verify that the product −∂ −∂ −∂ −∂ 1+κn(0)e 1+κn+1(0)e 1+κn+2(0)e 1+κn+3(0)e −∂ −∂ −∂ −∂ = 1+λn(0)e 1 − λn(0)e λn (0)e 1+λn (0)e is annihilated by Sn. This follows by an easy calculation with the use of the relation

λn(0)σn(0) = λn (0)σn(2) thus completing the proof. To state the counterpart of Proposition 11.3.1 for the dual Yangian (see Proposi- tion 11.3.2 below), note that the Harish-Chandra images calculated in Sections 11.1 and 11.2 can be obtained from each other by the replacements ←→ + λi(u) λi (u),i=1,...,N. Therefore, the above deﬁnitions and properties of the screening operators associated with the Yangian Y(gN ) are easily carried over to the dual Yangian by replacing the variables as follows: → + λi(a) λi (a),i=1,...,N, and → + − σi(a) σi (a κ + i),i=1,...,n, 210 11. YANGIAN CHARACTERS IN TYPES B, C AND D

∈ C + for all a . Now we consider the algebra of polynomials in variables λi (a) with + + i =1,...,N and a ∈ C and denote by L = L (gN ) its quotient by the relations + + − + + − ∈ C (11.50) λi (a) λi (a + κ i)=λi+1(a) λ(i+1) (a + κ i),a, − for i =0, 1,...,n 1ifgN = o2n or sp2n,andfori =0, 1,...,n if gN = o2n+1, + + where λ0 (a)=λ0 (a) = 1; cf. (11.46). L+ L+ + For i =1,...,nconsider the free left -module i with the generators σi (a), C L+ where a runs over and denote by i its quotient by the relations + + + + − ∈ C (11.51) λi (a) σi (a +1)=λi+1(a) σi (a),i=1,...,n 1,a , together with1 + + + + λn (a) σn (a +1/2) = λn+1(a) σn (a)forgN = o2n+1, + + + + − (11.52) λn (a) σn (a +1)=λn+1(a) σn (a 1) for gN = sp2n, + + + + λn−1(a) σn (a +2)=λn+1(a) σn (a +1) for gN = o2n. ∈{ } + L+ → L+ For every i 1,...,n deﬁne a linear operator Si : i satisfying the Leibniz rule (10.52). For i =1,...,n− 1set ⎧ + + ⎪−λ (a) σ (a +1) for j = i ⎪ i i ⎪ + + ⎨⎪λi+1(a) σi (a)forj = i +1 S+ : λ+(a) → + + − i j ⎪λi (a) σi (a κ + i)forj = i ⎪ ⎪− + + − ⎪ λ (a) σ (a κ + i +1) for j =(i +1) ⎩⎪ (i+1) i 0forj = i, i,i+1, (i +1).

+ The action of Sn is deﬁned separately for each type. + + → Case gN = o2n+1. Sn : λj (a) 0ifjn and + →− + + + λn (a) λn (a) σn (a +1/2) + σn (a +1) + → + + − + λn+1(a) λn+1(a) σn (a) σn (a +1) + → + + + λn (a) λn (a) σn (a)+σn (a +1/2) . + + → Case gN = sp2n. Sn : λj (a) 0ifjn and + →− + + λn (a) λn (a) σn (a +1) + → + + − λn (a) λn (a) σn (a 1). + + → − − Case gN = o2n. Sn : λj (a) 0ifj(n 1) and + →− + + λn−1(a) λn−1(a) σn (a +2) + →− + + λn (a) λn (a) σn (a +2) + → + + λn (a) λn (a) σn (a +1) + → + + λ(n−1) (a) λ(n−1) (a) σn (a +1).

1 + + One could replace σn (a) → σn (a ± 1) for types Cn and Dn, respectively, to make the conditions analogous to (11.47). This would not aﬀect the properties which we discuss below. 11.4. BIBLIOGRAPHICAL NOTES 211

+ L+ One easily verifies that the action of the operators Si on is well-defined. The i-th screening operator + L+ →L+ Si : i + L+ →L+ is now defined as the composition of Si and the projection i i . Define + + the subalgebra Rep Y (gN ) of Yangian characters in L as the intersection of the kernels of the screening operators: n + + (11.53) Rep Y (gN )= ker Si . i=1 Following Theorem 11.2.1, for g = o define the polynomial χ+ ∈L+ by N 2n+1 m χ+ = λ+ (0) λ+ (1) ...λ+ (m − 1) m i1 i2 im Ni1···im1 with the condition that n + 1 occurs among the summation indices i1,...,im at most once. Taking Theorem 11.2.2 into account, for gN = o2n define the polynomial + ∈L+ χm by χ+ = λ+ (0) λ+ (1) ...λ+ (m − 1) m i1 i2 im Ni1···im1 with the condition that n and n do not occur simultaneously among the summation indices i1,...,im. Following Theorem 11.2.3, for gN = sp2n define the polynomial + ∈L χm by χ+ = λ+ (0) λ+ (−1) ...λ+ (−m +1) m i1 i2 im 2ni1>···>im1 with the condition that if for any i both i and i occur among the summation indices as i = ir and i = is for some 1 r~~···>im1 + This allows us to extend the definition of χm to all values m =1, 2,...,2n +2. + Proposition . + ∈ 11.3.2 In all three cases we have χm Rep Y (gN ). 11.4. Bibliographical notes~~

The RT T presentation of the Yangian Y(gN ) and its representations were studied in the papers by Arnaudon et al. [6], [7]andGuayet al. [61], [62]. Explicit isomorphisms between various presentations of Y(gN ) were constructed recently in [63]and[83]. In Section 11.1 we followed [111]. The proofs of Lemma 11.1.6 and Corollary 11.1.9 are taken from the paper by Kuniba, Okado, Suzuki and Yamada [100]. The screening operators originate in the paper by Frenkel and Reshetikhin [51]; see also Frenkel and Mukhin [48]. We followed [113] for explicit formulas in the Yangian context.

~~CHAPTER 12~~

~~Classical W-algebras~~

Classical W-algebras W(g,f) are commutative algebras equipped with Poisson brackets and derivations. The word ‘classical’ refers to the fact that they are ‘classical limits’ of a certain family of vertex algebras known as the affine W-algebras so that these limits possess the structures of Poisson vertex algebras. The algebras W(g,f) are associated with finite-dimensional simple Lie algebras g over C and nilpotent elements f ∈ g. For the applications below, we will only be interested in the case where f is a regular (principal) nilpotent and therefore suppress f from the notation by denoting the algebra by W(g). Recall that the center Z(g) of the universal enveloping algebra is isomorphic to the subalgebra U(h)W of W -invariants in U(h); see (4.7). An affine version of the Harish-Chandra isomorphism between the center z(g) of the affine vertex algebra at the critical level (as defined in Chapter 6) and the classical W-algebra W(Lg) will be discussed in Chapter 13 below (by Lg we denote the Langlands dual Lie algebra of g). This chapter is meant to review some properties of the algebras W(g) and construct their generators. We begin with a description of the classical W-algebras in the framework of Poisson vertex algebras following De Sole, Kac and Valeri [26], [27], [28] and Kac [88] (see Sections 12.1 and 12.2 below), then give their definition via screening operators and show the equivalence of the definitions via the Chevalley isomorphism (Section 12.3).

12.1. Poisson vertex algebras We will use the term differential algebra for a unital commutative associative algebra V equipped with a derivation ∂. Recall also Lie conformal algebras as introduced in Definition 6.1.5. Definition 12.1.1. A Poisson vertex algebra is a differential algebra V =(V,∂) endowed with a C-linear map

~~V⊗V→C[λ] ⊗V,a⊗ b →{aλb}, called the λ-bracket, such that (V,∂,{λ}) is a Lie conformal algebra and the Leibniz rule~~

~~(12.1) {aλbc} = {aλb}c + b{aλc} holds for all a, b, c ∈V.~~

Suppose that g is a simple Lie algebra and let J1,...,Jd be a basis of g.Consider the diﬀerential algebra V(g) which is deﬁned as the algebra of polynomials V C (r) (r) | (0) (12.2) (g)= [J1 ,...,Jd r =0, 1, 2,...] with Ji = Ji,

~~213 214 12. CLASSICAL W-ALGEBRAS~~

(r) (r+1) equipped with the derivation ∂ deﬁned by ∂ (Ji )=Ji for all i =1,...,d and r 0. Fix a non-degenerate symmetric invariant bilinear form , on g and introduce the λ-bracket on V(g) by setting {X Y } =[X, Y ]+λX, Y for X, Y ∈ g, λ and extending to V(g) by sesquilinearity a, b ∈V(g) :

{∂aλ b} = − λ {a λ b}, {a λ ∂b} =(λ + ∂){a λ b}, skewsymmetry {a λ b} = −{b −λ−∂ a}, and the Leibniz rule (12.1); cf. Definition 6.1.5. One verifies that this λ-bracket equips the differential algebra (V(g),∂) with a Poisson vertex algebra structure and V(g)isknownastheaffine Poisson vertex algebra (associated with g). Remark 12.1.2. The affine Poisson vertex algebra V(g)canbeshowntobeob- tained from the affine vertex algebra Vκ(g) defined in Section 6.2 via a quasiclassical limit;see[88].

Introduce the standard Chevalley generators ei,hi,fi with i =1,...,n of the simple Lie algebra g of rank n. The linear span h of the generators h1,...,hn is a Cartan subalgebra of g, while the ei and fi generate the respective nilpotent subalgebras n+ and n−.LetA =[aij] be the Cartan matrix of g so that the deﬁning relations of g take the form

~~[ei,fj ]=δijhi, [hi,hj]=0,~~

~~[hi,ej ]=aij ej , [hi,fj ]=−aij fj , together with the Serre relations~~

~~1−aij 1−aij (ad ei) ej =0, (ad fi) fj =0,i= j. There exists a diagonal matrix~~

~~(12.3) D =diag[1,...,n] with positive rational entries such that the matrix B = D−1A is symmetric. Nor- malize the symmetric invariant bilinear form on g so that~~

~~(12.4) ei,fj = δij i. The element f ∈ g deﬁned by~~

f = f1 + ···+ fn is a regular nilpotent element of g. Introduce an sl2-subalgebra of g with the standard basis elements e, f, h satisfying the commutation relations (6.12) and set x = h/2. Consider the ad x-eigenspace decomposition of g, (12.5) g = g(i), g(i) = {Y ∈ g | [x, Y ]=iY}. i∈Z As with (9.19), the subspace g(i) is zero unless i is within the region determined (−1) (1) (0) by the Coxeter number: −hg 0 12.1. POISSON VERTEX ALGEBRAS 215

Set p = n− ⊕ h and introduce the projection map πp : g → p with the kernel n+. We will extend the notation (12.2) to an arbitrary subspace a ⊂ g so that V(a) will denote the corresponding differential subalgebra of V(g). Elements of V(a)are polynomials in the variables Y (r) with Y ∈ a and r 0. Define the differential algebra homomorphism ρ : V(g) →V(p) by setting

~~(12.6) ρ(X)=πp(X)+f,X,X∈ g. Definition 12.1.3. The classical W-algebra W(g) is deﬁned by~~

W(g)={P ∈V(p) | ρ{X λ P } =0 forall X ∈ n+}. Theorem 12.1.4. The subspace W(g) ⊂V(p) is a diﬀerential subalgebra of V(p). Moreover, W(g) is a Poisson vertex algebra equipped with the λ-bracket

{a λ b}ρ = ρ{a λ b},a,b∈W(g). To describe the structure of the classical W-algebra W(g), choose a subspace U ⊂ g complementary to [f,g], (12.7) g =[f,g] ⊕ U, which is compatible with the grading (12.5) in the sense that U is the direct sum of the subspaces U ∩ g(i).Since (12.8) g =[f,g] ⊕ ge, a possible universal choice is U = ge, the centralizer of e in g. Observe that the map (i) (i−1) adf : g → g is surjective for i 0 so that we have the inclusion n− ⊂ [f,g]. Therefore, for any choice of U we have U ⊂ h ⊕ n+. Since the form (12.4) is invariant and nondegenerate, the orthogonal complement to [f,g]ing coincides with the centralizer gf of f in g. Hence, by taking the orthogonal complements in the decomposition (12.7) we get f ⊕ ⊥ g = g Ug . Note that the centralizer gf is contained in p so that this implies the decomposition p = gf ⊕ U ⊥, whereweset ⊥ ∩ ⊥ { ∈ | } (12.9) U = p Ug = v p v, U =0 . This induces another direct sum decomposition, (12.10) V(p)=V(gf ) ⊕U ⊥, where U ⊥ denotes the differential algebra ideal of V(p) generated by U ⊥.Welet π = πgf denote the projection π : V(p) →V(gf ) to the first summand in (12.10) with the kernel U ⊥. Theorem 12.1.5. The restriction of π to the classical W-algebra W(g) ⊂V(p) defines a differential algebra isomorphism (12.11) π : W(g) →V(gf ). 216 12. CLASSICAL W-ALGEBRAS

~~Theorem 12.1.5 implies that for any element q ∈ gf there exists a unique element w(q) ∈W(g)oftheformw(q)=q + r with r ∈U ⊥.~~

12.2. Generators of W(g) We will need some general determinant formulas to be used for all classical types. Let A be a square matrix of the form ⎡ ⎤ a11 a12 00... 0 ⎢ ⎥ ⎢ a21 a22 a23 0 ... 0 ⎥ ⎢ ⎥ ⎢ a31 a32 a33 a34 ... 0 ⎥ (12.12) A = ⎢ ⎥ ⎢ ...... ⎥ ⎣ ⎦ aN−11 aN−12 aN−13 ...... aN−1 N aN1 aN2 aN3 ...... aNN with entries in a ring. If the entries a12, a23,...,aN−1 N belong to the center of the ring then one easily veriﬁes that the column-determinant of A coincides with its row-determinant and we will set (12.13) det A = sgn σ · aσ(1) 1 ...aσ(N) N = sgn σ · a1 σ(1) ...aNσ(N).

σ∈SN σ∈SN In particular, if the ring contains the identity and the matrix A has the form ⎡ ⎤ a11 100... 0 ⎢ ⎥ ⎢ a21 a22 10... 0 ⎥ ⎢ ⎥ ⎢ a31 a32 a33 1 ... 0 ⎥ (12.14) A = ⎢ ⎥ ⎢ ...... ⎥ ⎣ ⎦ aN−11 aN−12 aN−13 ...... 1 aN1 aN2 aN3 ...... aNN then N−1 (12.15) det A = (−1)N−k−1 a a ...a . i1 i0+1 i2 i1+1 ik+1 ik+1 k=0 0=i0

Returning to more general matrices (12.12), denote by Di (respectively, Di) the determinant of the i × i submatrix of A corresponding to the ﬁrst (respectively, last) i rows and columns. We suppose that D0 = D0 =1. Lemma 12.2.1. Fix p ∈{0, 1,...,N}. Then for the determinant of the matrix (12.12) we have p N − j+i det A = Dp DN−p + ( 1) Dj−1 aij DN−i, j=1 i=p+1 where aij = aij ajj+1 aj+1 j+2 ...ai−1 i for i>j. Proof. The formula follows easily from the deﬁnition of the determinant (12.13). In the applications which we consider below, the central elements a12, a23,...,aN−1 N turn out to be invertible. Then the lemma is reduced to the particular case of matrices (12.14) and it is immediate from the explicit formula (12.15). 12.2. GENERATORS OF W(g) 217

Type AN−1. It will be convenient to work with the reductive Lie algebra g = glN instead of the simple Lie algebra slN in type A. We will use its standard basis elements Eij, i, j =1,...,N. The elements E11,...,ENN span a Cartan subalgebra of glN which we denote by h. The respective subsets of basis elements Eij with ijspan the nilpotent subalgebras n+ and n−. The subalgebra p = n− ⊕ h is then spanned by the elements Eij with i j. Take the principal nilpotent element f in the form

f = E21 + E32 + ···+ ENN−1 and complete it to the sl2-triple {e, f, h} by N−1 N e = i(N − i) Eii+1 and h = (N − 2i +1)Eii. i=1 i=1 W W The classical -algebra (glN ) is introduced by Definition 12.1.3, and Theo- rems 12.1.4 and 12.1.5 hold for the reductive Lie algebra glN in the same form. We will be working with the algebra of differential operators V(p)⊗C[∂], where the commutation relations are given by (r) − (r) (r+1) ∂Eij Eij ∂ = Eij . In other words, ∂ will be regarded as a generator of this algebra rather than the derivation on V(p). For any element g ∈V(p) and any nonnegative integer r the element g(r) = ∂ r(g) coincides with the constant term of the differential operator ∂ rg so that (12.16) g(r) = ∂ rg 1, assuming that ∂ 1=0. As in Example 9.1.7, the nondegenerate invariant symmetric bilinear form on glN is defined by ∈ X, Y =trXY, X,Y glN , where X and Y are understood as N × N matrices over C. Consider the determinant (12.13) of the matrix with entries in V(p) ⊗ C[∂], ⎡ ⎤ ∂ + E11 100... 0 ⎢ ⎥ ⎢ E21 ∂ + E22 10... 0 ⎥ ⎢ ⎥ ⎢ E31 E32 ∂ + E33 1 ... 0 ⎥ (12.17) det ⎢ ⎥ ⎢ ...... ⎥ ⎣ ⎦ EN−11 EN−12 EN−13 ...... 1 EN1 EN2 EN3 ...... ∂+ ENN and write it as a differential operator N N−1 ∂ + w1 ∂ + ···+ wN ,wi ∈V(p).

Theorem 12.2.2. All elements w1,...,wN belong to the classical W-algebra W (r) (r) (glN ). Moreover, the elements w1 ,...,wN with r =0, 1,... are algebraically W independent and generate the algebra (glN ).

~~Proof. Denote the determinant in (12.17) by DN .Foreach1 k~~

~~(12.19) DN = DN−1 (∂ + ENN) − DN−2 ENN−1 + DN−3 ENN−2~~

N−2 N−1 + ···+(−1) D1 EN 2 +(−1) D0 EN 1. Hence, using the properties of the λ-bracket we get { } − N−i+1 + − N−i + ρ Eii+1 λDN =( 1) Di−1 ENi+1 +( 1) Di−1 ENi+1 =0 for all i =1,...,N − 2. Furthermore,

~~{ } + − + − + − ρ EN−1 N λDN = DN−1 DN−2 (∂ + ENN) DN−2 (EN−1 N−1 ENN + λ) + ··· − N−2 + − N−1 + + DN−3 EN−1 N−2 + +( 1) D1 EN−12 +( 1) D0 EN−11~~

+ which is zero due to relation (12.19) applied to the determinant DN−1 instead of { } − DN .Sinceρ Eii+1 λDN =0foralli =1,...,N 1, we may conclude that { } ∈ ρ X λDN =0forallX n+ so that all elements w1,...,wN belong to the W V subalgebra (glN )of (p). To verify the second claim, apply Theorem 12.1.5. We can take

~~U =spanof {E11,E12,...,E1N } so that ⊥ U =spanof {Eij | N i j>1}.~~

To calculate the images of the elements w1,...,wN under the isomorphism (12.11) write them first as differential polynomials in the variables E11,E21,...,EN1 modulo the differential algebra ideal of V(p) generated by U ⊥. This amounts to replacing all the elements Eij with N i j>1 in the determinant (12.17) by zero. By (12.15) the reduced determinant equals

~~N−1 N−1 N−1 (∂ + E11)∂ − E21 ∂ + ···+(−1) EN1~~

m−1 ⊥ so that wm ≡ (−1) Em 1 mod U . Hence, for the images with respect to the isomorphism (12.11) we get m−1 π : wm → (−1) Em 1 + Em+1 2 + ···+ ENN−m+1 for m =1,...,N. Since the images π(w1),...,π(wN ) form a basis of the centralizer f glN , the second claim follows. 12.2. GENERATORS OF W(g) 219

Type Bn. We will use the same presentation of the classical Lie algebras in types B, C and D as in Sections 2.2 and 5.1. The elements F11,...,Fnn span a Cartan subalgebra of oN for N =2n or N =2n + 1, which we denote by h. The respective subsets of elements Fij with ijspan the nilpotent subalgebras n+ and n−. The subalgebra p = n− ⊕h is then spanned by the elements Fij with i j. We will be working with the algebra of diﬀerential operators V(p)⊗C[∂], where the commutation relations are given by (r) − (r) (r+1) (12.20) ∂Fij Fij ∂ = Fij . For any element g ∈V(p) and any nonnegative integer r the element g(r) coincides with the constant term of the diﬀerential operator ∂ rg as in (12.16). Take the principal nilpotent element f ∈ o2n+1 in the form

~~f = F21 + F32 + ···+ Fn+1 n.~~

~~The sl2-triple is now formed by the elements {e, f, h} with n n (12.21) e = i(2n − i +1)Fii+1 and h =2 (n − i +1)Fii. i=1 i=1~~

The nondegenerate invariant symmetric bilinear form on o2n+1 is now deﬁned by 1 X, Y = tr XY, X,Y ∈ o , 2 2n+1 where X and Y are understood as matrices over C which are skew-symmetric with respect to the antidiagonal. Consider the determinant (12.13) of the matrix with entries in V(p) ⊗ C[∂], ⎡ ⎤ ∂ + F11 1 ... 00 0... 0 ⎢ ⎥ ⎢ F21 ∂ + F22 ... 00 0... 0 ⎥ ⎢ ⎥ ⎢ .. ⎥ ⎢ ...... ⎥ ⎢ ⎥ ⎢ Fn 1 Fn 2 ... ∂+ Fnn 10... 0 ⎥ ⎢ ⎥ det ⎢ Fn+1 1 Fn+1 2 ... Fn+1 n ∂ −1 ... 0 ⎥ ⎢ ⎥ ⎢ Fn 1 Fn 2 ... 0 Fn n+1 ∂ + Fn n ... 0 ⎥ ⎢ ⎥ ⎢ . ⎥ ⎢ ...... ⎥ ⎣ ⎦ F2 1 0 ...... F2 n+1 F2 n ... −1 0 F1 2 ...... F1 n+1 F1 n ... ∂+ F1 1 which has the form

~~2n+1 2n−1 2n−2 (12.22) ∂ + w2 ∂ + w3 ∂ + ···+ w2n+1,wi ∈V(p).~~

Theorem 12.2.3. All elements w2,w3,...,w2n+1 belong to the classical W- W (r) (r) (r) algebra (o2n+1). Moreover, the elements w2 ,w4 ,...,w2n with r =0, 1,... are algebraically independent and generate the algebra W(o2n+1). Proof. The argument is similar to the proof of Theorem 12.2.2. Denote the determinant by D and let Di (respectively, Di)denotethei×i minor corresponding to the ﬁrst (respectively, last) i rows and columns. We suppose that D0 = D0 =1. 220 12. CLASSICAL W-ALGEBRAS

~~Lemma 12.2.1 implies the expansion n+1 n−j+1 (12.23) D = Dn ∂ Dn + (−1) Dj−1 Fk j Dk−1. j,k=1~~

To prove the ﬁrst part of the theorem, note that the elements Fij with 1 i, j n span a subalgebra of o2n+1 isomorphic to the Lie algebra gln. Hence, by Theo- rem 12.2.2, if 1 i n − 1then { } { } ρ Fii+1 λ Dk =0 and ρ Fii+1 λ Dk =0 for all 1 k n with k = i.Moreover, { } − + { } ρ Fii+1 λ Di = Di−1 and ρ Fii+1 λ Di = Di−1, where, as before, P + = P (∂ + λ) for any polynomial P = P (∂) ∈V(p) ⊗ C[∂]. Hence, for any k ∈{1,...,n+1} and k = i, i +1wehave − ρ Fii+1 λ (Di Fk i+1 Dk−1 Di−1 Fk i Dk−1) =0. Similarly, for any j ∈{1,...,n+1} and j = i, i +1wehave ρ Fii+1 λ (Dj−1 Fi j Di−1 + Dj−1 F(i+1) j Di) =0 and − ρ Fii+1 λ (Di−1 F(i+1) i Di Di Fi i+1 Di−1) =0. { } These relations imply ρ Fii+1 λ D = 0. Finally, performing similar calculations we get n−1 { } + + − + ρ Fnn+1 λ D = Dn (∂ + λ) Dn−1 + Dn Fk n Dk−1 Dn (Fnn + λ) Dn−1 k=1 n−1 − + − n−j+1 + − + Dn−1 ∂ Dn + ( 1) Dj−1 Fnj Dn Dn−1 (Fnn + λ) Dn. j=1 + Applying Lemma 12.2.1 to the determinants Dn and Dn, we get the relations n−1 + + − n−j + (12.24) Dn = Dn−1 (∂ + λ + Fnn)+ ( 1) Dj−1 Fnj j=1 and n−1 (12.25) Dn =(∂ + Fn n ) Dn−1 + Fk n Dk−1 k=1 { } which imply that ρ Fnn+1 λ D = 0. This shows that all elements w2,...,w2n+1 belong to the subalgebra W(o2n+1)ofV(p). For the proof of the second part, apply Theorem 12.1.5. The odd powers e, e3,...,e2n−1 of the matrix e given in (12.21) form a basis of the centralizer e o2n+1. Therefore, using (12.8) we obtain that (12.7) holds for the subspace

~~U =spanof {F12,F14,...,F12n} so that ⊥ U =spanof {Fij with 1 i j>1andF2k−11 with k =1,...,n}.~~

~~Now calculate the images of the elements w2,w4,...,w2n under the isomorphism (12.11). As a ﬁrst step, write each element w2m as a diﬀerential polynomial in the 12.2. GENERATORS OF W(g) 221~~

⊥ variables F21,F41,...,F2n 1 modulo the diﬀerential ideal U . Formula (12.15) and expansion (12.23) imply that modulo U ⊥ the determinant D equals n n−2 n−1 n n−2 D ≡ ∂ − F21 ∂ + ···+(−1) Fn1 ∂ ∂ − ∂ F21 −···−Fn1 n+1 n k−1 k−3 + (−1) Fk1 ∂ − ∂ F21 −···−Fk−11 k=2 n+1 n−j+1 j−1 j−3 j−2 + (−1) ∂ − F21 ∂ + ···+(−1) Fj−11 F1j , j=2 where we keep all variables Fp1 with p 2. It is clear from this expression, that if ⊥ 2m n then modulo the diﬀerential ideal U , the element w2m equals −2F2m 1 (s) plus a linear combination of the elements Fq 1 with s + q =2m and s>0, and (t) the products Fp 1 Fq 1 with p + q + t =2m,whereallp, q, s, t are even. Similarly, if n 2m n +1,thenw2m equals 2(−1) F2m 1 plus a linear combination of the same form as above. On the other hand, the odd powers f,f3,...,f2n−1 of the matrix f form a basis f 2m−1 ± ⊥ of the centralizer o2n+1.Observethatf coincides with F2m 1 modulo U . (r) (r) (r) Therefore, we may conclude that the images of the elements w2 ,w4 ,...,w2n with r =0, 1,... under the isomorphism (12.11) are algebraically independent V f generators of the algebra (o2n+1) thus completing the proof.

Type Dn. We use the same notation for the generators of the Lie algebra o2n as for type Bn above. We will work with the algebra of pseudo-diﬀerential operators V(p) ⊗ C((∂−1)), where the relations are given by (12.20) and ∞ −1 (r) − s (r+s) −s−1 ∂ Fij = ( 1) Fij ∂ . s=0

~~Take the principal nilpotent element f ∈ o2n in the form~~

~~f = F21 + F32 + ···+ Fnn−1 + Fn n−1.~~

~~The sl2-triple is formed by the elements {e, f, h} with~~

~~− n2 n2 − n (12.26) e = i(2n − i − 1) F + F − + F − and ii+1 2 n 1 n n 1 n i=1 n−1 h =2 (n − i) Fii. i=1~~

The nondegenerate invariant symmetric bilinear form on o2n is deﬁned by 1 X, Y = tr XY, X,Y ∈ o , 2 2n where X and Y are understood as matrices over C which are skew-symmetric with respect to the antidiagonal. Consider the following (2n+1)×(2n+1) matrix with entries in V(p)⊗C((∂−1)), 222 12. CLASSICAL W-ALGEBRAS

⎡ ⎤ ∂ + F11 1 ... 00 0... 0 ⎢ ⎥ ⎢ F21 ∂ + F22 ... 00 0... 0 ⎥ ⎢ ⎥ ⎢ .. ⎥ ⎢ ...... ⎥ ⎢ ⎥ ⎢Fn1 − Fn1 Fn2 − Fn2 ... ∂+ Fnn 0 −2∂ ... 0 ⎥ ⎢ − ⎥ ⎢ 00... 0 ∂ 1 0 ... 0 ⎥ ⎢ ⎥ ⎢ Fn 1 Fn 2 ... 00∂ + Fn n ... 0 ⎥ ⎢ ⎥ ⎢ . ⎥ ⎢ ...... ⎥ ⎣ ⎦ F2 1 0 ...... 0 F2n − F2n ... −1 0 F1 2 ...... 0 F1 n − F1 n ... ∂+ F1 1 where all entries in the row and column n + 1 are zero, except for the (n +1,n+1) −1 entry which equals ∂ .The(n, j)entriesareFnj − Fn j for j =1,...,n− 1, the (n, n)entryis∂ + Fnn and the (n, n +2)entryis −2∂, while the remaining entries in row n are zero. The remaining nonzero entries in column n +2areFk n − Fk n for k =1, 2,...,n− 1 which occur in the respective rows 2n − k +2,and ∂ + Fn n which occurs in row n +2. One easily verifies that the column-determinant and row-determinant of this matrix coincide, so that the determinant (12.13) is well-defined and we denote it by D. Applying the simultaneous column expansion along the first n columns and using Lemma 12.2.1, one derives that it can be written in the form n −1 n−j (12.27) D = Dn ∂ Dn +2 (−1) Dj−1 Fk j Dk−1, j,k=1 where Di (respectively, Di) denotes the i × i minor corresponding to the first (respectively, last) i rows and columns. We suppose that D0 = D0 =1.Write n n−1 n−2 Dn = ∂ + y1 ∂ + y2 ∂ + ···+ yn,

~~n n−1 n−2 Dn = ∂ + ∂ y¯1 + ∂ y¯2 + ···+¯yn, for certain uniquely determined elements yi, y¯i ∈V(p). i Lemma 12.2.4. We have y¯i =(−1) yi for all i =1,...,n.~~

Proof. Replace ∂ by −∂ in the minor Dn and multiply each row by −1. The lemma can then be equivalently stated as the identity n n n−1 n−2 (−1) Dn = ∂ + ∂ y1 + ∂ y2 + ···+ yn. ∂ →−∂ The left hand⎡ side is the determinant ⎤ ∂ + Fnn 100... 0 ⎢ ⎥ ⎢Fnn−1 − Fn n−1 ∂ + Fn−1 n−1 10 ... 0 ⎥ ⎢ ⎥ det ⎢ ...... ⎥ . ⎣ ⎦ Fn 2 − Fn 2 Fn−12 ...... ∂+ F22 1 Fn 1 − Fn 1 Fn−11 ...... F21 ∂ + F11 The claim now follows from the observation that this determinant coincides with the image of Dn under the anti-automorphism of the algebra V(p) ⊗ C[∂]whichis the identity on the generators Fij and ∂. 12.2. GENERATORS OF W(g) 223

~~By Lemma 12.2.4, the pseudo-diﬀerential operator D canbewrittenas 2n−1 2n−3 2n−4 n −1 (12.28) D = ∂ + w2 ∂ + w3 ∂ + ···+ w2n−1 +(−1) yn ∂ yn for certain elements wi ∈V(p).~~

Theorem 12.2.5. The coeﬃcients w2,w3,...,w2n−1 and yn belong to the clas- W W (r) (r) (r) (r) sical -algebra (o2n). Moreover, the elements w2 ,w4 ,...,w2n−2,yn with r =0, 1,... are algebraically independent and generate the algebra W(o2n). Proof. { } − The relation ρ Fii+1 λ D =0for1 i n 1 follows by the same calculations as in the proof of Theorem 12.2.3. Furthermore, let σ =(nn)bethe permutation of the set of indices {1,...,2n} which swaps n and n = n +1 and leaves all other indices ﬁxed. The mapping

(12.29) ς : Fij → Fσ(i) σ(j) defines an involutive automorphism of the Lie algebra o2n. It also extends to an involutive automorphism of the Poisson vertex algebra V(o2n). We claim that all coefficients of the pseudo-differential operator D are ς-invariant. Indeed, let us apply the following operations on the rows and columns of the given matrix. Replace row n +2bythesumofrowsn and n + 2. Then replace column n by the sum of columns n and n + 2. Finally, multiply row n and column n +2by−1. As a result, we get the image of the matrix with respect to the involution (12.29). On the other hand, the determinant D remains unchanged. This proves the relation { } ρ Fn−1 n λ D = 0. This shows that all coefficients of the operator D belong to the subalgebra W(o2n). Note that the minor Dn canbewrittenintheform

~~(12.30) Dn = Dn−1 (∂ + Fnn) − Dn−2 (Fnn−1 − Fn n−1)~~

n−2 n−1 + ···+(−1) D1 (Fn 2 − Fn 2)+(−1) D0 (Fn 1 − Fn 1). Repeating the calculations used in the proof of Theorem 12.2.2, we ﬁnd that { } − ρ Fii+1 λ Dn =0fori =1,...,n 1. Furthermore, + + + { } − − ρ Fn−1 n λDn = Dn−1 Dn−2 (∂ + Fnn)+Dn−2 (Fn−1 n−1 + Fnn + λ) − + ··· − n−1 + − n + Dn−3 Fn−1 n−2 + +( 1) D1 Fn−12 +( 1) D0 Fn−11. + Applying relation (12.19) to the determinant Dn−1 we get + { } − ρ Fn−1 n λDn = 2 Dn−2 ∂.

{ } This implies ρ Fn−1 n λ yn = 0 so that the constant term yn of the diﬀerential operator Dn belongs to W(o2n). The second part of the theorem will follow from Theorem 12.1.5. The odd powers e, e3,...,e2n−3 of the matrix e given in (12.26) together with the element − e F1n F1n form a basis of the centralizer o2n. Hence (12.8) implies that (12.7) holds for the subspace U deﬁned by

~~U =spanof {F12,F14,...,F1 n,F1 n+1,F1 n+3,...F12n−1}, if n is even, and by~~

~~U =spanof {F12,F14,...,F1 n−1,F1n − F1n+1,F1 n+2,F1 n+4,...,F12n−1}, ⊥ f if n is odd. The complementary subspace U to o2n in p is then given by (12.9). 224 12. CLASSICAL W-ALGEBRAS~~

As in the proof of Theorem 12.2.3, expansion (12.27) implies that modulo U ⊥ the determinant D equals n n−2 n−1 D ≡ ∂ − F21 ∂ + ···+(−1) (Fn1 − Fn1) −1 n n−2 × ∂ ∂ − ∂ F21 −···−(Fn1 − Fn1) n n−1 k−1 k−3 +2 (−1) Fk1 ∂ − ∂ F21 −···−Fk−11 k=2 n n−j j−1 j−3 j−2 +2 (−1) ∂ − F21 ∂ + ···+(−1) Fj−11 F1j , j=2 where we keep all variables Fp1 with p 2. Moreover, due to (12.30) we also n−1 have yn ≡ (−1) (Fn1 − Fn1). Therefore, if 2m n − 1 then modulo the dif- ⊥ ferential ideal U , the element w2m equals −2F2m 1 plus a linear combination (s) (t) of the elements Fq 1 with s + q =2m and s>0, and the products Fp 1 Fq 1 with p + q + t =2m,whereallp, q, s, t are even. Next we look at the cases of even and odd n separately. If n is even, then ⊥ modulo the differential ideal U , the coefficient wn equals −2(Fn1 + Fn1)plusa linear combination of the same form as above with 2m = n.Ifn +2 2m 2n − 2 then w2m equals −4F2m+1 1 plus a similar linear combination. If n is odd and ⊥ n +1 2m 2n − 2 then modulo the differential ideal U , the coefficient w2m equals 4F2m+1 1 plus a linear combination of the same kind. The odd powers f,f3,...,f2n−3 of the matrix f together with the element f ⊥ − Fn1 Fn 1 form a basis of the centralizer o2n. Observe that modulo U ,theodd 2m−1 n power f coincides with F2m 1 if 2mn. n−1 ⊥ Furthermore, if n is even, then f coincides with Fn 1 + Fn1 modulo U .There- (r) (r) (r) fore, we may conclude that the images of the elements w2 ,w4 ,...,w2n−2 and (r) yn with r =0, 1,... under the isomorphism (12.11) are algebraically independent V f generators of the algebra (o2n) which completes the proof.

Type Cn. We will use the same presentation of the Lie algebra sp2n as in Sections 2.2 and 5.1. The elements F11,...,Fnn span a Cartan subalgebra of sp2n whichwedenotebyh. The respective subsets of elements Fij with ij span the nilpotent subalgebras n+ and n−. The subalgebra p = n− ⊕ h is then spanned by the elements Fij with i j. We will work with the algebra of diﬀerential operators V(p) ⊗ C[∂], where the commutation relations are given by (r) − (r) (r+1) ∂Fij Fij ∂ = Fij . For any element g ∈V(p) and any nonnegative integer r the element g(r) coincides with the constant term of the diﬀerential operator ∂ rg as in (12.16). ∈ Take the principal nilpotent element f sp2n in the form 1 f = F + F + ···+ F − + F . 21 32 nn 1 2 n n The sl2-triple is formed by the elements {e, f, h} with − n1 n2 n e = i(2n − i) F + F and h = (2n − 2i +1)F . ii+1 2 nn ii i=1 i=1 12.2. GENERATORS OF W(g) 225

1 The invariant symmetric bilinear form on sp2n is deﬁned by 1 X, Y = tr XY, X,Y ∈ sp , 2 2n where X and Y are understood as 2n × 2n symplectic matrices over C. Consider the determinant (12.13) of the matrix with entries in V(p) ⊗ C[∂], ⎡ ⎤ ∂ + F11 1 ... 000... 0 ⎢ ⎥ ⎢ F21 ∂ + F22 ... 000... 0 ⎥ ⎢ ⎥ ⎢ .. ⎥ ⎢ ...... ⎥ ⎢ ⎥ ⎢ Fn 1 Fn 2 ... ∂+ Fnn 10... 0 ⎥ det ⎢ ⎥ ⎢ Fn 1 Fn 2 ... Fn n ∂ + Fn n −1 ... 0 ⎥ ⎢ ⎥ ⎢ ...... ⎥ ⎣ ⎦ F2 1 F2 2 ... F2 n F2 n ...... −1 F1 1 F1 2 ... F1 n F1 n ...... ∂+ F1 1 which has the form 2n 2n−2 2n−3 (12.31) ∂ + w2 ∂ + w3 ∂ + ···+ w2n,wi ∈V(p).

Theorem 12.2.6. All coeﬃcients w2,w3,...,w2n belong to the classical W- W (r) (r) (r) algebra (sp2n). Moreover, the elements w2 ,w4 ,...,w2n with r =0, 1,... are W algebraically independent and generate the algebra (sp2n).

Proof. Denote the determinant by D and let Di (respectively, Di)denotethe i × i minor corresponding to the ﬁrst (respectively, last) i rows and columns. We suppose that D0 = D0 = 1. Lemma 12.2.1 implies the expansion n n−j+1 (12.32) D = Dn Dn + (−1) Dj−1 Fk j Dk−1. j,k=1 − { } If 1 i n 1 then the relation ρ Fii+1 λ D = 0 follows by the same calculation as in the proof of Theorem 12.2.3. Furthermore,

~~n−1 + + { } − + ρ Fnn λ D = 2Dn−1 Dn +2Dn Dn−1 +2 Dn−1 Fk n Dk−1 k=1 n−1 − + − n−j+1 + 2 Dn−1 (2Fnn + λ) Dn−1 +2 ( 1) Dj−1 Fnj Dn−1. j=1~~

This is zero since the relations (12.24) and (12.25) are valid for the case of sp2n as W V well. Thus, all elements w2,w3,...,w2n belong to the subalgebra (sp2n)of (p). Now apply Theorem 12.1.5. The odd powers e, e3,...,e2n−1 of the matrix e e form a basis of the centralizer sp2n. Therefore, using (12.8) we obtain that (12.7) holds for the subspace

~~U =spanof {F12,F14,...,F12n} so that ⊥ U =spanof {Fij with 1 i j>1andF2k−11 with k =1,...,n}.~~

~~1Note the additional factor 1/2 as compared to the normalized Killing form of Section 8.3. 226 12. CLASSICAL W-ALGEBRAS~~

Represent each of the coefficients w2m as a differential polynomial in the variables ⊥ F21,F41,...,F2n 1 modulo the differential ideal U . Formula (12.15) and expansion (12.32) imply that modulo U ⊥ the determinant D equals n n−2 n−1 n n−2 D ≡ ∂ − F21 ∂ + ···+(−1) Fn1 ∂ − ∂ F21 −···−Fn1 n n k−1 k−3 + (−1) Fk1 ∂ − ∂ F21 −···−Fk−11 k=1 n n−j+1 j−1 j−3 j−2 + (−1) ∂ − F21 ∂ + ···+(−1) Fj−11 F1j , j=2 where we keep all variables Fp1 with p 2. Hence, if 2m n then modulo the ⊥ differential ideal U , the element w2m equals −2F2m 1 plus a linear combination (s) (t) of the elements Fq 1 with s + q =2m and s>0, and the products Fp 1 Fq 1 with p + q + t =2m,whereallp, q, s, t are even. Similarly, if n +1 2m,thenw2m n n equals 2(−1) F2m 1 or (−1) F2n 1 depending on whether m<2n or m =2n, plus a linear combination of the same form. As with type B,theoddpowersf,f3,...,f2n−1 of the matrix f form a basis f 2m−1 ± ⊥ of the centralizer sp2n.Thematrixf coincides with F2m 1 modulo U . (r) (r) (r) Therefore, the images of the elements w2 ,w4 ,...,w2n with r =0, 1,... under the isomorphism (12.11) are algebraically independent generators of the algebra V f (sp2n) thus completing the proof. 12.3. Chevalley projection Return to the general settings related to an arbitrary simple Lie algebra g as in Section 12.1. The differential algebra V(h) is the algebra of polynomials in (r) r the variables hj = ∂ (hj) with r =0, 1,... and j =1,...,n. Consider the homomorphism of differential algebras (12.33) φ : V(p) →V(h) which is defined on the generators as the projection p → h with the kernel n−. Recall that due to Theorem 12.1.4 the classical W-algebra W(g) is a differential subalgebra of V(p). Theorem 12.3.1. The restriction of the homomorphism φ to the subalgebra W W (g) is injective. Hence, the differential algebra (g) is isomorphic to the subalgebra W(g)=φ W(g) of V(h). Proof (for classical types). We will show that the images of the algebraically independent generators of W(g) constructed in Section 12.2 are algebraically independent elements of V(h). We will employ the following general lemma.

Lemma 12.3.2. Suppose that polynomials v1,...,vn in the n variables h1,...,hn (r) (r) are algebraically independent. Then all derivatives v1 ,...,vn with r =0, 1,... are algebraically independent elements of V(h). Proof. It is suﬃcient to demonstrate that for any nonnegative integer p the (r) (r) elements v1 ,...,vn with r =0, 1,...,p are algebraically independent polynomi- (r) (r) als in the variables h1 ,...,hn with r =0, 1,...,p. We will use the Jacobian 12.3. CHEVALLEY PROJECTION 227 criterion for algebraic independence [71, Proposition 3.10]. By the criterion, the (r) (r) polynomials v1 ,...,vn are algebraically independent if and only if the Jacobian is nonzero; that is, ∂v(r) (12.34) det i =0 . (s) ∂hj Let us order the rows and columns of the Jacobian in accordance with the ordering (r) ≺ (s) (r) ≺ (s) on the variables and polynomials deﬁned by hi hj and vi vj if r

Now consider the classical types separately. W W Case AN−1. The generators w1,...,wN of the classical -algebra (glN ) were produced in Theorem 12.2.2. Their images under the homomorphism (12.33) are the elements wm ∈V(h) found from the relation

N N−1 (∂ + E11) ...(∂ + ENN)=∂ + w1 ∂ + ···+ wN . (r) (r) We want to show that all polynomials w1 ,...,wN with r =0, 1,... are algebraically independent. It suﬃces to do this for their top degree components with V (r) respect to the grading on (h) deﬁned by setting deg Eii =1foralli and r. However, the top degree component of wm coincides with the m-th elementary symmetric polynomial (2.10) in the variables E11,...,ENN. Since the elementary symmetric polynomials are algebraically independent, the desired property of the (r) (r) polynomials w1 ,...,wN follows from Lemma 12.3.2.

~~Case Bn. The images wm ∈V(h) of the coeﬃcients wm of the diﬀerential operator (12.22) under the homomorphism (12.33) are found from the relation~~

~~(∂ + F11) ...(∂ + Fnn) ∂ (∂ − Fnn) ...(∂ − F11)~~

~~2n+1 2n−1 2n−2 = ∂ + w2 ∂ + w3 ∂ + ···+ w2n+1.~~

V (r) As with type A above, deﬁne the grading on (h) by setting deg Fii =1.We k ﬁnd that for k =1,...,nthe top degree component of (−1) w2k coincides with the 2 2 k-th elementary symmetric polynomial in the variables F11,...,Fnn. Hence, the (r) (r) (r) polynomials w2 , w4 ,...,w2n with r =0, 1,... are algebraically independent by Lemma 12.3.2. 228 12. CLASSICAL W-ALGEBRAS

Case Dn. The images of the coeﬃcients w2,w3,...,w2n−1,yn of the operator D given in (12.28) under the homomorphism (12.33) are the respective elements w2, w3,...,w2n−1, yn ∈V(h) found from the relation

~~−1 (∂ + F11) ...(∂ + Fnn) ∂ (∂ − Fnn) ...(∂ − F11)~~

~~2n−1 2n−3 2n−4 n −1 = ∂ + w2 ∂ + w3 ∂ + ···+ w2n−1 +(−1) yn ∂ yn. In particular,~~

yn =(∂ + F11) ...(∂ + Fnn)1. k Similar to type Bn above, the top degree component of (−1) w2k coincides with 2 2 the k-th elementary symmetric polynomial in the variables F11,...,Fnn for k = 1,...,n− 1. Furthermore, the top degree component of yn is F11 ...Fnn.These components are well-known to be algebraically independent so the proof is completed by an application of Lemma 12.3.2.

~~Case Cn. The images wm ∈V(h) of the coeﬃcients wm of the diﬀerential operator (12.31) under the homomorphism (12.33) are found from the relation~~

~~(∂ + F11) ...(∂ + Fnn)(∂ − Fnn) ...(∂ − F11)~~

2n 2n−2 2n−3 = ∂ + w2 ∂ + w3 ∂ + ···+ w2n. k As with type Bn, the top degree component of (−1) w2k coincides with the k-th 2 2 elementary symmetric polynomial in the variables F11,...,Fnn for all k =1,...,n. (r) (r) (r) By Lemma 12.3.2 the polynomials w2 , w4 ,...,w2n with r =0, 1,... are algebraically independent. W →V The injective homomorphism φ : (g) (h) provided by Theorem 12.3.1 is known as the Miura transformation. Its image W(g)=φ W(g) can be regarded as an alternative presentation of the classical W-algebra. It is this presentation which we will need to describe the Harish-Chandra images of the Segal–Sugawara vectors in Chapter 13. An alternative description of the subalgebra W(g) ⊂V(h) can be given by in terms of screening operators as we discuss in the next section.

12.4. Screening operators For each i =1,...,n introduce the screening operator ◦ V →V Vi : (h) (h) by the formula ∞ n ∂ (12.35) V ◦ = V ◦ a , i ir ji (r) r=0 j=1 ∂hj where A =[aij] is the Cartan matrix of the simple Lie algebra g and the coeﬃcients ◦ V Vir are elements of (h) found by the relation ∞ ∞ − V ◦ zr h(m 1) zm (12.36) ir =exp − i . r! m! r=0 m=1 i

~~The positive rational numbers i are given in (12.3). Their particular values for the classical types will be chosen below. 12.4. SCREENING OPERATORS 229~~

Proposition 12.4.1. The image φ(P ) of any element P ∈W(g) under the homomorphism (12.33) is annihilated by all screening operators, ◦ Vi φ(P )=0,i=1,...,n. Hence, n W ⊂ ◦ (12.37) (g) ker Vi . i=1 Proof. It will be convenient to work with an equivalent affine version of the differential algebra V(g). Recall that the affine Kac–Moody algebra g is defined as the central extension (6.5). Consider the quotient S(g)/ I of the symmetric algebra S(g) by its ideal I generated by the subspace g[t]andtheelement K − −1 −1 1. This quotient can be identified with the symmetric algebra S t g[t ] ,asa ∼ − − vector space. We will identify the differential algebras V(g) = S t 1g[t 1] via the isomorphism (12.38) X(r) → r! X[−r − 1],X∈ g,r 0, −1 −1 so that the derivation ∂ will correspond to the derivation T of S t g[t ] defined ∼ − − in (6.20). Similarly, we will identify the differential algebras V(p) = S t 1p[t 1] . By Definition 12.1.3, if an element P ∈V(p) belongs to the subalgebra W(g), { } then ρ ei λ P =0foralli =1,...,n. Now observe that, regarding P as an element ∼ − − of the g[t]-module S(g)/ I = S t 1g[t 1] , we can write2 ∞ λr {e P } = e [r] P. i λ r! i r=0 We have the following relations in g, ei[r],fi[−s − 1] = hi[r − s − 1] + rδr, s+1 i K, ei[r],hj[−s − 1] = −aji ei[r − s − 1]. Moreover, for each positive root α = α we also have i − − − − ei[r],e−α[ s 1] = ci(α) e−α+αi [r s 1] for a certain constant ci(α), if α − αi is a root; otherwise the commutator is zero. Hence, recalling the definition (12.6) of the homomorphism ρ, we can conclude that the condition that P belongs to the subalgebra W(g) implies the relations (12.39) e [r] P =0 forall i =1,...,n and r 0, i −1 −1 where ei[r] is the operator on S t p[t ] given by ∞ ∂ ∂ n ∂ e [r]= h [r − s − 1] + r − a i i ∂f [−s − 1] i ∂f [−r] ji ∂h [−r − 1] s=r i i j=1 j ∞ − − ∂ + ci(α) e−α+αi [r s 1] , ∂e− [−s − 1] + α α∈Δ ,α= αi s=r − and e−α+αi is understood as being equal to zero, if α αi is not a root. Denote the ◦ (m−1) − generating function in z introduced in (12.36) by Vi (z) and replace hi /(m 1)!

2This module structure diﬀers from the one described by (6.18) since the central element K now takes the value 1 in the quotient. 230 12. CLASSICAL W-ALGEBRAS with hi[−m]form 1 in accordance with (12.38). We have the relation for its derivative, ∞ h [−m]zm−1 V ◦(z)=V ◦(z) − i . i i m=1 i Taking the coeﬃcient of zp−1 with p 1 we get the relations

~~◦ p−1 V V ◦ (12.40) p ip + ir h [r − p]=0. i p! r! i r=0 By (12.39) the element P has the property~~

~~∞ V ◦ (12.41) ir e [r] P =0. r! i r=0~~

Note that by (12.40) all the diﬀerentiations ∂/∂fi[−s − 1] with s 0 will cancel in the expansion of the left hand side of (12.41). Moreover, the elements of the − − form e−α+αi [r s 1] occurring in the expansion of ei[r] will vanish under the projection (12.33). Therefore, (12.41) implies that the image φ(P ) with respect to this projection satisﬁes the relation

~~∞ V ◦ n ∂ ir a φ(P )=0 r! ji ∂h [−r − 1] r=0 j=1 j which is equivalent to~~

~~∞ n ∂ V ◦ a φ(P )=0, ir ji (r) r=0 j=1 ∂hj ◦ that is, Vi φ(P ) = 0, as claimed.~~

The inclusion (12.37) is in fact an equality which can be proved by comparing the Hilbert–Poincar´e series of both subalgebras of V(h); see [46, Chapter 8]. We omit this computation and state this result without a proof.

Theorem 12.4.2. The restriction of the homomorphism φ to W(g) yields an isomorphism φ : W(g) → W(g), where W(g) is the subalgebra of V(h) which consists of the elements annihilated by ◦ all screening operators Vi , n W ◦ (g)= ker Vi . i=1

In the rest of this section we will give a description of the generators of the classical W-algebras and screening operators by regarding W(g) as a subalgebra of the symmetric algebra S t−1h[t−1] via the isomorphism (12.38). 12.4. SCREENING OPERATORS 231

~~Type AN−1. For i =1,...,N and all r 0 introduce new variables~~

1 (r) μ [−r − 1] = E ,i= N − i +1. i r! ii Hence (r) − − − − − − hi = r! μi [ r 1] μ(i+1) [ r 1] ,i=1,...,N 1, so that for the partial derivatives over new variables we have ∂ ∂ ∂ = −r! + r! (r) (r) ∂μi [−r − 1] ∂hi−1 ∂hi with out-of-range terms omitted. The Cartan matrix is of the size (N −1)×(N −1), ⎡ ⎤ 2 −10... 00 ⎢ ⎥ ⎢−12−1 ... 00⎥ ⎢ ⎥ ⎢ 0 −12... 00⎥ A = ⎢ ⎥ ⎢ .. ⎥ ⎢ ...... ⎥ ⎣ 000... 2 −1 ⎦ 000... −12 and the diagonal matrix D given in (12.3) is the identity matrix. Therefore, the sum in (12.35) equals − N1 ∂ ∂ ∂ ∂ r! a = r! − +2 − ji (r) (r) (r) (r) j=1 ∂hj ∂hi−1 ∂hi ∂hi+1 ∂ ∂ = − ∂μi [−r − 1] ∂μ(i+1) [−r − 1] ◦ − so that the screening operator (12.35) takes the form Vi = V(i+1) ,where ∞ ∂ ∂ (12.42) V = V − ,i=1,...,N − 1, i ir ∂μ [−r − 1] ∂μ [−r − 1] r=0 i i+1 and the coefficients Vir are found from the expansion ∞ ∞ μ [−m] − μ [−m] (12.43) V zr =exp i i+1 zm. ir m r=0 m=1 W In the new variables the elements w1,..., wN of (glN ) become the respective −1 −1 coefficients E1,...,EN in the expansion in S t h[t ] ⊗ C[τ], N N−1 (12.44) τ + μN [−1] ... τ + μ1[−1] = τ + E1 τ + ···+ EN , wherefortheelementτ we have the relations (12.45) τ,μi[r] = −rμi[r − 1], implied by (7.1). We will also need the operator T =adτ which is the derivation of the algebra S t−1h[t−1] so that T 1 = 0 and it is defined on the generators by

~~(12.46) Tμi[r]=−rμi[r − 1].~~

To write a more explicit formula for Em, we need to recall standard noncommutative versions of symmetric functions. Suppose that x1,...,xN are variables which can be thought of as elements of a not necessarily commutative associative algebra. 232 12. CLASSICAL W-ALGEBRAS

~~The (generally, noncommutative) complete and elementary symmetric functions in the variables x1,...,xN are deﬁned by the respective formulas~~

~~(12.47) hm(x1,...,xN )= xi1 ...xim , ··· i1 im~~

~~(12.48) em(x1,...,xN )= xi1 ...xim ,~~

i1>···>im for m 1, and h0(x1,...,xN )=e0(x1,...,xN ) = 1. If the variables do commute, the notation agrees with (2.10) and (2.22). Lemma 12.4.3. The complete and elementary symmetric functions are related by the formulas m − m−k (12.49) hm = ( 1) ea1 ...eak , k=1 a1+···+ak=m m − m−k (12.50) em = ( 1) ha1 ...hak , k=1 a1+···+ak=m with the second sums taken over positive integers ai,whereha = ha(x1,...,xN ) and ea = ea(x1,...,xN ). Proof. Writing the deﬁnitions (12.47) and (12.48) in terms of generating functions we get ∞ m −1 −1 hm q =(1− qx1) ...(1 − qxN ) , m=0 N m em q =(1+qxN ) ...(1 + qx1). m=0 This implies that ∞ ∞ N −1 N k m a k a hm q = 1+ ea (−q) =1+ (−1) ea (−q) . m=0 a=1 k=1 a=1 Therefore (12.49) follows by taking the coeﬃcients of qm on both sides. By swapping hm and em we get (12.50) by the same argument.

Now return to the expansion (12.44). Let us specialize the variables by setting xi = τ + μi[−1] for i =1,...,N and write the elementary symmetric function as a polynomial in τ. Proposition 12.4.4. For m =1,...,N we have m N − k (12.51) e τ + μ [−1],...,τ + μ [−1] = E τ m−k. m 1 N m − k k k=0

~~In particular, Em coincides with the constant term of the polynomial (12.51), (12.52) Em = em τ + μ1[−1],...,τ + μN [−1] 1, assuming τ 1=0. 12.4. SCREENING OPERATORS 233~~

Proof. Note that relations (12.45) will remain valid under the replacement τ → u + τ for a (commutative) variable u. Hence (12.44) implies N N−k u + τ + μN [−1] ... u + τ + μ1[−1] = Ek (u + τ) . k=0 Since the left hand side equals N N−m em τ + μ1[−1],...,τ + μN [−1] u , m=0 the desired relation follows by taking the coeﬃcients of uN−m on both sides. Example 12.4.5. We have E1 = μ1[−1] + ···+ μN [−1], N E2 = μi[−1]μj [−1] + (N − j)μj[−2]. i>j j=1 −1 −1 By analogy with (12.52), for all m 0 introduce elements Hm ∈ S t h[t ] by taking the constant terms of the complete symmetric functions, Hm = hm τ + μ1[−1],...,τ + μN [−1] 1. Example 12.4.6. We have H1 = μ1[−1] + ···+ μN [−1], N H2 = μi[−1]μj [−1] + jμj [−2]. ij j=1 The following is a counterpart of Proposition 12.4.4. Proposition 12.4.7. For all m 1 we have m N + m − 1 h τ + μ [−1],...,τ + μ [−1] = H τ m−k. m 1 N m − k k k=0

Proof. If u is a variable, commuting with all the xi, then the following identity holds for the noncommutative complete symmetric functions (12.47), m N + m − 1 (12.53) h (u + x ,...,u+ x )= h (x ,...,x )um−k. m 1 N m − k k 1 N k=0 Indeed, calculating the generating function for the sequence on the right hand side we get ∞ m N + m − 1 h (x ,...,x )um−k qm m − k k 1 N m=0 k=0 ∞ ∞ N + m − 1 = h (x ,...,x )qk um−k qm−k. k 1 N m − k k=0 m=k This equals ∞ qk h (x ,...,x ) =(1− qu− qx )−1 ...(1 − qu− qx )−1 k 1 N (1 − qu)N+k 1 N k=0 234 12. CLASSICAL W-ALGEBRAS and so coincides with the generating function of the sequence on the left hand side of (12.53). Now specialize the variables by xi = τ + μi[−1] for i =1,...,N and introduce H(m) the coeﬃcients k by m − − H(m) m−k hm τ + μ1[ 1],...,τ + μN [ 1] = k τ k=0 H(k) H so that k = k. The relation (12.53) gives m m N + m − 1 k H(m) (u + τ)m−k = H(k) τ k−p um−k. k m − k p k=0 k=0 p=0 By taking the coeﬃcients of um−kτ 0 we get N + m − 1 H(m) = H , k m − k k as required.

Corollary 12.4.8. Each of the families r r T Em and T Hm with m =1,...,N and r =0, 1,... is algebraically independent and generates the W W ⊂ −1 −1 classical -algebra (glN ) S t h[t ] . Proof. The claim for the first family follows from the type A part of the proof of Theorem 12.3.1. Furthermore, relations (12.49) and (12.51) imply that each element Hm is a differential polynomial in the elements E1,...,EN and so W r H belongs to (glN ). The algebraic independence of the family T m is deduced from Lemma 12.3.2 by taking into account the algebraic independence of the (commutative) complete symmetric polynomials h1,...,hN in N variables. Note that the classical W-algebra W(slN ) associated with the special linear Lie W E algebra slN can be obtained as the quotient of (glN )bytherelation 1 =0. Although this follows from Proposition 12.4.1, it is also possible to verify directly that all elements Em are annihilated by the screening operators Vi defined in (12.42). This is implied by the following relations for the operators on S t−1h[t−1] . Lemma 12.4.9. For i =1,...,N − 1 we have Vi T = T + μi[−1] − μi+1[−1] Vi. Proof. −1 −1 Since both Vi and T are derivations of S t h[t ] so is their commutator. Therefore, it suffices to check the relation [Vi,T]= μi[−1] − μi+1[−1] Vi on the generators μj [−r]. This holds trivially for j = i, i +1sowetakej = i (the case j = i + 1 will only differ by an additional sign). Using the notation (12.43) we get

~~Vi Tμi[−r]=rVi μi[−r − 1] = rVir. On the other hand, T + μi[−1] − μi+1[−1] Vi μi[−r]= T + μi[−1] − μi+1[−1] Vir−1. 12.4. SCREENING OPERATORS 235~~

Hence, denoting the generating function (12.43) by Vi(z), we come to verifying the relation ∂z Vi(z)= T + μi[−1] − μi+1[−1] Vi(z). It does hold since both sides are equal to ∞ m−1 Vi(z) μi[−m] − μi+1[−m] z , m=1 thus completing the proof.

As a consequence of Lemma 12.4.9 and the property T =adτ, we obtain the corresponding relations for the operators on S t−1h[t−1] ⊗ C[τ], (12.54) Vi τ = τ + μi[−1] − μi+1[−1] Vi,i=1,...,N − 1, where τ is regarded as the operator of left multiplication by τ and Vi acts as the identity operator on C[τ]. For each i the relation Vi τ + μN [−1] ... τ + μ1[−1] =0 then follows easily. It reduces to the particular case N =2wherewehave V1 τ + μ2[−1] τ + μ1[−1] = τ + μ1[−1] − μ2[−1] V1 + μ2[−1] V1 − 1 τ + μ1[−1] = τ + μ1[−1] V1 τ + μ1[−1] − τ + μ1[−1] =0.

Type Bn. For i =1,...,n and all r 0 introduce new variables 1 μ [−r − 1] = − F (r). i r! ii (r) − − − − − − Hencewehavetherelationshi = r! μi[ r 1]+μi+1[ r 1] for i =1,...,n 1 (r) and hn = −2r! μn[−r − 1]. The Cartan matrix of type Bn has the form ⎡ ⎤ 2 −10... 00 ⎢ ⎥ ⎢−12−1 ... 00⎥ ⎢ ⎥ ⎢ 0 −12... 00⎥ A = ⎢ ⎥ ⎢ .. ⎥ ⎢ ...... ⎥ ⎣ 000... 2 −1 ⎦ 000... −22 so that the entries of the diagonal matrix D =diag[1,...,n] in (12.3) are found by

1 = ···= n−1 =1 and n =2. Therefore, n ∂ ∂ ∂ r! a = − + ,i=1,...,n− 1, ji (r) ∂μi[−r − 1] ∂μi+1[−r − 1] j=1 ∂hj and n ∂ ∂ r! a = − . jn (r) ∂μn[−r − 1] j=1 ∂hj 236 12. CLASSICAL W-ALGEBRAS

◦ − This implies that the screening operators (12.35) take the form Vi = Vi,where ∞ ∂ ∂ V = V − , i ir ∂μ [−r − 1] ∂μ [−r − 1] r=0 i i+1 for i =1,...,n− 1, and ∞ ∂ (12.55) V = V ; n nr ∂μ [−r − 1] r=0 n the coeﬃcients Vir are found from the expansions ∞ ∞ μ [−m] − μ [−m] V zr =exp i i+1 zm,i=1,...,n− 1 ir m r=0 m=1 and ∞ ∞ μ [−m] V zr =exp n zm. nr m r=0 m=1 In the new variables the elements w2,...,w2n+1 ofW(o2n+1) become the re- −1 −1 spective coeﬃcients E2,...,E2n+1 in the expansion in S t h[t ] ⊗ C[τ], τ − μ1[−1] ... τ − μn[−1] τ τ + μn[−1] ... τ + μ1[−1]

2n+1 2n−1 2n−2 = τ + E2 τ + E3 τ + ···+ E2n+1. By (12.52) we have Em = em τ + μ1[−1],...,τ + μn[−1],τ,τ − μn[−1],...,τ − μ1[−1] 1. The relation (12.56) Vi τ − μ1[−1] ... τ − μn[−1] τ τ + μn[−1] ... τ + μ1[−1] =0 − can be veriﬁed directly for i =1,...,n 1inthesamewayasforglN with the use of (12.54). Furthermore, we also have Vn τ = τ + μn[−1] Vn, which is deduced in the same way as (12.54). Therefore, V τ − μ [−1] τ τ + μ [−1] = τV − 1 τ τ + μ [−1] n n n n n = τ τ + μn[−1] τ +2μn[−1] Vn, thus implying (12.56) for i = n. As with type A, relations (12.49) and (12.51) imply that for all m 1the elements Hm = hm τ + μ1[−1],...,τ + μn[−1],τ,τ − μn[−1],...,τ − μ1[−1] 1 belong to W(o2n+1). The same argument as for Corollary 12.4.8 together with the type B part of the proof of Theorem 12.3.1 implies the following. Corollary 12.4.10. Each of the families r r T Em and T Hm with m =2, 4,...,2n and r =0, 1,... is algebraically independent and generates −1 −1 the classical W-algebra W(o2n+1) ⊂ S t h[t ] . 12.4. SCREENING OPERATORS 237

Type Dn. By using the determinant D = D(∂) given in (12.28) deﬁne the elements em ∈V(p) ⊗ C[∂] as the coeﬃcients of the formal power series in q, ∞ m 2 n−1 −1 em q = q D(∂ + q ). m=0

~~Denote this series by e(q). Furthermore, deﬁne elements hm ∈V(p) ⊗ C[∂]asthe coeﬃcients of the series ∞ m −1 h(q)= hm q ,h(q)=e(−q) . m=0~~

By Theorem 12.2.5, all coeﬃcients of the diﬀerential operators em and hm belong to W(o2n). Their images under the homomorphism (12.33) are described as follows. Set aii = ∂ + Fii.Wehave −1 φ : e(q) → (1 + qa11) ...(1 + qann)(1+q∂) (1 + qan n ) ...(1 + qa1 1 ) and hence −1 −1 −1 −1 φ : h(q) → (1 − qa1 1 ) ...(1 − qan n ) (1 − q∂)(1− qann) ...(1 − qa11) .

~~Observe that since ann + an n =2∂,wehavetherelation −1 −1 1 −1 −1 (1 − qa ) (1 − q∂)(1− qa ) = (1 − qa ) +(1− qa ) . n n nn 2 nn n n~~

Therefore, the images of the elements hm can be written explicitly as 1 k1 kn kn−1 k1 (12.57) φ(hm)= a11 ...an n an−1 n−1 ...a11 2 ··· k1 + +k1=m 1 k − k1 (n 1) kn k1 + a11 ...a(n−1) (n−1) ann ...a11. 2 ··· k1 + +k1=m To embed the classical W-algebra into the symmetric algebra S t−1h[t−1] ,for all i =1,...,n and r 0 introduce new variables 1 μ [−r − 1] = − F (r). i r! ii Hence (r) − − − − − − hi = r! μi[ r 1] + μi+1[ r 1] ,i=1,...,n 1, and (r) − − − − − hn = r! μn−1[ r 1] + μn[ r 1] . The Cartan matrix has the form ⎡ ⎤ 2 −10... 000 ⎢ ⎥ ⎢−12−1 ... 000⎥ ⎢ ⎥ ⎢ 0 −12... 000⎥ ⎢ ⎥ ⎢ .. ⎥ A = ⎢ ...... ⎥ ⎢ ⎥ ⎢ 000... 2 −1 −1 ⎥ ⎣ 000... −120⎦ 000... −102 238 12. CLASSICAL W-ALGEBRAS so that n ∂ ∂ ∂ r! a = − + ,i=1,...,n− 1, ji (r) ∂μi[−r − 1] ∂μi+1[−r − 1] j=1 ∂hj and n ∂ ∂ ∂ r! a = − − . jn (r) ∂μn−1[−r − 1] ∂μn[−r − 1] j=1 ∂hj The diagonal matrix D in (12.3) is the identity matrix. Therefore, the screening ◦ − operators (12.35) take the form Vi = Vi,where ∞ ∂ ∂ V = V − , i ir ∂μ [−r − 1] ∂μ [−r − 1] r=0 i i+1 for i =1,...,n− 1, and ∞ ∂ ∂ (12.58) Vn = Vnr + ; ∂μ − [−r − 1] ∂μ [−r − 1] r=0 n 1 n the coefficients Vir are found from the expansions ∞ ∞ μ [−m] − μ [−m] V zr =exp i i+1 zm,i=1,...,n− 1 ir m r=0 m=1 and ∞ ∞ μ − [−m]+μ [−m] V zr =exp n 1 n zm. nr m r=0 m=1 The elements w2, w3,...,w2n−1 and yn of the algebra W(o2n) become the E E E E ◦ respective coefficients 2, 3,..., 2n−1 and n in the expansion of the pseudo- differential operator −1 τ − μ1[−1] ... τ − μn[−1] τ τ + μn[−1] ... τ + μ1[−1]

2n−1 E 2n−3 E 2n−4 ··· E − n E ◦ −1 E ◦ = τ + 2 τ + 3 τ + + 2n−1 +( 1) n τ n . In particular, E ◦ − − − − (12.59) n = τ μ1[ 1] ... τ μn[ 1] 1. The identity −1 (12.60) Vi τ − μ1[−1] ... τ − μn[−1] τ τ + μn[−1] ... τ + μ1[−1] =0 is veriﬁed with the use of (12.54) and the additional relations −1 −1 Vi τ = τ + μi[−1] − μi+1[−1] Vi,i=1,...,n− 1, and −1 −1 (12.61) Vn τ = τ + μn−1[−1] + μn[−1] Vn. 12.4. SCREENING OPERATORS 239

To perform the calculation for the case i = n in (12.60), we may assume that n =2. We have −1 V2 τ − μ1[−1] τ − μ2[−1] τ τ + μ2[−1] τ + μ1[−1] −1 = τ + μ2[−1] V2 − 1 τ − μ2[−1] τ τ + μ2[−1] τ + μ1[−1] −1 = τ + μ2[−1] τ + μ1[−1] V2 − 2τ τ τ + μ2[−1] τ + μ1[−1] .

Furthermore, applying the operator V2 we ﬁnd V2 τ + μ2[−1] τ + μ1[−1] = τ + μ1[−1] + 2μ2[−1] V2 +1 τ + μ1[−1] =2 τ + μ1[−1] + μ2[−1] and so by (12.61), −1 V2 τ τ + μ2[−1] τ + μ1[−1] =2 thus completing the calculation. The relations Vi τ − μ1[−1] ... τ − μn[−1] 1=0,i=1,...,n, are veriﬁed with the use of (12.54). Using the notation (12.47), we can write the constant terms of the elements (12.57) in the new variables in the form H 1 − − − − − − (12.62) m = 2 hm τ + μ1[ 1],...,τ + μn−1[ 1],τ μn[ 1],...τ μ1[ 1] 1 1 − − − − − − + 2 hm τ + μ1[ 1],...,τ + μn[ 1],τ μn−1[ 1],...τ μ1[ 1] 1. To summarize, we have the following corollary of the type D part of the proof of Theorem 12.3.1. Corollary 12.4.11. Each of the families rE rE rE rE ◦ T 2,T 4,...,T 2n−2,T n and rH rH rH rE ◦ T 2,T 4,...,T 2n−2,T n with r running over the set of nonnegative integers is algebraically independent and −1 −1 generates the classical W-algebra W(o2n) ⊂ S t h[t ] . −1 −1 Type Cn. To embed the classical W-algebra into S t h[t ] ,fori =1,...,n and all r 0 introduce new variables 1 μ [−r − 1] = − F (r). i r! ii (r) − − − − − − We have the relations hi = r! μi[ r 1] + μi+1[ r 1] for i =1,...,n 1 (r) and hn = −r! μn[−r − 1]. The Cartan matrix has the form ⎡ ⎤ 2 −10... 00 ⎢ ⎥ ⎢−12−1 ... 00⎥ ⎢ ⎥ ⎢ 0 −12... 00⎥ A = ⎢ ⎥ ⎢ .. ⎥ ⎢ ...... ⎥ ⎣ 000... 2 −2 ⎦ 000... −12 240 12. CLASSICAL W-ALGEBRAS so that the entries of the diagonal matrix D =diag[1,...,n] are found by

1 = ···= n−1 =1 and n =1/2. Hence, n ∂ ∂ ∂ r! a = − + ,i=1,...,n− 1, ji (r) ∂μi[−r − 1] ∂μi+1[−r − 1] j=1 ∂hj and n ∂ 2∂ r! a = − . jn (r) ∂μn[−r − 1] j=1 ∂hj − ◦ − Therefore, for i =1,...,n 1 the screening operators (12.35) are Vi = Vi,where ∞ ∂ ∂ V = V − , i ir ∂μ [−r − 1] ∂μ [−r − 1] r=0 i i+1 ◦ − and Vn = 2Vn with ∞ ∂ (12.63) V = V ; n nr ∂μ [−r − 1] r=0 n the coeﬃcients Vir are found from the expansions ∞ ∞ μ [−m] − μ [−m] V zr =exp i i+1 zm,i=1,...,n− 1 ir m r=0 m=1 and ∞ ∞ 2μ [−m] V zr =exp n zm. nr m r=0 m=1 W The elements w2,...,w2n of the algebra (sp2n) become the respective coef- −1 −1 ﬁcients E2,...,E2n in the expansion in S t h[t ] ⊗ C[τ], τ − μ1[−1] ... τ − μn[−1] τ + μn[−1] ... τ + μ1[−1]

2n 2n−2 2n−3 = τ + E2 τ + E3 τ + ···+ E2n. By (12.52) we have Em = em τ + μ1[−1],...,τ + μn[−1],τ − μn[−1],...,τ − μ1[−1] 1. The relation (12.64) Vi τ − μ1[−1] ... τ − μn[−1] τ + μn[−1] ... τ + μ1[−1] =0 − is veriﬁed for i =1,...,n 1inthesamewayasforglN with the use of (12.54). In the case i = n we have Vn τ = τ +2μn[−1] Vn, so that Vn τ − μn[−1] τ + μn[−1] = τ + μn[−1] Vn − 1 τ + μn[−1] = τ + μn[−1] τ +3μn[−1] Vn, and (12.64) with i = n also follows. 12.5. BIBLIOGRAPHICAL NOTES 241

By (12.49) and (12.51), for all m 1 the elements (12.65) Hm = hm τ + μ1[−1],...,τ + μn[−1],τ − μn[−1],...,τ − μ1[−1] 1 W belong to (sp2n). As with Corollary 12.4.8, the following is implied by the type C part of the proof of Theorem 12.3.1. Corollary 12.4.12. Each of the families r r T Em and T Hm with m =2, 4,...,2n and r =0, 1,... is algebraically independent and generates W W ⊂ −1 −1 the classical -algebra (sp2n) S t h[t ] . 12.5. Bibliographical notes Classical W-algebras W(g) were defined by Drinfeld and Sokolov [33]andwere used to introduce equations of the KdV type for arbitrary simple Lie algebras g.For their connections to the affine W-algebras see the review by Arakawa [4]. In the case W g = glN the algebra (g) is isomorphic to the Adler–Gelfand–Dickey algebra [1], [56]. For the definition of the classical W-algebras via screening operators see [46, Chapter 8]. In Section 12.2 we followed [115], where generators of W(g)in type G2 were also given. More general results describing generators of the classical W-algebras W(g,f) for arbitrary nilpotent elements f and the Poisson brackets are given by De Sole, Kac and Valeri [27], [28]. The functions (12.47) and (12.48) can be regarded as specializations of noncommutative symmetric functions in the sense of the general theory developed by Gelfand et al. [57].

~~CHAPTER 13~~

~~Aﬃne Harish-Chandra isomorphism~~

We will now discuss an affine version of the Harish-Chandra isomorphism (4.7) where the center Z(g) of the universal enveloping algebra U(g) is replaced with the center Z(g) of the completed universal enveloping algebra U−h∨ (g)atthecritical level. The role of the algebra of W -invariants in U(h) will now be played by a completion of the classical W-algebra associated with the Langlands dual Lie algebra Lg (corresponding to the transposed Cartan matrix of g). This affine Harish-Chandra isomorphism turns out to be largely determined by an isomorphism between the Feigin–Frenkel center z(g) and the classical W-algebra W(Lg). Theorem 13.1.1 stated below originates in [39] and its detailed proof is given in Frenkel’s book [46, Theorem 4.3.2]. Together with [46, Theorem 4.3.6] these are principal results of the book. In fact, these theorems state much more than just algebra isomorphisms for the centers. The classical W-algebras can be understood as algebras of functions on geometric objects known as opers and the affine Harish-Chandra isomorphisms in both versions are coordinate-independent; they are equivariant with respect to changes of variables. The proofs in [46] rely on the properties of the Wakimoto modules over g and the fact that elements of Z(g) act by scalar multiplication in these modules. We will not go beyond the algebraic statement of the affine Harish-Chandra isomorphism and avoid using Wakimoto modules (although we will rely on Theo- rem 13.1.1 in type C to calculate the images of the Segal–Sugawara vectors). As before, we will only work with the Lie algebras g of classical types and adopt a more direct approach based on the Yangian version of the Harish-Chandra isomorphism, as considered in Chapters 10 and 11; cf. [51, Section 8]. Nonetheless, we will return to the Wakimoto modules in Chapter 15 and calculate the eigenvalues of the Sugawara operators constructed in Chapters 7 and 8 which act in these modules.

~~13.1. Feigin–Frenkel centers and classical W-algebras~~

As in Section 4.2, suppose that g = n− ⊕ h ⊕ n+ is a triangular decomposition of a simple Lie algebra g. Regard h as a subalgebra of g via the embedding taking H ∈ h to H[0]. The adjoint action of h on t−1g[t−1] extends to the universal enveloping algebra and we have a natural analogue of the homomorphism (4.6) for the h-centralizer, h (13.1) f :U t−1g[t−1] → U t−1h[t−1] which is the projection to the ﬁrst summand in the direct sum decomposition −1 −1 h −1 −1 −1 −1 h −1 −1 −1 −1 U t g[t ] =U t h[t ] ⊕ U t g[t ] ∩ U t g[t ] t n−[t ] .

~~243 244 13. AFFINE HARISH-CHANDRA ISOMORPHISM~~

~~The second summand is the kernel of the projection which coincides with −1 −1 h −1 −1 −1 −1 U t g[t ] ∩ t n+[t ]U t g[t ] .~~

Note that the roles of n− and n+ are interchanged as compared to the projection (4.6). This is necessary for the definition of the homomorphism to be consistent with the traditional formulas for the Wakimoto modules. It is clear that the derivation T on the algebra U t−1g[t−1] defined in (6.13) preserves the h-centralizer and that the homomorphism f commutes with T . Recall that the Feigin–Frenkel center z(g) is the center of the vertex algebra −1 −1 h V−h∨ (g) and we can regard z(g) as a commutative subalgebra of U t g[t ] ;see Section 6.2. Using Theorem 12.3.1 and the isomorphism (12.38), we will regard the classical − − ∼ W-algebra as a subalgebra of U t 1h[t 1] = V(h) and simply write W(g)instead of W(g). We will identify the Cartan subalgebras of g and its Langlands dual Lg via a natural isomorphism so that W(Lg) will be viewed as a subalgebra of U t−1h[t−1] . The next theorem provides an affine version of the Harish-Chandra isomorphism for z(g). Theorem 13.1.1. The restriction of the homomorphism (13.1) to the subalgebra z(g) yields an isomorphism (13.2) z(g) →W(Lg). For the classical types (with the exception of type C), the theorem will follow from the calculation of the Harish-Chandra images of the families of algebraically independent generators of z(g) constructed in Chapters 7 and 8. We will show separately for types A, B and D that these images coincide with the algebraically independent generators of W(Lg) described in Chapter 12. In type C we will rely on Theorem 13.1.1 for the calculation of the images. As before, we will use the element τ as defined in (7.1) and consider the natural extension of the homomorphism (13.1) to the homomorphism h (13.3) f :U t−1g[t−1] ⊗ C[τ] → U t−1h[t−1] ⊗ C[τ], which is identical on C[τ].

~~Type AN−1. Take g = glN and use the notation of Section 7.1. Recall also the noncommutative complete and elementary symmetric polynomials deﬁned in (12.47) and (12.48). Set~~

μi[−r]=Eii[−r],i=1,...,N and r =1, 2,.... Proposition 13.1.2. Under the homomorphism (13.3) we have (m) − − (13.4) tr1,...,m A τ + E[ 1]1 ... τ + E[ 1]m → − − em τ + μ1[ 1],...,τ + μN [ 1] , and (m) − − (13.5) tr1,...,m H τ + E[ 1]1 ... τ + E[ 1]m → − − hm τ + μ1[ 1],...,τ + μN [ 1] . 13.1. FEIGIN–FRENKEL CENTERS AND CLASSICAL W-ALGEBRAS 245

Proof. We argue as in the proof of Proposition 4.6.1. As we pointed out in Lemma 7.1.2, τ + E[−1] is a Manin matrix. By Remark 3.2.3, for a Manin matrix M we have (m) · tr1,...,m A M1 ...Mm = sgn σ Miσ(1) i1 ...Miσ(m) im .

~~Ni1>···>im1 σ∈Sm Taking M = τ + E[−1] we ﬁnd that the image of the product~~

Miσ(1) i1 ...Miσ(m) im under the homomorphism f is zero unless σ is the identity permutation. In that case the image is − − Mi1 i1 ...Mim im = τ + μi1 [ 1] ... τ + μim [ 1] which proves (13.4). To verify (13.5), we use the following counterpart of (3.18) provided by Remark 3.2.3:

~~(m) tr1,...,m H M1 ...Mm 1 = Mim iσ(m) ...Mi1 iσ(1) , α1! ...αN ! Ni1···im1 σ∈Sm where αi is the multiplicity of index i ∈{1,...,N} in the multiset {i1,...,im}.~~

The image of the product Mim iσ(m) ...Mi1 iσ(1) under the homomorphism f is zero unless σ stabilizes the multiset {i1,...,im}. The desired formula follows by taking into account the number of permutations which stabilize the multiset; cf. the proof of Proposition 4.7.1. Note that (13.5) also follows from (13.4) due to the MacMahon Master Theorem (Theorem 3.2.1). Corollary 13.1.3. Under the homomorphism (13.3) we have (13.6) f :cdet τ + E[−1] → τ + μN [−1] ... τ + μ1[−1] and ∞ m (13.7) f : qm tr τ + E[−1] m=0 N −1 −1 → 1 − q τ + μ1[−1] ··· 1 − q τ + μi[−1] i=1 × 1 − q τ + μi−1[−1] ··· 1 − q τ + μ1[−1] , where q is an independent variable. Proof. Relation (13.6) is immediate from (7.10) and (13.4), while (13.7) is a consequence of the Newton identity (3.39) for the Manin matrix M = τ +E[−1]. Corollary 13.1.4. The restriction of the homomorphism (13.1) to the subalgebra z(glN ) yields an isomorphism →W (13.8) z(glN ) (glN ).

Hence, Theorem 13.1.1 holds for g = glN . 246 13. AFFINE HARISH-CHANDRA ISOMORPHISM Proof. − By Theorem 7.1.4, the coefficients φ1,...,φN of cdet τ +E[ 1] form a complete set of Segal–Sugawara vectors for glN . This means that the elements r T φm with m =1,...,N and r =0, 1,... are algebraically independent generators of z(glN ). By Corollary 13.1.3 their respective images under the homomorphism f rE are the elements T m as defined in (12.44). However, by Corollary 12.4.8 those W W images are algebraically independent generators of the classical -algebra (glN ). Hence, the homomorphism f induces the isomorphism (13.8). We will now give an alternative proof of a slightly different form of (13.4) (and hence of Proposition 13.1.2). This argument relies on the calculations of Harish- + Chandra images for the dual Yangian Y (glN ) and it is this approach which we will be able to generalize to the other classical types below. In the following we use the notation of Section 10.5. Proposition 13.1.5. Under the homomorphism (13.3) we have (m) f :tr1,...,m A ∂u + E(u)+1 ... ∂u + E(u)+ m → em ∂u + μ1(u),...,∂u + μN (u) , where ∞ − − r μi(u)= μi[ r 1]u . r=0 Proof. Due to the consistency of the definitions of the homomorphisms (10.24) and (13.1) and the arguments of Section 10.5, it is sufficient to calculate the classical limit of the Harish-Chandra image of the element (10.40). This element equals (10.42) and so by Corollary 10.2.2, the image is given by m N − k − (−1)k λ+ (u) ···λ+ (u − k +1)e k∂u m − k i1 ik k=0 Ni1>···>ik1 which coincides with − − 1 − λ+ (u)e ∂u ... 1 − λ+ (u)e ∂u . i1 im Ni1>···>im1 − − + ∂u By the definition (10.25), the classical limit of 1 λi (u)e coincides with the expression ∂u + μi(u) thus completing the proof. Remark 13.1.6. Note that relation (13.4) is equivalent to Proposition 13.1.5. Furthermore, the corresponding counterparts of relations (13.5), (13.6) and (13.7) are obtained by replacing τ with ∂u,thematrixE[−1] with E(u)+,andμi[−1] with μi(u). The equivalence is a consequence of the vertex algebra structure on the vacuum module V−N (glN ) as outlined in Section 6.6. Applying the state-field cor- ⊂ respondence map to elements of z(glN ) V−N (glN ), we get formal Laurent series in z whose coefficients are endomorphisms of V−N (glN ). For the elements under discussion, these series are given by formulas which coincide with those given in (7.23), (7.24), (7.25) and (7.26) for the completed universal enveloping algebra. In accordance with the vacuum axioms, the application, say, of (7.23) to the vacuum vector ∈ 1 V−N (glN ) yields formal power series in z which appear in Proposition 13.1.5 (with the variable changed from z to u). Moreover, the evaluations of the series at u = 0 recover the original elements of z(glN ) in (13.4) and such evaluations are consistent with the Harish-Chandra homomorphism. 13.1. FEIGIN–FRENKEL CENTERS AND CLASSICAL W-ALGEBRAS 247

~~Type Bn. Now take g = oN with odd N =2n + 1 and use the notation of Section 8.1. Consider the Segal–Sugawara vectors provided by Theorem 8.1.6. Set~~

μi[−r]=Fii[−r],i=1,...,n and r =1, 2,... and recall the notation (12.47). Proposition 13.1.7. Under the homomorphism (13.3) the image of the polynomial (m) γm(N)tr1,...,m S τ + F [−1]1 ... τ + F [−1]m equals hm τ + μ1[−1],...,τ + μn[−1],τ − μn[−1],...τ − μ1[−1] . Proof. We will apply the argument similar to the one used in the proof of Proposition 13.1.5. As in Section 10.5, extend the ascending ﬁltration on the + (−r) − dual Yangian Y (oN ) deﬁned by deg tij = r to the algebra of formal series + Y (oN )[[u, ∂u]] by setting deg u =1anddeg ∂u = −1 so that the associated graded −1 −1 algebra is isomorphic to U t oN [t ] [[u, ∂u]]. The element − − (m) − + ∂u − + ∂u (13.9) γm(N)tr1,...,m S 1 T1 (u)e ... 1 Tm (u)e has degree −m and its image in the graded algebra coincides with (m) (13.10) γm(N)tr1,...,m S ∂u + F (u)+1 ... ∂u + F (u)+ m , where ∞ r−1 F (u)+ = F [−r]u . r=1 Write (13.9) in the form m − γ (N)tr S(m) (−1)k T +(u) ...T+(u − k +1)e k∂u . m 1,...,m i1 ik k=0 1i1<···

Applying the conjugation by the longest permutation in Sk and using the cyclic property of trace we get (k) + + − + + − (k) tr1,...,k S T1 (u) ...Tk (u k +1)=tr1,...,k Tk (u) ...T1 (u k +1)S which equals (k) + − + tr1,...,k S T1 (u k +1)...Tk (u) by (11.11) and the defining relations (11.37) for the dual Yangian. Hence by The- orem 11.2.1, the Harish-Chandra image of the expression (13.9) is found by m N + m − 2 k + + −k∂u (−1) γk(N) λ (u − k +1)...λ (u)e m − k i1 ik k=0 Ni1···ik1 with the condition that n + 1 occurs among the summation indices i1,...,ik at most once. We can rewrite the image in the form m N + m − 2 k + −∂u + −∂u (13.12) (−1) γk(N) λ (u)e ...λ (u)e m − k j1 jk k=0 1j1···jkN with the condition that n + 1 occurs among the summation indices j1,...,jk at most once. The next step is to express (13.12) in terms of the new variables − − + ∂u (13.13) νi(u)=1 λi (u)e ,i=1,...,N. This is done by a combinatorial argument as shown in the following lemma. Lemma . − m+1 N/2−2 13.1.8 The expression (13.12) multiplied by 2( 1) N+m−2 equals m − N/2 2 − r a1 an an a1 ( 1) − ν1(u) ...νn(u) νn (u) ...ν1 (u) + N + r 3 ··· r=0 a1+ +a1 =r m N/2 − 2 − r a1 an − ( 1) − ν1(u) ...νn(u) νn+1(u) 2 N + r 3 ··· − r=1 a1+ +a1 =r 1 a a × νn (u) n ...ν1 (u) 1 , where a1,...,a1 run over nonnegative integers. Proof. The statement is verified by substituting (13.13) into both terms and calculating the coefficients of the sum − − (13.14) λ+ (u)e ∂u ...λ+ (u)e ∂u i1 ik 1i1···ikN for all 0 k m,wheren + 1 occurs among the summation indices i1,...,ik at most once. Note the following expansion formula for the noncommutative complete symmetric functions (12.47), which is a consequence of (12.53), r p + r − 1 (13.15) h (1 − x ,...,1 − x )= (−1)k h (x ,...,x ). r 1 p r − k k 1 p k=0

+ −∂u Take xi = λi (u)e with i =1,...,n,n ...,1 and apply (13.15) with p =2n to the first term in the expression of the lemma. Using a similar expansion for the 13.1. FEIGIN–FRENKEL CENTERS AND CLASSICAL W-ALGEBRAS 249 second term we find that the coefficient of the sum (13.14) in the entire expression will be found as m N/2 − 2 N + r − 3 N/2 − 2 N/2+m − 1 (−1)r−k = , N + r − 3 r − k N + k − 3 m − k r=k which coincides with N/2 − 2 N + m − 2 2(−1)m−k+1 γ (N) , k N + m − 2 m − k as claimed.

Denote the expression in Lemma 13.1.8 by Am. Since the degree of the element (13.9) is −m, its Harish-Chandra image (13.12) and the expression Am also have degree −m. Observe that the terms in the both sums of Am are independent of m so that Am+1 = Am + Bm+1,where N/2 − 2 B =(−1)m+1 m+1 N + m − 2 a1 an a a × ν1(u) ...νn(u) νn (u) n ...ν1 (u) 1 ··· a1+ +a1 =m+1 N/2 − 2 − m+1 a1 an − +( 1) − ν1(u) ...νn(u) νn+1(u) 2 N + m 2 ··· a1+ +a1 =m a a × νn (u) n ...ν1 (u) 1 .

Since Am+1 has degree −m−1, its component of degree −m is zero, and so the sum of the homogeneous components of degree −m of Am and Bm+1 is zero. However, each element νi(u) has degree −1 with the top degree component equal to ∂u+μi(u), where ∞ r (13.16) μi(u)= μi[−r − 1] u . r=0

This implies that the component of Am of degree −m equals the component of degree −m of the term − N/2 2 − m+1 a1 an an a1 2( 1) − ν1(u) ...νn(u) νn (u) ...ν1 (u) . N + m 2 ··· a1+ +a1 =m Taking into account the constant factor used in Lemma 13.1.8, we can conclude that the Harish-Chandra image of (13.10) is given by the noncommutative complete symmetric function (13.17) hm ∂u + μ1(u),...,∂u + μn(u),∂u + μn (u),...,∂u + μ1 (u) .

The proof is completed by setting u = 0 in the coeﬃcients of the polynomials in ∂u which appear in (13.10) and (13.17); cf. Remark 13.1.6. Corollary 13.1.9. The restriction of the homomorphism (13.1) to the subalgebra z(o2n+1) yields an isomorphism →W (13.18) z(o2n+1) (sp2n).

~~Hence, Theorem 13.1.1 holds for g = o2n+1. 250 13. AFFINE HARISH-CHANDRA ISOMORPHISM~~

Proof. By Theorem 8.1.9, the elements φ22,φ44,...,φ2n 2n form a complete r set of Segal–Sugawara vectors for o2n+1. Therefore, the elements T φmm with m =2, 4,...,2n and r =0, 1,... are algebraically independent generators of z(o2n+1). By Proposition 13.1.7, their respective images under the homomorphism f rH are the elements T m as deﬁned in (12.65). By Corollary 12.4.12 these images are W W algebraically independent generators of the classical -algebra (sp2n). Hence, the homomorphism f induces the isomorphism (13.18).

~~Type Dn. Take g = oN with N =2n and use the notation of Section 8.1. Consider the Segal–Sugawara vectors provided by Theorem 8.1.6. Set~~

~~μi[−r]=Fii[−r],i=1,...,n and r =1, 2,....~~

~~Proposition 13.1.10. Under the homomorphism (13.3) the image of the polynomial (m) γm(N)tr1,...,m S τ + F [−1]1 ... τ + F [−1]m equals~~

~~1 − − − − − − 2 hm τ + μ1[ 1],...,τ + μn−1[ 1],τ μn[ 1],...τ μ1[ 1] 1 − − − − − − + 2 hm τ +μ1[ 1],...,τ+μn[ 1],τ μn−1[ 1],...τ μ1[ 1] .~~

Proof. We repeat the beginning of the proof of Proposition 13.1.7 with N now taking the even value 2n, up to the application of the Yangian character formula. This time we apply Theorem 11.2.2 to conclude that the Harish-Chandra image of the expression (13.9) is found by m 2n + m − 2 k + + −k∂u (−1) γk(2n) λ (u − k +1)...λ (u)e m − k i1 ik k=0 2ni1···ik1 with the condition that n and n do not occur simultaneously among the summation indices i1,...,ik. Rewrite the image in the form m 2n + m − 2 k + −∂u + −∂u (13.19) (−1) γk(2n) λ (u)e ...λ (u)e m − k j1 jk k=0 1j1···jk2n with the condition that n and n do not occur simultaneously among the summation indices j1,...,jk. Introducing new variables by the same formulas (13.13) we come to the Dn series counterpart of Lemma 13.1.8, where we use the notation

~~− 2n + r − 2 1 c = − . r n − 1 13.1. FEIGIN–FRENKEL CENTERS AND CLASSICAL W-ALGEBRAS 251~~

Lemma 13.1.11. The expression (13.19) multiplied by 2cm equals a1 a 2cm ν1(u) ...ν1 (u) 1 ··· a1+ +a1 =m a =a =0 n n a1 a + cm ν1(u) ...ν1 (u) 1 ··· a1+ +a1 =m only one of an and an is zero m rcr − a1 a1 − ν1(u) ...ν1 (u) n + r 1 ··· r=1 a1+ +a1 =r an=an =0 m (n − 1) cr a1 a1 + − ν1(u) ...ν1 (u) , n + r 1 ··· r=1 a1+ +a1 =r only one of an and an is zero where a1,...,a1 run over nonnegative integers. Proof. Substitute (13.13) into the expression given in the lemma and calculate the coeﬃcients of the sum − − (13.20) λ+ (u)e ∂u ...λ+ (u)e ∂u . i1 ik 1i1···ik2n The argument splits into two cases, depending on whether neither of n and n occurs among the summation indices i1,...,ik in (13.20) or only one of them occurs. The application of the expansion formula (13.15) brings this to a somewhat lengthy but straightforward calculation with binomial coeﬃcients which we will omit.

Let Am denote the four-term expression in Lemma 13.1.11. This expression equals 2cm times the Harish-Chandra image of (13.9) and so Am has degree −m. Hence, the component of degree −m of the expression Am+1 is zero. On the other hand, each element νi(u) has degree −1 with the top degree component equal to ∂u + μi(u), where μi(u) is deﬁned in (13.16). This implies that the component of degree −m in the sum of the third and fourth terms in Am is zero. Therefore, the component of Am of degree −m equals the component of degree −m in the sum of the ﬁrst and the second terms. Taking into account the constant factor 2cm,we conclude that the Harish-Chandra image of (13.10) equals the component of degree −m of the sum 1 a1 a a1 a ν1(u) ...ν1 (u) 1 + ν1(u) ...ν1 (u) 1 ··· 2 ··· a1+ +a1 =m a1+ +a1 =m an=an =0 only one of an and an is zero and hence coincides with 1 − − (13.21) 2 hm ∂u + μ1(u),...,∂u + μn−1(u),∂u μn(u),...∂u μ1(u) 1 − − + 2 hm ∂u + μ1(u),...,∂u + μn(u),∂u μn−1(u),...∂u μ1(u) .

~~To complete the proof set u = 0 in the coeﬃcients of the polynomials in ∂u which appear in (13.10) and (13.21); cf. Remark 13.1.6. 252 13. AFFINE HARISH-CHANDRA ISOMORPHISM~~

Recall the additional generator Pfaffian Pf F [−1] of the Feigin–Frenkel center z(o2n) defined by (8.6). Proposition 13.1.12. The image of the element Pf F [−1] under the homomorphism (13.1) equals f :PfF [−1] → μ1[−1] − τ ... μn[−1] − τ 1. Proof. By using the notation (12.59) we can state the proposition equivalently − → − n E ◦ in the form f :PfF [ 1] ( 1) n . We will use the automorphism of the Lie −1 −1 algebra t o2n[t ] defined on the generators by → (13.22) Fkl[r] Fk˜ ˜l[r], where k → k˜ is the involution on the set {1,...,2n} such that n → n, n → n and k → k for all k = n, n. Note that this automorphism takes Pf F [−1] to − − E ◦ →−E◦ −1 −1 Pf F [ 1]. Similarly, n n with respect to the automorphism of t h[t ] induced by (13.22). r Now apply Corollary 12.4.11 and observe that the elements T H2k with r 0 and k =1,...,n− 1 are stable under the automorphism (13.22). The Harish- Chandra image f Pf F [−1] is a polynomial in the generators of W(o2n) and its degree with respect to the variables μ1[−1],...,μn[−1] does not exceed n.There- − E ◦ fore, f Pf F [ 1] must be proportional to n . Comparing the coefficients of the product μ1[−1] ...μn[−1] in each of these two polynomials we can conclude that − → − n E ◦ f :Pf F [ 1] ( 1) n as required.

~~As with types A and B we have the following.~~

~~Corollary 13.1.13. The restriction of the homomorphism (13.1) to the subalgebra z(o2n) yields an isomorphism~~

~~(13.23) z(o2n) →W(o2n).~~

Hence, Theorem 13.1.1 holds for g = o2n. Proof. r − By Theorem 8.1.9, the elements T φmm with m =2, 4,...,2n 2to- gether with T r Pf F [−1] for r =0, 1,... are algebraically independent generators of z(o2n). By Propositions 13.1.10 and 13.1.12 their images under the homomor- rH − n rE ◦ phism f are the elements T m as deﬁned in (12.62) and ( 1) T n , respectively. By Corollary 12.4.11 these images are algebraically independent generators of the classical W-algebra W(o2n). Hence, the homomorphism f induces the isomorphism (13.23).

~~Type Cn. Now let g = spN with N =2n and use the notation of Section 8.3. Consider the Segal–Sugawara vectors provided by Theorem 8.3.2. Set~~

μi[−r]=Fii[−r],i=1,...,n and r =1, 2,... and recall the noncommutative elementary symmetric polynomials given in (12.48). By Proposition 8.3.7, expression (8.36) admits an equivalent form which is well- deﬁned for all values 1 m 2n + 1. We will keep the same notation (8.36) for all these values with the understanding that the equivalent form is used for m>n. 13.1. FEIGIN–FRENKEL CENTERS AND CLASSICAL W-ALGEBRAS 253

Proposition 13.1.14. For all 1 m 2n +1 the image of the polynomial − (m) − − (13.24) γm( 2n)tr1,...,m S τ + F [ 1]1 ... τ + F [ 1]m under the homomorphism (13.3) equals em τ + μ1[−1],...,τ + μn[−1],τ,τ − μn[−1],...τ − μ1[−1] . Proof. First we will prove the statement under the assumption m n.Then it will be extended to all values m 2n + 1 by using an argument similar to that of Section 5.5. + As for type B, extend the ascending ﬁltration on the dual Yangian Y (sp2n) (−r) − + deﬁned by deg tij = r to the algebra of formal series Y (sp2n)[[u, ∂u]] by setting deg u =1anddeg ∂u = −1 so that the associated graded algebra is isomorphic to −1 −1 U t sp2n[t ] [[u, ∂u]]. The element − − − (m) − + ∂u − + ∂u (13.25) γm( 2n)tr1,...,m S 1 T1 (u)e ... 1 Tm (u)e has degree −m and its image in the graded algebra coincides with − (m) (13.26) γm( 2n)tr1,...,m S ∂u + F (u)+1 ... ∂u + F (u)+ m , where ∞ r−1 F (u)+ = F [−r]u . r=1 Expand (13.25) as m − γ (−2n)tr S(m) (−1)k T +(u) ...T+(u − k +1)e k∂u . m 1,...,m i1 ik k=0 1i1<······>ik1 254 13. AFFINE HARISH-CHANDRA ISOMORPHISM

Lemma 13.1.15 . Suppose that m n. The expression (13.28) multiplied by − m 2 n−m+1 2( 1) n+1 equals m 2n − r +2 (13.29) (−1)r n +1 r=0 − − × 1 − κ+(u)e ∂u ... 1 − κ+(u)e ∂u . i1 ir 2n+2i1>···>ir 1 Proof. Use the expansion formula for the noncommutative elementary symmetric functions (12.48), r p − k e (1 − x ,...,1 − x )= (−1)k e (x ,...,x ). r 1 p r − k k 1 p k=0 − κ+ ∂u Taking xi = i (u)e with i =1,...,2n + 2, and simplifying the expression involving binomial coefficients, we find that the coefficient of the sum − − (13.30) κ+(u)e ∂u ...κ+ (u)e ∂u i1 ik 2n+2i1>···>ik1 in (13.29) equals n − k 2n − k +2 (−1)m−k m − k n +1 which coincides with 2n − m +1 2n − k +1 2(−1)m−k γ (−2n) k n +1 m − k as claimed.

The expression (13.29) is well-deﬁned for m n + 1 and we denote it by Am. The proof of Lemma 13.1.15 shows that if m = n + 1 then the coeﬃcient of the sum (13.30) in An+1 is zero for all 0 k n. Furthermore, by the second part of Corollary 11.2.4, the sum (13.30) is zero for k = n + 1. Therefore, An+1 =0. The degree of the element (13.25) is −m and so, for m n the expression Am also has degree −m. Hence, the component of degree −m of the expression Am+1 is zero; this holds for m = n as well, because An+1 =0.WehaveAm+1 = Am +Bm+1, where 2n − m +1 B =(−1)m+1 m+1 n +1 − − × 1 − κ+(u)e ∂u ... 1 − κ+ (u)e ∂u . i1 im+1 2n+2i1>···>im+11

The sum of the homogeneous components of degree −m of Am and Bm+1 is zero. − − κ+ ∂u − However, each factor 1 i (u)e with i = n +1,n+2 hasdegree 1 with the top degree component equal to ∂u + μi(u), where μi(u) is deﬁned in (13.16). Therefore, the components of degree −m in Bm+1 can only come from the terms corresponding to the sets i1 > ···>im+1 containing indices n +1orn +2.Inthe cases where only one of these indices occurs, the sum simpliﬁes due to the relations (11.42), since − − − κ+ ∂u − κ+ ∂u 1 n+1(u)e +1 n+2(u)e =2. 13.2. YANGIAN CHARACTERS AND CLASSICAL W-ALGEBRAS 255

In the cases where both indices occur, by (11.42) we have − − − − κ+ ∂u − κ+ ∂u − + + − 2 ∂u (13.31) 1 n+2(u)e 1 n+1(u)e =1 λn (u)λn (u 1)e . + + Using (11.50), we can express λn (u)=λn+1(u)asthefraction + + + λ (u − n)λ (u − n +1)...λ − (u − 2) (13.32) λ+ (u)= 1 2 n 1 . n+1 + − − + − + − λ1 (u n 1)λ2 (u n) ...λn (u 2) Hence the product (13.31) has degree −1 with the top degree component equal to 2∂u. Thus, taking Lemma 13.1.15 into account, we can conclude that the Harish- Chandra image of (13.26) equals (13.33) em ∂u + μ1(u),...,∂u + μn(u),∂u,∂u − μn(u),...∂u − μ1(u) . The argument is completed by setting u = 0 in the coeﬃcients of the polynomials in ∂u which appear in (13.26) and (13.33); cf. Remark 13.1.6. To complete the proof of the proposition, we will rely on Theorem 13.1.1 in type C [46, Theorem 8.1.5], which states that the restriction of the homomorphism (13.1) to the subalgebra z(sp2n) yields an isomorphism →W (13.34) z(sp2n) (o2n+1).

For a ∈{0, 1,...,m } the coefficient φma in the expansion (8.36) is a homogeneous −1 −1 element of U t sp2n[t ] of degree a with respect to the grading defined by deg Fij[−r]=r. Since the homomorphism (13.1) preserves the grading, by Corol- lary 12.4.10 the image of φma under the isomorphism (13.34) is a polynomial in r the generators T Em with m =2, 4,...,2n and r =0, 1,... with r + m a.For a fixed value of m and varying values of n the coefficients of the polynomial are rational functions in n. Therefore, they are uniquely determined by infinitely many values of n m. This allows us to extend the range of n for the Harish-Chandra image of (13.24) to all values n (m − 1)/2 for which it is defined.

13.2. Yangian characters and classical W-algebras Here we will discuss an alternative way to see that the image of the restriction of the homomorphism (13.1) to the subalgebra z(g) is contained in the classical W-algebra W(Lg). The idea is suggested by the alternative proof of Theorem 7.1.3 given in Section 10.5. Namely, for g = glN the generators of z(glN ) are recovered via a classical limit procedure from some elements of the subalgebra of invariants V + z( cri). The Harish-Chandra images of those elements are polynomials in λi (u + a) calculated in Corollary 10.2.2, while Proposition 10.6.3 states that the images belong to the intersection of the kernels of the screening operators. Therefore, to W see that the Harish-Chandra images of elements of z(glN )belongto (glN ), we can + just verify that the classical limits of the polynomials in λi (u + a) are annihilated by all screening operators Vi introduced in Section 12.2. In this sense, the operators + Vi are thus recovered as classical limits of the screening operators Si . Type A. Consider the algebra L+ introduced in (10.49) and embed it into the algebra of formal power series in the variables μi[−r − 1] with i =1,...,N and r =0, 1,... by setting ∞ + → − − − r (13.35) λi (a) 1 μi[ r 1] a . r=0 256 13. AFFINE HARISH-CHANDRA ISOMORPHISM

+ Define the degrees of the variables by deg μi[−r − 1] = −r − 1. Given A ∈L , consider the corresponding formal power series in the variables μi[−r − 1] and take its homogeneous component A of the maximum degree. This component is a polynomial and so we have a map + (13.36) gr : L → C μi[−r − 1] | i =1,...,N, r =0, 1,... ,A → A. Note its property which is immediate from the definition: (13.37) gr(AB)=gr(A)gr(B). + Recall the subalgebra Rep Y (glN ) defined in (10.53). Proposition 13.2.1. The image of the restriction of the map (13.36) to the + W W subalgebra Rep Y (glN ) is contained in the classical -algebra (glN ) and so it defines a map + →W gr : Rep Y (glN ) (glN ).

Proof. Similar to (13.35), introduce variables σi[−r − 1] by the expansion ∞ + → − − r (13.38) σi (a) σi[ r 1] a r=0 and set deg σi[−r − 1] = −r − 1. Regarding a as a formal variable in (10.50) and (10.51), write the screening operators in terms of the new variables. Explicitly, for i =1,...,N − 1set ⎧ ⎪ 1 − μ [−1] σ [−k − 1] for j = i ⎨⎪ i i ◦ − → k 0 (13.39) Si : μj [ 1] ⎪− 1 − μi+1[−1] σi[−1] for j = i +1 ⎩⎪ 0forj = i, i +1 and T r (13.40) S ◦ : μ [−r − 1] → S ◦ μ [−1] ,r 1, i j r! i j ◦ where the derivation T isdeﬁnedin(12.46).TheactionofSi then extends to the entire algebra of formal power series in the variables μi[−r − 1] via the Leibniz rule as in (10.52). ∈ + + − Now suppose that A Rep Y (glN )sothatSi A =0foralli =1,...,N 1. ◦ Denote by A the corresponding formal power series in the variables μi[−r −1]. By ◦ L+ the deﬁnition of the operators Si , their restriction to the subalgebra coincides + ◦ ◦ with the action of the respective operators Si . Therefore, Si A =0.Takingthe top degree component A of A◦ we can write ◦ ◦ Si A = Si A + lower degree terms, where the operator S is given by i ⎧ ⎨⎪σi[−r − 1] for j = i

(13.41) Si : μj [−r − 1] → −σ [−r − 1] for j = i +1 ⎩⎪ i 0forj = i, i +1. On the other hand, write relations (10.50) in the form − ∂z + − − + (13.42) 1 μi(z) e σi (z) 1 μi+1(z) σi (z)=0, 13.2. YANGIAN CHARACTERS AND CLASSICAL W-ALGEBRAS 257 where ∞ r μj (z)= μj [−r − 1] z r=0 and we replaced a by a formal variable z. Setting deg z =1anddeg∂z = −1take the top degree components in these relations to get + − + σi (z)= μi(z) μi+1(z) σi (z). Hence for the images under S we have i Si : μi(z) → exp μi(z)−μi+1(z) dz, μi+1(z) →−exp μi(z)−μi+1(z) dz, and Si : μj (z) → 0forj = i, i + 1. However, due to (12.42) and (12.43) this coincides with the action of the operator Vi on the series μk(z). Thus, we may conclude that if an element A ∈L+ is annihilated by all operators + Si , then its image A under the map (13.36) is annihilated by all operators Vi. Together with Theorem 12.4.2 this completes the proof.

Types B,C and D. Now we let g = gN and use the notation of Section 11.3. + L We will construct a map gr : Rep Y (gN ) →W( gN ) and describe its properties. + + First, embed L = L (gN ) into the algebra of formal power series in variables μi[−r − 1] with i =1,...,N and r =0, 1,... by using (13.35) and taking the quotient by the corresponding relations (11.50). Define the degrees of the variables + by deg μi[−r − 1] = −r − 1. Given A ∈L , consider the corresponding formal power series in the variables μi[−r − 1] and take its homogeneous component A of the maximum degree. This component is a polynomial and so we have a map + (13.43) gr : L → C μi[−r − 1] | i =1,...,N, r =0, 1,... ,A → A. + Note its property (13.37). Recall the subalgebra Rep Y (gN ) defined in (11.53). Proposition 13.2.2. The image of the restriction of the map (13.43) to the + L subalgebra Rep Y (gN ) is contained in W( gN ) and so it defines a map + L gr : Rep Y (gN ) →W( gN ). Proof. The proof is quite similar to that of Proposition 13.2.1 so we only point out the changes to be made. Introduce variables σi[−r − 1] by (13.38) and − − − − ◦ set deg σi[ r 1] = r 1. Define operators Si for i =1,...,n as in (13.39) and (13.40), where the quotient is now taken by the respective relations (11.51) and (11.52) written in terms of the μi[−r − 1] with a understood as a variable. Since ◦ relations (11.51) are identical to (10.50), the argument for the operators Si with i =1,...,n− 1 follows the same steps as for type A. To complete the proof for the ◦ operator Sn , consider the three cases separately. Case g = o2n+1. As with (13.41), the corresponding operator Sn is now given by 2σn[−r − 1] for j = n (13.44) Sn : μj [−r − 1] → 0forj = n. Relations (11.50) imply λ+(a − n +1)λ+(a − n +2)...λ+(a) λ+ (a)= 1 2 n . n+1 + − + − + − λ1 (a n +1/2) λ2 (a n +3/2) ...λn (a 1/2) 258 13. AFFINE HARISH-CHANDRA ISOMORPHISM

+ − Now use (11.52) and write λi (a)=1 μi(a)fori =1,...,nto get the corresponding analogue of (13.42). As a result, we get the equation + + σn (z)=2μn(z) σn (z) so that for the images under S we have n

Sn : μn(z) → 2exp2 μn(z) dz, and Sn : μj (z) → 0forj = n. This coincides with the action of the operator 2Vn associated with sp2n on the series μn(z), as deﬁned in (12.63). Hence, if an element ∈L+ + A is annihilated by all operators Si , then its image A under the map (13.43) is annihilated by all operators Vi associated with sp2n which is the Langlands dual Lie algebra of o2n+1. Case g = sp . Similar to (13.44), we have 2n σn[−r − 1] for j = n Sn : μj [−r − 1] → 0forj = n. By (13.32), + + + λ (a − n)λ (a − n +1)...λ − (a − 2) λ+ (a)= 1 2 n 1 . n+1 + − − + − + − λ1 (a n 1)λ2 (a n) ...λn (a 2) + − Write λi (a)=1 μi(a)fori =1,...,n and use (11.52) to get the equation + + σn (z)=μn(z) σn (z). Hence, for the images under S we have n

Sn : μn(z) → exp μn(z) dz, and Sn : μj (z) → 0forj = n. This coincides with the action of the operator Vn associated with o2n+1 on the series μn(z); see (12.55). Therefore, if an element ∈L+ + A is annihilated by all operators Si , then its image A under the map (13.43) is annihilated by all operators Vi associated with o2n+1 which is the Langlands dual Lie algebra of sp2n. Case g = o . Similar to (13.41), we have 2n σn[−r − 1] for j = n − 1,n Sn : μj [−r − 1] → 0forj = n − 1,n. We derive from (11.50) that + + + λ (a − n +2)λ (a − n +3)...λ − (a) λ+ (a)= 1 2 n 1 . n+1 + − + − + λ1 (a n +1)λ2 (a n +2)...λn (a) Write λ+(a)=1− μ (a)fori =1,...,n and use (11.52) to get the equation i i + + σn (z)= μn−1(z)+μn(z) σn (z). Hence, for the images under S we have n Sn : μn−1(z) → exp μn−1(z)+μn(z) dz, μn(z) → exp μn−1(z)+μn(z) dz, and Sn : μj (z) → 0forj = n − 1,n. This coincides with the action of the operator Vn associated with o2n on the series μn−1(z)andμn(z), as given in (12.58). Thus, 13.3. HARISH-CHANDRA IMAGES OF SUGAWARA OPERATORS 259

~~∈L+ + if an element A is annihilated by all operators Si , then its image A under the map (13.43) is annihilated by all operators Vi associated with o2n which is Langlands self-dual.~~

13.3. Harish-Chandra images of Sugawara operators By Proposition 6.6.1, generators of the center Z(g) of the completed universal enveloping algebra U−h∨ (g) at the critical level can be obtained by the application of the state-field correspondence map (6.35) to a complete set of Segal–Sugawara vectors. We will use this fact to calculate the images of the generators of Z(g)in the classical types which were produced in Sections 7.2 and 8.5, under an affine version of the Harish-Chandra isomorphism involving the center Z(g). Given a triangular decomposition g = n− ⊕ h ⊕ n+, choose an arbitrary basis of h and root bases of n− and n+. Consider the corresponding basis of the affine Kac–Moody algebra g (see (6.5)) comprised of K and the elements of the form X[r] with r ∈ Z,whereX runs over the basis elements of g. Introduce any linear ordering ≺ on the basis elements to satisfy the following conditions. First, each basis element of t−1g[t−1] should precede each basis element of g[t] and the ordering on the corresponding basis elements of g should be consistent with the conditions −1 −1 −1 −1 −1 −1 n−[t] ≺ h[t] ≺ n+[t]andt n+[t ] ≺ t h[t ] ≺ t n−[t ] indicating the ordering between the basis elements belonging to the subspaces of g. By the Poincaré–Birkhoff–Witt theorem, any element x ∈ U(g) can be written as a unique linear combination of ordered monomials in the basis elements of g.Set h = h[t, t−1] ⊕ CK and denote by x0 ∈ U(h) the component of the linear combination representing the element x, where each monomial does not contain any basis elements X[r] with X ∈ n− ⊕ n+. The linear map θ : x → x0 defines the projection θ :U(g) → U(h). Consider now the universal enveloping algebra at the critical level (i.e., take the quotient of U(g) by the ideal generated by K + h∨) and extend θ by continuity to get the projection θ : U−h∨ (g) → U−h∨ (h), where U−h∨ (h) denotes the completion of U−h∨ (h) at the critical level defined as in (6.33). Set Π = S h[t, t−1] and define the completion Π of the commutative algebra Π as the inverse limit (13.45) Π = lim Π/I ,p>0, ←− p where Ip denotes the ideal of Π generated by all elements H[r] with H ∈ h and r p. The linear map η :U−h∨ (h) → Π which takes each ordered monomial to the same monomial regarded as an element of the commutative algebra Π, defines a vector space isomorphism. It extends to an isomorphism of the respective completed vector spaces η : U−h∨ (h) → Π. Thus we get a linear map (13.46) f : U−h∨ (g) → Π 260 13. AFFINE HARISH-CHANDRA ISOMORPHISM defined as the composition f = η ◦ θ. The next proposition provides an analogue of the Harish-Chandra homomorphism for the completed universal enveloping algebra. Proposition 13.3.1. The restriction of the map (13.46) to the center Z(g) of the algebra U−h∨ (g) is a homomorphism of commutative algebras (13.47) f :Z(g) → Π.

Proof. For x, y ∈ Z(g)setx0 = f(x)andy0 = f(y). Write y as a (possibly in- ﬁnite) linear combination of ordered monomials in the basis elements of g. Suppose that − − m = Xia [ ra 1] Yjb [rb],ra,rb 0, a b is an ordered monomial which occurs in the linear combination. Note its weight property

(13.48) wt Xia + wt Yjb =0 a b implied by the centrality of y,wheretheweightα =wtX ∈ h∗ of a basis vector X ∈ g is defined by the usual condition [H, X]=α(H) X for all H ∈ h. Suppose that m ∈ ker f.Sincex is in the center, we have − − xm = Xia [ ra 1] x Yjb [rb]. a b To write xm as a linear combination of ordered monomials we will only need to use the commutation relations in g[t]andthoseint−1 g[t−1]. Since they are weight- preserving, we derive that xm ∈ ker f. Hence a nonzero contribution to the image f(xy) can only come from f(xy0), that is, from expressions of the form − − Hia [ ra 1] x Hjb [rb],ra,rb 0, a b ∈ where Hia ,Hjb h.Ifp is an ordered monomial which occurs in the linear combination representing x and f(p) = 0, then applying property (13.48) to the monomial p we conclude that − − → f : Hia [ ra 1] p Hjb [rb] 0. a b Finally, observe that the Lie subalgebras h[t]andt−1 h[t−1]ofh are abelian which proves that f(xy)=x0y0. The homomorphism (13.47) gives rise to a an analogue of the Harish-Chandra isomorphism as follows. Consider the homomorphism of commutative algebras S t−1h[t−1] → Π[[ z,z−1]], defined on the generators by the rule ∂ r H[−r − 1] → z H(z),H∈ h,r 0, r! where H(z)= H[p]z−p−1. p∈Z 13.3. HARISH-CHANDRA IMAGES OF SUGAWARA OPERATORS 261 In particular, given any element w ∈W(g) ⊂ S t−1h[t−1] of the classical W- algebra W(g) we obtain a formal Laurent series −n−1 (13.49) w → w[n] z ,w[n] ∈ Π. n∈Z r Suppose that {T wi} is a family of algebraically independent generators of W(g); see Chapter 12. We define the subalgebra W (g) ⊂ Π as the completion of the algebra of polynomials in all variables wi [n] with respect to the topology which defines Π in (13.45); that is, the topology is determined by the system of neighborhoods of 0 formed by the ideals Ip. With this notation we have the following version of the Harish-Chandra isomorphism; see [46, Theorem 4.3.6]. Theorem 13.3.2. The homomorphism (13.47) is injective. Its image coincides with W (Lg) so that we have an isomorphism (13.50) f :Z(g) → W (Lg) of commutative algebras. It is clear from the construction that the isomorphisms described in Theo- rems 13.1.1 and 13.3.2 are consistent in the sense that the following diagram commutes: ∼ z(g) −−−−= →W(Lg) ⏐ ⏐ ⏐ ⏐ ∼ Z(g) −−−−= → W (Lg), where the vertical arrows indicate the maps constructed in Section 6.6 and in (13.49). This allows us to find the Harish-Chandra images of the generators of Z(g) in the classical types. The formulas stated below for type AN−1 follow from Proposition 13.1.2 and Corollary 13.1.3. Type AN−1. Consider the generators of the center Z(glN ) produced in Sec- tion 7.2. We will use the noncommutative complete and elementary symmetric polynomials defined in (12.47) and (12.48). Set −r−1 μi[r]=Eii[r]andμi(z)= μi[r]z r∈Z for i =1,...,N. Proposition 13.3.3. Under the isomorphism (13.50) we have :tr A(m) ∂ + E(z) ... ∂ + E(z) : 1,...,m z 1 z m → em ∂z + μ1(z),...,∂z + μN (z) , :tr H(m) ∂ + E(z) ... ∂ + E(z) : 1,...,m z 1 z m → hm ∂z + μ1(z),...,∂z + μN (z) ,

→ :cdet ∂z + E(z) : ∂z + μN (z) ... ∂z + μ1(z) , 262 13. AFFINE HARISH-CHANDRA ISOMORPHISM and ∞ m m q :tr ∂z + E(z) : m=0 N −1 −1 → 1 − q ∂z + μ1(z) ··· 1 − q ∂z + μi(z) i=1 × 1 − q ∂z + μi−1(z) ··· 1 − q ∂z + μ1(z) .

Now let gN be the classical Lie algebra of type Bn, Cn or Dn and recall the generators of the center Z(gN ) produced in Section 8.5. The following Harish- Chandra images of the Sugawara operators are found from the respective formulas of Propositions 13.1.7, 13.1.10, 13.1.12 and 13.1.14.

~~Type Bn. Set −r−1 (13.51) μi[r]=Fii[r]andμi(z)= μi[r]z r∈Z for i =1,...,N.TakeN =2n +1.~~

Proposition 13.3.4. Under the isomorphism (13.50) the image of the series (m) : γm(N)tr1,...,m S ∂z + F (z)1 ... ∂z + F (z)m : equals − − hm ∂z + μ1(z),...,∂z + μn(z),∂z μn(z),...∂z μ1(z) .

Type Dn. Now take N =2n and use the notation (13.51). Proposition 13.3.5. Under the isomorphism (13.50) the image of the series (m) : γm(N)tr1,...,m S ∂z + F (z)1 ... ∂z + F (z)m : equals 1 − − 2 hm ∂z + μ1(z),...,∂z + μn−1(z),∂z μn(z),...∂z μ1(z) 1 − − + 2 hm ∂z + μ1(z),...,∂z + μn(z),∂z μn−1(z),...∂z μ1(z) .

~~Recall the noncommutative Pfaﬃan deﬁned by (8.51).~~

~~Proposition 13.3.6. Under the isomorphism (13.50) we have Pf F (z) → μ1(z) − ∂z ... μn(z) − ∂z 1, where we assume ∂z 1=0. 13.4. HARISH-CHANDRA IMAGES OF CASIMIR ELEMENTS 263~~

Type Cn. Take N =2n and use the notation (13.51). Proposition 13.3.7. For all 1 m 2n +1 the image of the series − (m) : γm( 2n)tr1,...,m S ∂z + F (z)1 ... ∂z + F (z)m : under the homomorphism (13.3) equals em ∂z + μ1(z),...,∂z + μn(z),∂z,∂z − μn(z),...∂z − μ1(z) .

13.4. Harish-Chandra images of Casimir elements As we pointed out in Section 6.5, generators of the center of the universal enveloping algebra U(g) can be obtained from complete sets of Segal–Sugawara vectors via evaluation homomorphisms; see Proposition 6.5.2. Furthermore, the homomorphism (13.1) induces the Harish-Chandra homomorphism (13.52) U(g)h → U(h) h which is the projection whose kernel is the two-sided ideal U(g) ∩ U(g)n−.More precisely, for any nonzero z ∈ C we have the commutative diagram h f U t−1g[t−1] −−−−→ U t−1h[t−1] ⏐ ⏐ ⏐ ⏐ z z .

~~U(g)h −−−−→ U(h)~~

Note that for a ﬁxed triangular decomposition g = n− ⊕ h ⊕ n+, the roles of the subalgebras n− and n+ in (4.6) and (13.52) are swapped. One way to apply the results of Section 13.1 to the calculation of the Harish- Chandra images of Casimir elements with respect to the isomorphism (4.7) is to twist the homomorphism (13.52) by taking its composition with the Chevalley involution which is the involutive automorphism of g deﬁned in terms of the Chevalley presentation by

ei →−fi,fi →−ei,hi →−hi; see Section 12.1 for the deﬁning relations. An alternative way is to interpret the images of Casimir elements under the homomorphism (13.52) as their eigenvalues in the lowest weight representations of g similar to (4.15) and (5.9). We will take the latter approach and consider the classical types separately.

Type AN−1. Given an N-tuple of complex numbers μ =(μ1,...,μN ), the ◦ corresponding irreducible lowest weight representation L (μ)oftheLiealgebraglN is generated by a nonzero vector ζ ∈ L◦(μ) (the lowest vector) such that

~~Eij ζ =0 for N i>j 1, and~~

Eii ζ = μi ζ for N i 1. ∈ ◦ Any element z Z(glN )actsinL (μ) by multiplying each vector by a scalar f(z) which coincides with the image of z under the homomorphism (13.52) with g = glN , where μi is identiﬁed with the element Eii ∈ h. ◦ The representation L (μ)ofglN is ﬁnite-dimensional if and only if

~~μi+1 − μi ∈ Z + for i =1,...,N − 1. 264 13. AFFINE HARISH-CHANDRA ISOMORPHISM~~

If these conditions hold, then L◦(μ) is isomorphic to a highest weight representation, ◦ ∼ (13.53) L (μ) = L(λ), as deﬁned in Section 4.2, where λ =(λ1,...,λN ) with λi = μN−i+1 for i =1,...,N. We can now derive the following versions of Propositions 4.6.1 and 4.7.1, where we regard z as a variable and embed the universal enveloping algebra into the tensor ⊗ C −1 product U(glN ) [z ,∂z]. Corollary 13.4.1. For the images under the Harish-Chandra isomorphism (4.15) we have (m) − −1 − −1 χ :tr1,...,m A ∂z + E1 z ... ∂z + Em z → − −1 ··· − −1 ∂z + λi1 z ∂z + λim z

~~1i1<···~~

Ni1···im1 Proof. Using the homomorphism (6.30) and taking into account the property (6.31), we obtain from Proposition 13.1.2 that under the homomorphism (13.52), (m) − −1 − −1 tr1,...,m A ∂z + E1 z ... ∂z + Em z → − −1 − −1 em ∂z + μ1z ,..., ∂z + μN z , and (m) − −1 − −1 tr1,...,m H ∂z + E1 z ... ∂z + Em z → − −1 − −1 hm ∂z + μ1z ,..., ∂z + μN z . We will regard these images as the eigenvalues of the Casimir elements in ﬁnite- dimensional lowest weight representations L◦(μ). Hence by (13.53), their eigenvalues in L(λ) are found by replacing μi with λN−i+1 for all i =1,...,N. Since the Harish-Chandra image of any Casimir element is determined by its eigenvalues in the ﬁnite-dimensional representations, the claim follows.

Note that the formulas of Corollary 13.4.1 are respectively equivalent to those of Propositions 4.6.1 and 4.7.1. This is implied by the relations m −1 −1 (13.54) z (−∂z + E1 z ) ...(−∂z + Em z )=(u + E1 + m − 1) ...(u + Em) and m − −1 ··· − −1 − ··· (13.55) z ( ∂z + λi1 z ) ( ∂z + λim z )=(u + λi1 + m 1) (u + λim ), where u = −z∂z so that zu =(u +1)z. For the second formula we also use Lemma 4.5.4. We derive the following from the ﬁrst formula of Corollary 13.1.3 by the same argument as in the proof of Corollary 13.4.1. 13.4. HARISH-CHANDRA IMAGES OF CASIMIR ELEMENTS 265

~~Corollary 13.4.2. For the image under the Harish-Chandra isomorphism (4.15) we have χ :cdet −∂ + Ez−1 → −∂ + λ z−1 ... −∂ + λ z−1 . z z 1 z N~~

This was also established in Corollary 4.6.2 in an equivalent form. The following formulas are alternatives to Corollary 4.8.2 as they deal with a modiﬁed family of Gelfand invariants. Corollary 13.4.3. Suppose that m is a positive integer. For the image under the Harish-Chandra isomorphism (4.15) we have −1 m χ :tr −∂z + Ez N m → − −1 ··· − −1 ∂z + λi1 z ∂z + λik z ··· ··· i=1 k=0 i1 ik i

Now let gN be the orthogonal Lie algebra oN or symplectic Lie algebra spN .In the notation of Section 5.1, for any n-tuple of complex numbers μ =(μ1,...,μn), the corresponding irreducible lowest weight representation L◦(μ)oftheLiealgebra ◦ gN is generated by a nonzero vector ζ ∈ L (μ) (the lowest vector) such that

~~(13.56) Fij ζ =0 for N i>j 1, and~~

(13.57) Fii ζ = λi ζ for n i 1. ◦ Type Bn. If the representation L (μ) is finite-dimensional then it is isomorphic to a highest weight representation, ◦ ∼ L (μ) = L(λ), as defined in Section 5.1, where λ =(λ1,...,λn) with λi = −μi for i =1,...,n. We will extend the range of the subscripts by setting λi = −λi for i =1,...,n.We will now give an alternative way to find the Harish-Chandra images of the Casimir elements constructed in Section 5.4. Corollary 13.4.4. For the image under the Harish-Chandra isomorphism (5.9) we have (m) − −1 − −1 χ : γm(N)tr1,...,m S ∂z + F1 z ... ∂z + Fm z −1 −1 −1 −1 → hm −∂z − λ1z ,...,−∂z − λnz , −∂z + λnz ,...−∂z + λ1z . 266 13. AFFINE HARISH-CHANDRA ISOMORPHISM

Moreover, for k 1 the image of the Casimir element (2k) − − γ2k(N)tr1,...,2k S (F1 + k 1) ...(F2k k) under the Harish-Chandra isomorphism is given by − − − (13.58) (λi1 k)(λi2 k +1)...(λi2k + k 1), 1i1···i2k1 summed over the multisets {i1,...,i2k} with entries from {1,...,n,n,...,1 }. Proof. As in the proof of Corollary 13.4.1, the first part follows from Propo- sition 13.1.7 by the application of the homomorphism (6.30) and replacing μi with −λi. Furthermore, using the respective counterparts of (13.54) and (13.55), we find that the Harish-Chandra image of the polynomial (m) − γm(N)tr1,...,m S u + F1 + m 1 ... u + Fm equals − − − (u λi1 + m 1) ...(u λim ) 1i1···im1 which can also be written as − (u + λi1 ) ...(u + λim + m 1). 1i1···im1 For m =2k set u = −k to get the second part of the corollary. Comparing this result with Proposition 5.4.1, we find that the polynomial 2 2 | (13.58) coincides with the factorial complete symmetric polynomial hk(l1,...,ln a). This equality can be verified directly by using the characterization theorem for the factorial symmetric functions (Proposition 4.3.2). Namely, both elements are sym- 2 2 metric polynomials in l1,...,ln of degree k, and their top degree components are 2 2 both equal to the complete symmetric polynomial hk(l1,...,ln). It remains to ver- 2 2 | ify that each of the elements (13.58) and hk(l1,...,ln a)vanisheswhen(λ1,...,λn) is specialized to a partition with λ1 + ···+ λn

The polynomial (13.59) coincides with the factorial complete symmetric poly- 2 2 | nomial hk(l1,...,ln a) which occurs as the Harish-Chandra image in Proposi- tion 5.4.1. This equality can also be veriﬁed directly with the use of Proposi- tion 4.3.2. More general identities follow by comparing the formulas of Corollar- ies 5.4.2 and 13.4.5.

◦ Type Cn. Use the definition of the lowest weight representation L (μ)ofsp2n given in (13.56) and (13.57). If this representation is finite-dimensional then it is isomorphic to a highest weight representation, ◦ ∼ L (μ) = L(λ), as defined in Section 5.1, where λ =(λ1,...,λn) with λi = −μi for i =1,...,n. We will extend the range of the subscripts by setting λ0 =0andλi = −λi for i =1,...,n. By the results of Section 5.5, the expression (5.37) admits an equivalent form which is well-defined for all values 1 m 2n + 1. We will keep using the same notation for this expression for all these values of m. Corollary 13.4.6. For 1 m 2n +1 the image of the differential operator − (m) − −1 − −1 γm( 2n)tr1,...,m S ∂z + F1 z ... ∂z + Fm z under the Harish-Chandra isomorphism (5.9) equals −1 −1 −1 −1 em −∂z − λ1z ,...,−∂z − λnz , −∂z, −∂z + λnz ,...−∂z + λ1z . Moreover, if m =2k is even, then the image of the Casimir element − (2k) − γ2k( 2n)tr1,...,2k S (F1 + k) ...(F2k k +1) under the Harish-Chandra isomorphism is given by − (13.60) (λi1 + k) ...(λi2k k +1), 1i1<···

By Proposition 5.5.4, the polynomial (13.60) coincides with the factorial ele- − k 2 2 | mentary symmetric polynomial ( 1) ek(l1,...,ln a). This also follows from the characterization theorem for the factorial symmetric functions (Proposition 4.3.2). Corollary 5.5.5 implies some more general identities which follow by comparing the Harish-Chandra images of the same expressions.

13.5. Bibliographical notes The Harish-Chandra images of generators of z(g) were calculated in [23]for type A andin[111]fortypesB, C and D. A direct proof of Proposition 13.1.12 can be found in Rozhkovskaya [137]. The calculations with the screening operators follow [113], but re-written in the context of the dual Yangians. CHAPTER 14

~~Higher Hamiltonians in the Gaudin model~~

In their seminal work [40], Feigin, Frenkel and Reshetikhin established a connection between the center z(g) of the aﬃne vertex algebra at the critical level and higher order Hamiltonians in the Gaudin model. They showed that any element of S ∈ z(g) gives rise to a Hamiltonian and used the Wakimoto modules over the aﬃne Kac–Moody algebra g to calculate the eigenvalues of such Hamiltonians on the Bethe vectors. In this section we apply these results and their generalizations [42] to get formulas for the action of the higher Gaudin Hamiltonians on tensor products of representations of the Lie algebra g of a classical type in an explicit form and calculate the corresponding eigenvalues of the Bethe vectors. We will rely on the constructions of generators of z(g) obtained in Sections 7.1, 8.1 and 8.3, and their Harish-Chandra images found in Section 13.1.

~~14.1. Bethe ansatz equations~~

Recall the standard Chevalley generators ei,hi,fi with i =1,...,n of a simple Lie algebra g of rank n. The generators hi form a basis of the Cartan subalgebra h of g, while the ei and fi generate the respective nilpotent subalgebras n+ and n−. Let A =[aij] be the Cartan matrix of g so that the deﬁning relations of g take the form

~~[ei,fj ]=δijhi, [hi,hj]=0,~~

~~[hi,ej ]=aij ej , [hi,fj ]=−aij fj , together with the Serre relations~~

1−aij 1−aij (ad ei) ej =0, (ad fi) fj =0,i= j. Given any element χ ∈ g∗ and a nonzero z ∈ C, the mapping −1 −1 r (14.1) U t g[t ] → U(g),X[r] → Xz + δr,−1 χ(X), for X ∈ g and r<0, defines an algebra homomorphism; cf. (9.29). Using the coassociativity of the standard coproduct on U t−1g[t−1] defined by Δ:Y → Y ⊗ 1+1⊗ Y, Y ∈ t−1g[t−1], for any 1 we get the homomorphism ⊗ (14.2) U t−1g[t−1] → U t−1g[t−1] as an iterated coproduct map. Now fix distinct complex numbers z1,...,z and let u be a complex parameter. Applying homomorphisms of the form (14.1) to the tensor factors in (14.2), we get another homomorphism (14.3) Ψ : U t−1g[t−1] → U(g)⊗ ,

269 270 14. HIGHER HAMILTONIANS IN THE GAUDIN MODEL given by r ⊗ Ψ:X[r] → Xa(za − u) + δr,−1 χ(X) ∈ U(g) , a=1 ⊗(a−1) ⊗(−a) where Xa =1 ⊗ X ⊗ 1 . We will twist this homomorphism by the involutive anti-automorphism (14.4) ς :U t−1g[t−1] → U t−1g[t−1] ,X[r] →−X[r],X∈ g, to get the anti-homomorphism (14.5) Φ : U t−1g[t−1] → U(g)⊗ , deﬁned as the composition Φ = Ψ ◦ ς.Sincez(g) is a commutative subalgebra of −1 −1 U t g[t ] , the image of z(g) under Φ is a commutative subalgebra A(g)χ of ⊗ U(g) , depending on the chosen parameters z1,...,z, but it does not depend on u; cf. Section 9.2. ∗ Given λ ∈ h , the Verma module Mλ is the quotient of U(g) by the left ideal generated by n+ and the elements hi −λ(hi) with i =1,...,n. We denote the image ∗ of 1 in Mλ by 1λ. For any weights λ1,...,λ ∈ h consider the tensor product of ⊗ ⊗ the Verma modules Mλ1 ... Mλ . We will now describe common eigenvectors for the commutative subalgebra A(g)χ in this tensor product. For a set of distinct complex numbers w1,...,wm with wi = zj and a collection (multiset) of labels i1,...,im ∈{1,...,n} introduce the Bethe vector i1 im ∈ ⊗ ⊗ φ(w1 ,...,wm ) Mλ1 ... Mλ by the formula a k 1 (14.6) φ(wi1 ,...,wim )= f 1 , 1 m − ir λk wjk wjk (I1,...,I) k=1 s=1 s s+1 r∈Ik summed over all ordered partitions I1 ∪ I2 ∪··· ∪I of the set {1,...,m} into ordered subsets Ik = {jk,jk,...,jk } with the products taken from left to right, 1 2 ak where w k := zk for s = ak. js+1 Now suppose that χ ∈ h∗. We regard χ as a functional on g which vanishes on n+ and n−. The system of the Bethe ansatz equations takes the form λ (ˇα ) α (ˇα ) i ij − is ij (14.7) = χ(ˇαij ),j=1,...,m, wj − zi wj − ws i=1 s= j ∗ where αl ∈ h andα ˇl ∈ h with l =1,...,n denote the simple roots and coroots, respectively. Given the above parameters, introduce the homomorphism from U t−1h[t−1] to the algebra of rational functions in u by the rule: ∂ r : H[−r − 1] → u H(u),H∈ h,r 0, r! where m λ (H) αi (H) (14.8) H(u)= a − j − χ(H). u − z u − w a=1 a j=1 j Let S ∈ z(g) be a Segal–Sugawara vector. The composition ◦ f of this homomorphism with the isomorphism (13.2) takes S to a rational function f(S) in u. 14.2. GAUDIN HAMILTONIANS AND EIGENVALUES 271

Furthermore, we regard the image Φ(S)ofS under the anti-homomorphism (14.5) ⊗ ⊗ as an operator in the tensor product of Verma modules Mλ1 ... Mλ .The following is essentially a reformulation of Theorems 6.5 and 6.7 from [42]; in the case χ =0theresultgoesbackto[40, Theorem 3]. Theorem 14.1.1. Suppose that the Bethe ansatz equations (14.7) are satisﬁed. i1 im If the Bethe vector φ(w1 ,...,wm )is nonzero, then it is an eigenvector for the operator Φ(S) with the eigenvalue f(S) .

In what follows we will rely on the results of Chapter 13 to give explicit formulas for the operators Φ(Si) and their eigenvalues f(Si) on the Bethe vectors for complete sets of Segal–Sugawara vectors S1,...,Sn in all classical types.

~~14.2. Gaudin Hamiltonians and eigenvalues~~

Type AN−1. We need to calculate the images of the Segal–Sugawara vectors constructed in Section 7.1 under the involution (14.4). We extend it to the algebra −1 −1 ⊗ C C U t glN [t ] [τ] with the action on [τ] as the identity map. Lemma 14.2.1. For the images with respect to the involution ς we have (m) − − tr1,...,m A τ + E[ 1]1 ... τ + E[ 1]m → (m) − − − − tr1,...,m A τ E[ 1]1 ... τ E[ 1]m , (m) − − tr1,...,m H τ + E[ 1]1 ... τ + E[ 1]m → (m) − − − − tr1,...,m H τ E[ 1]1 ... τ E[ 1]m , m m (14.9) tr τ + E[−1] → tr τ − E t[−1] , and (14.10) cdet τ + E[−1] → cdet τ − E t[−1] , where t denotes the standard matrix transposition. Proof. The left hand side of the ﬁrst relation equals a linear combination of expressions of the form (m) k (14.11) tr1,...,m A E[r1]a1 ...E[rp]ap τ with 1 a1 < ···

~~E[r]a E[s]b − E[s]b E[r]a = PabE[r + s]b − E[r + s]b Pab (m) (m) (m) for a~~

For the image of (7.10) under ς we get tr A(N) τ − E[−1] ... τ − E[−1] 1,...,N 1 N (N) − t − − t − =tr1,...,N A τ E [ 1]1 ... τ E [ 1]N , where we have applied the transposition t1 ...tN with respect to all copies of End CN and used the invariance of A(N) under this transposition. Since τ −E t[−1] is also a Manin matrix, the resulting expression coincides with cdet τ − E t[−1] . Finally, (14.9) follows from Theorem 3.2.10 which connects the coefficients of the polynomial in (7.9) with those of (7.11). With the parameters chosen as in Section 14.1, suppose that χ vanishes on the ⊕ ∗ subspace n− n+ of glN so that we can regard χ as an element of h .Set (E ) E (u)= ij a − χ(E ) ∈ U(gl )⊗ . ij u − z ij N a=1 a Consider the row-determinant rdet ∂u + E(u) of the matrix ∂u + E(u)= δij ∂u + Eij(u) ⊗ as a differential operator in ∂u with coefficients in U(glN ) ; see (3.25). Further- more, in accordance with (14.8), set m λ (E ) αi (Eii) E (u)= a ii − j − χ(E ). ii u − z u − w ii a=1 a j=1 j In all the following eigenvalue formulas for the Gaudin Hamiltonians we will assume that the Bethe ansatz equations (14.7) hold. Theorem 14.2.2. The eigenvalue of the operator rdet ∂u +E(u) on the Bethe vector (14.6) is found by i1 im E E i1 im rdet ∂u +E(u) φ(w1 ,...,wm )= ∂u + NN(u) ... ∂u + 11(u) φ(w1 ,...,wm ). Proof. To apply Theorem 14.1.1, we will find the image of the polynomial cdet τ + E[−1] under the anti-homomorphism Φ. Extend Φ to the map −1 −1 ⊗ C → ⊗ ⊗ C Φ:U t glN [t ] [τ] U(glN ) [∂u] by setting τ → ∂u. Note that by definition of the homomorphism (14.3) we have Ψ:E[−1] →−E(u) and so, by (14.10), t Φ : cdet τ + E[−1] → cdet ∂u + E (u) =rdet ∂u + E(u) . On the other hand, due to (13.6) we have ◦ f :cdet τ + E[−1] → ∂u + ENN(u) ... ∂u + E11(u) thus completing the proof. The following corollaries can be derived from Theorem 14.2.2 or verified in a similar way as above with the use of Lemma 14.2.1 and the Harish-Chandra images calculated in Section 13.1. 14.2. GAUDIN HAMILTONIANS AND EIGENVALUES 273

Corollary 14.2.3. The eigenvalues of the operators (m) tr1,...,m A ∂u + E(u)1 ... ∂u + E(u)m and (m) tr1,...,m H ∂u + E(u)1 ... ∂u + E(u)m on the Bethe vector (14.6) are found by the respective formulas em ∂u + E11(u),...,∂u + ENN(u) and hm ∂u + E11(u),...,∂u + ENN(u) . Corollary 14.2.4. The eigenvalue of the series ∞ k t k q tr ∂u + E (u) k=0 on the Bethe vector (14.6) is found by the formula

~~N −1 −1 1 − q ∂u + E11(u) ··· 1 − q ∂u + Eii(u) i=1 × 1 − q ∂u + Ei−1 i−1(u) ··· 1 − q ∂u + E11(u) .~~

Types Bn and Dn. Consider the Segal–Sugawara vectors for oN constructed in Section 8.1. We extend the involution (14.4) to the algebra −1 −1 U t oN [t ] ⊗ C[τ] with the action on C[τ] as the identity map.

Lemma 14.2.5. The element (8.15) is stable under ς. Moreover, in type Dn we have (14.12) ς :Pf F [−1] → (−1)n Pf F [−1]. Proof. The same argument as in the proof of Lemma 14.2.1 shows that the image of (8.15) under the involution ς equals (m) − − − − (14.13) γm(N)tr1,...,m S τ F [ 1]1 ... τ F [ 1]m . This is implied by the deﬁning relations

~~F [r]a F [s]b − F [s]b F [r]a =(Pab − Qab) F [r + s]b − F [r + s]b (Pab − Qab) for a~~

~~Following (14.8) deﬁne the rational functions in u by m λ (F ) αi (Fii) F (u)= a ii − j − χ(F ). ii u − z u − w ii a=1 a j=1 j~~

In the case g = o2n introduce the operator 1 Pf F (u)= sgn σ · F (u) ...F − (u). 2nn! σ(1) σ(2) σ(2n 1) σ(2n) σ∈S2n As before, we will assume that the Bethe ansatz equations (14.7) hold. Theorem 14.2.6. The eigenvalue of the operator (m) (14.14) γm(N)trS ∂u + F (u)1 ... ∂u + F (u)m on the Bethe vector (14.6) is found by hm ∂u + F11(u),...,∂u + Fnn(u),∂u −Fnn(u),...∂u −F11(u) for type Bn,andby 1 F F −F −F 2 hm ∂u + 11(u),...,∂u + n−1 n−1(u),∂u nn(u),...∂u 11(u) 1 F F −F −F + 2 hm ∂u + 11(u),...,∂u + nn(u),∂u n−1 n−1(u),...∂u 11(u) for type Dn. Moreover, the eigenvalue of the operator Pf F (u) in type Dn is given by F11(u) − ∂u ... Fnn(u) − ∂u 1. Proof. We apply Theorem 14.1.1 again and regard Φ as the map −1 −1 ⊗ Φ:U t oN [t ] ⊗ C[τ] → U(oN ) ⊗ C[∂u] such that τ → ∂u. By the definition of the homomorphism (14.3) we have Ψ:F [−1] →−F (u). Hence, using the equivalent formula (14.13) for the polynomial (8.15) we find that its image under Φ coincides with the operator (14.14). The proof of the first part of the theorem is completed by using the formulas for the Harish-Chandra images of (8.15) provided by Propositions 13.1.7 and 13.1.10. Finally, by Lemma 14.2.5, in type Dn, Φ:Pf F [−1] → Pf F (u) so that the last claim follows from Proposition 13.1.12.

Type Cn. Now we use the Segal–Sugawara vectors for sp2n constructed in −1 −1 ⊗ C Section 8.3. Extend the involution (14.4) to the algebra U t sp2n[t ] [τ] with the action on C[τ] as the identity map. Lemma 14.2.7. The element (8.36) is stable under ς. Proof. The proof is the same as for Lemma 14.2.5, which also provides an equivalent formula (m) γm(−2n)trS τ − F [−1]1 ... τ − F [−1]m for the polynomial (8.36). 14.3. BIBLIOGRAPHICAL NOTES 275

With the parameters chosen as in Section 14.1 suppose that χ vanishes on the ⊕ ∗ subspace n− n+ of sp2n so that we can regard χ as an element of h .Set (F ) F (u)= ij a − χ(F ) ∈ U(sp )⊗ . ij u − z ij 2n a=1 a In accordance with (14.8) deﬁne the rational functions in u by λ (F ) m α (F ) F (u)= a ii − ij ii − χ(F ). ii u − z u − w ii a=1 a j=1 j As before, we will assume that the Bethe ansatz equations (14.7) hold. Theorem 14.2.8. For any 1 m 2n +1 the eigenvalue of the operator (m) γm(−2n)trS ∂u + F (u)1 ... ∂u + F (u)m on the Bethe vector (14.6) is found by em ∂u + F11(u),...,∂u + Fnn(u),∂u,∂u −Fnn(u),...∂u −F11(u) . Proof. The argument relies on Proposition 13.1.14 so that the claim is derived from Theorem 14.1.1 and Lemma 14.2.7 as in the proof of Theorem 14.2.6.

14.3. Bibliographical notes For the origins of the formula (14.6) for the Bethe vectors see [9], [40], [142] and references therein. Theorem 14.2.2 was first proved in [118]isadifferentway. Our exposition follows [113]. The Bethe algebras of classical types were studied in a recent work by Lu, Mukhin and Varchenko [102] in connection with stratifications of self-dual Grassmannians.

~~CHAPTER 15~~

~~Wakimoto modules~~

The Wakimoto modules are representations of the aﬃne Kac–Moody algebras g in the Fock spaces M(g) associated with the Heisenberg algebras. Their construction relies on the existence of the vertex algebra homomorphism

V−h∨ (g) → M(g) from the vacuum module at the critical level to the Fock space M(g). This homomorphism can be ‘deformed’ to the vacuum module Vκ(g) to define the Wakimoto module of an arbitrary level κ ∈ C, as explained in the book by Frenkel [46, Chap- ter 6]. However, we will only consider such modules of the critical level κ = −h∨; see Section 6.2 for the definition of Vκ(g). The Wakimoto modules play a significant role in the proof of a principal theorem of the book [46] describing the center of the affine vertex algebra at the critical level. Their importance is explained by the fact that elements of the center Z(g)of the completed universal enveloping algebra U−h∨ (g) (i.e., the Sugawara operators constructed in Section 6.6) act as multiplication by scalars in the Wakimoto modules of the critical level. Our goal in this chapter is to connect the eigenvalues of the Sugawara operators with their Harish-Chandra images calculated in Section 13.3. We make this connection explicit by constructing the Wakimoto modules in all classical types and calculating the scalars. Explicit formulas for the action of generators of g in the Wakimoto modules depend on the free field realization of the Lie algebra g. We will start by reproducing these realizations in the classical types by following the general construction of [46, Chapter 5]. Closely related realizations of irreducible finite-dimensional representations of the classical groups were studied in the book by Zhelobenko [156].

~~15.1. Free ﬁeld realization of glN~~

Consider the group GLN of all N×N matrices over C with nonzero determinant. Denote by B− the subgroup of all lower-triangular matrices and by N+ the subgroup of upper-triangular matrices with 1’s on the diagonal. Elements of the Lie algebra N glN will act on the space of polynomial functions Fun + by ﬁrst order diﬀerential operators. We will identify Fun N+ with the space of polynomials ∼ Fun N+ = C[yij | 1 i

~~277 278 15. WAKIMOTO MODULES~~

~~Proposition 15.1.1. The formulas~~

~~i−1 N Eii → yri ∂ri − yis ∂is + χi, r=1 s=i+1 for i =1,...,N,and~~

N Eii+1 → ∂ii+1 + yi+1 s ∂is, s=i+2 i−1 i Ei+1 i → yri yii+1 − yri+1 ∂ri − yri+1 yii+1 ∂ri+1 r=1 r=1 N + yis ∂i+1 s + χi − χi+1 yii+1 s=i+2 − C for i =1,...,N 1, deﬁne a representation of glN in the space [yij ]. Proof. ∈ N By deﬁnition, the action of an element A glN on Fun + is given by d (15.1) (Af)(Y )= f X(t) ,Y∈N+,f∈ Fun N+, dt t=0 where X(t) ∈N+ is uniquely determined from the decomposition

−tA (15.2) e Y = X(t) Z(t),Z(t) ∈B−, which is valid for all values of t in a neighborhood of 0. The entries of the matrix X(t) can be found from the well-known formulas for the Gauss decomposition.

Lemma 15.1.2. Given a decomposition C = XZ of a N × N matrix C =[cij], where X ∈N+ and Z ∈B−,theentriesofX =[xij] are found by −1 cij cij+1 ... ciN cjj cjj+1 ... cjN cj+1 j cj+1 j+1 ... cj+1 N cj+1 j cj+1 j+1 ... cj+1 N xij = ...... ...... cNj cNj+1 ... cNN cNj cNj+1 ... cNN for all 1 i

~~−tA Now take A = Eii and apply Lemma 15.1.2 to the matrix C = e Y ,where Y ∈N+ and the (i, j)entryofY is the coordinate yij for all i~~

~~i−1 N Eii → yri ∂ri − yis ∂is. r=1 s=i+1 15.1. FREE FIELD REALIZATION OF glN 279~~

−tA Furthermore, taking A = Eii+1,observethate =1− tEii+1. Therefore, applying Lemma 15.1.2, we ﬁnd that the entries of the corresponding matrix X which diﬀer from the respective entries of Y , lie in row i and are given by xii+1 = yii+1 − t and xis = yis − tyi+1 s for s = i +2,...,N. Hence, Eii+1 is represented by the operator N Eii+1 →−∂ii+1 − yi+1 s ∂is. s=i+2

Similarly, taking A = Ei+1 i and applying Lemma 15.1.2, we ﬁnd that the entries of the matrix X which diﬀer from the respective entries of Y ,belongtorowi +1 or columns i and i + 1. They are given by the expressions

~~xri = yri (1 − tyii+1)+tyri+1 for r =1,...,i− 1,~~

~~yri+1 xri+1 = for r =1,...,i, 1 − tyii+1 and~~

~~xi+1 s = yi+1 s − tyis for s = i +2,...,N.~~

This implies that Ei+1 i acts by the formula i−1 i N Ei+1 i → −yri yii+1 + yri+1 ∂ri + yri+1 yii+1 ∂ri+1 − yis ∂i+1 s. r=1 r=1 s=i+2 C The above formulas deﬁne a representation of glN in the space [yij ]. Twist this representation with the automorphism of glN which changes the signs of the generators Eii+1 and Ei+1 i for i =1,...,N − 1 and leaves all Eii unchanged. We will then get the formulas given in the proposition with χ =0.Asdemonstrated in [46, Section 5.2], given any χ ∈ h∗, this representation can be extended in a way that each elements H ∈ h acts on the element 1 as multiplication by the scalar χ(H), and so the formulas for the extended action follow.

As shown in [46, Section 5.2], the representation of glN provided by Proposi- ∗ tion 15.1.1 is isomorphic to the contragredient Verma module Mχ. Example 15.1.3. For N = 2 the formulas of Proposition 15.1.1 deﬁne a repre- C sentation of gl2 in the space of polynomials in one variable [y12]:

~~E11 →−y12 ∂12 + χ1,E22 → y12 ∂12 + χ2, → →− 2 − E12 ∂12,E21 y12 ∂12 +(χ1 χ2)y12.~~

If χ1 − χ2 ∈/ Z + then this representation is irreducible. If χ1 − χ2 = k ∈ Z + then the polynomials in y12 of degree not exceeding k form a (k + 1)-dimensional irreducible submodule. Example . 15.1.4 TherepresentationoftheLiealgebragl3 in the space of polynomials C[y12,y13,y23] is given by the formulas

~~E11 →−y12 ∂12 − y13 ∂13 + χ1,~~

~~E22 → y12 ∂12 − y23 ∂23 + χ2,~~

E33 → y13 ∂13 + y23 ∂23 + χ3, 280 15. WAKIMOTO MODULES together with → →− 2 − E12 ∂12 + y23 ∂13,E21 y12 ∂12 + y13 ∂23 +(χ1 χ2)y12, and → → − − − 2 − E23 ∂23,E32 (y12 y23 y13)∂12 y13 y23 ∂13 y23 ∂23 +(χ2 χ3)y23.

~~15.2. Free ﬁeld realization of oN~~

~~Recall the classical group GN as deﬁned in (2.29). We point out an important property of the presentation of GN associated with the form (2.26).~~

Lemma 15.2.1. Suppose that a matrix C ∈ GN is written as a product C = XZ of an upper-triangular matrix X with 1’s on the diagonal and a lower-triangular matrix Z. Then both X and Z belong to GN . Proof. We have 1 = CC =(XZ)XZ = ZXXZ. Therefore, XX = (Z)−1Z−1.ThematrixX is upper-triangular with 1’s on the diagonal, while Z is lower-triangular. Therefore, XX =1andZZ = 1, so the claim follows.

Consider the special orthogonal group SON which is the subgroup of ON whose elements are matrices C with det C =1.NowB− will denote the subgroup of all lower-triangular matrices in SON ,andN+ the subgroup of SON which consists of upper-triangular matrices with 1’s on the diagonal. Due to the relation XX =1 satisﬁed by X ∈N+, we can identify Fun N+ with the space of polynomials ∼ Fun N+ = C[yij | 1 i

i−1 i−1 Fii → yri ∂ri − yri ∂ri − yis ∂is + χi r=1 s=i+1 for i =1,...,n, i−2 n Fii+1 → ∂ii+1 + yi+1 s ∂is − yi+1 s yi+1 s ∂ii−1 s=i+2 s=i+2 for i =1,...,n− 1,andFn−1 n → ∂n−1 n , together with i−1 Fi+1 i → yri yii+1 −yri+1 ∂ri + yri−1 yii+1 +yri ∂ri−1 −yri yii+1 ∂ri r=1 i i−2 n − yri+1 yii+1 ∂ri+1 + yis ∂i+1 s − yisyis ∂ii−1 + χi − χi+1 yii+1 r=1 s=i+2 s=i+2 15.2. FREE FIELD REALIZATION OF oN 281 for i =1,...,n− 1,and

n−2 Fn n−1 → yrn−1 yn−1 n − yrn ∂rn−1 + yrn yn−1 n + yrn+2 ∂rn r=1 n−1 − yrn+2 yn−1 n ∂rn+2 − yrn yn−1 n ∂rn + χn−1 + χn yn−1 n r=1 deﬁne a representation of o2n in the space C yij | 1 i

Proof. TheactionofanelementA ∈ o2n on Fun N+ is given by the general formulas (15.1) and (15.2). We will apply Lemma 15.1.2 to the matrix C = e−tA Y . Here Y ∈N+ and we let yij be the (i, j)entryofY for all 1 i

Now let A = Fii+1 for i ∈{1,...,n− 1} and apply Lemma 15.1.2 again. The entries of the corresponding matrix X which diﬀer from the respective entries of Y are given by xii+1 = yii+1 − t and xis = yis − tyi+1 s for s = i +2,...,i − 1. Hence, Fii+1 is represented by the operator i−1 Fii+1 →−∂ii+1 − yi+1 s ∂is. s=i+2

~~We need to express yi+1 i−1 in terms of the admissible coordinates on N+.By taking the (i, i)entriesonbothsidesofYY =1weget~~

i−1 (15.3) 2yii + yisyis =0,i=1,...,n. s=i+1 Hence, i−2 n Fii+1 →−∂ii+1 − yi+1 s ∂is + yi+1 s yi+1 s ∂ii−1. s=i+2 s=i+2 −tA Similarly, for A = Fn−1 n we have e =1− tFn−1 n and so the only entry of the corresponding matrix X which differs from the respective entry of Y is xn−1 n = yn−1 n − t. Hence, Fn−1 n →−∂n−1 n . Now turn to the generators A = Fi+1 i for i ∈{1,...,n− 1}. By Lemma 15.1.2, we find that the entries of the matrix X which differ from the respective entries of Y aregivenbytheexpressions

~~yri xri = yri (1 − tyii+1)+tyri+1,xri = 1+tyi−1 i for r =1,...,i− 1,~~

~~yri+1 xri−1 = yri−1 (1 + tyi−1 i ) − tyri ,xri+1 = 1 − tyii+1 282 15. WAKIMOTO MODULES for r =1,...,i,and xi+1 s = yi+1 s − tyis for s = i +2,...,i − 2.~~

Taking into account (15.3) to express yii in terms of the admissible coordinates, we conclude that Fi+1 i acts by the formula i−1 Fi+1 i →− yri yii+1−yri+1 ∂ri+ yri−1 yii+1+yri ∂ri−1−yri yii+1 ∂ri r=1 i i−2 n + yri+1 yii+1 ∂ri+1 − yis ∂i+1 s + yisyis ∂ii−1. r=1 s=i+2 s=i+2

The calculation of the action of Fn n−1 is not essentially diﬀerent from the previous one; the resulting formula is obtained from the action of Fnn−1 by swapping the indices n and n. Finally, twist this representation of o2n with the automorphism which changes the signs of the generators Fii+1 and Fi+1 i for i =1,...,n−1, as well as Fn n−1 and Fn−1 n , but leaves all Fii unchanged. The action is then extended to an arbitrary χ ∈ h∗ as in [46, Section 5.2] which yields the required formulas.

~~Example 15.2.3. The action of o4 in the space of polynomials C[y12,y13]given by Proposition 15.2.2 takes the form~~

~~F11 →−y12 ∂12 − y13 ∂13 + χ1,F22 → y12 ∂12 − y13 ∂13 + χ2,~~

F12 → ∂12,F13 → ∂13, →− 2 − →− 2 F21 y12 ∂12 +(χ1 χ2)y12,F31 y13 ∂13 +(χ1 + χ2)y13. This representation is isomorphic to the tensor product of two representations of sl2 obtained by the restriction from the gl2-action provided by Example 15.1.3.

~~Example 15.2.4. TheactionoftheLiealgebrao6 in the space of polynomials C[y12,y13,y14,y15,y23,y24]isgivenby~~

~~F11 →−y12 ∂12 − y13 ∂13 − y14 ∂14 − y15 ∂15 + χ1,~~

~~F22 → y12 ∂12 − y15 ∂15 − y23 ∂23 − y24 ∂24 + χ2,~~

~~F33 → y13 ∂13 − y14 ∂14 + y23 ∂23 − y24 ∂24 + χ3, for the elements of the Cartan subalgebra, together with~~

F23 → ∂23,F24 → ∂24,F12 → ∂12 + y23 ∂13 + y24 ∂14 − y23 y24 ∂15, and →− 2 − − F21 y12 ∂12 y13 y14 ∂15 + y13 ∂23 + y14 ∂24 +(χ1 χ2)y12, F32 → y12 y23 − y13 ∂12 − y13 y23 ∂13 + y14 y23 + y15 ∂14 − − 2 − y15 y23 ∂15 y23 ∂23 +(χ2 χ3)y23, F42 → y12 y24 − y14 ∂12 − y14 y24 ∂14 + y13 y24 + y15 ∂13 − − 2 y15 y24 ∂15 y24 ∂24 +(χ2 + χ3)y24. 15.2. FREE FIELD REALIZATION OF oN 283

Now suppose that N is odd, N =2n + 1, and ﬁx an element χ =(χ1,...,χn) n of C which is understood as a functional on the Cartan subalgebra h of o2n+1 spanned by the basis elements F11,...,Fnn. Proposition 15.2.5. The formulas i−1 i−1 Fii → yri ∂ri − yri ∂ri − yis ∂is + χi, r=1 s=i+1

− − i2 1 i2 F → ∂ + y ∂ − y y ∂ − ii+1 ii+1 i+1 s is 2 i+1 s i+1 s ii 1 s=i+2 s=i+2 for i =1,...,n, together with i−1 Fi+1 i → yri yii+1 −yri+1 ∂ri + yri−1 yii+1 +yri ∂ri−1 −yri yii+1 ∂ri r=1 − − i i2 1 i2 − y y ∂ + y ∂ − y y ∂ − + χ −χ y ri+1 ii+1 ri+1 is i+1 s 2 is is ii 1 i i+1 ii+1 r=1 s=i+2 s=i+2 for i =1,...,n− 1,and n−1 Fn+1 n → yrn ynn+1 − yrn+1 ∂rn − yrn+2 ynn+1 ∂rn+2 + yrn+2 ∂rn+1 r=1 1 − y2 ∂ + χ y 2 nn+1 nn+1 n nn+1 deﬁne a representation of o in the space 2n+1 C yij | 1 i

~~tyrn+2 xrn+1 = yrn+1 − for r =1,...,n, 1 − tynn+1/2 and yrn+2 − xrn+2 = 2 for r =1,...,n 1. 1 − tynn+1/2 284 15. WAKIMOTO MODULES~~

Therefore, for the action of Fn+1 n we get n−1 Fn+1 n → yrn+1 − yrn ynn+1 ∂rn + yrn+2 ynn+1 ∂rn+2 − yrn+2 ∂rn+1 r=1 − ynn+2 ∂nn+1. The argument is completed in the same way as for Proposition 15.2.2.

~~Example 15.2.6. The action of o3 in the space of polynomials C[y12]takesthe form 1 F →−y ∂ + χ ,F → ∂ ,F →− y2 ∂ + χ y . 11 12 12 1 12 12 21 2 12 12 1 12~~

~~This representation is isomorphic to the representation of sl2 obtained by the restriction from the gl2-action provided by Example 15.1.3 so that χ1 corresponds to (χ1 − χ2)/2.~~

~~Example 15.2.7. TheactionoftheLiealgebrao5 in the space of polynomials C[y12,y13,y14,y23]isgivenby~~

F11 →−y12 ∂12 − y13 ∂13 − y14 ∂14 + χ1,F22 → y12 ∂12 − y14 ∂14 − y23 ∂23 + χ2, together with 1 1 F → ∂ +y ∂ + y2 ∂ ,F →−y2 ∂ +y ∂ − y2 ∂ +(χ −χ )y , 12 12 23 13 2 23 14 21 12 12 13 23 2 13 14 1 2 12 and 1 F → ∂ ,F → y y − y ∂ − y y ∂ + y ∂ − y2 ∂ + χ y . 23 23 32 12 23 13 12 14 23 14 14 13 2 23 23 2 23

15.3. Free ﬁeld realization of sp2n B Consider the symplectic group Sp2n deﬁned in (2.29). Now − will denote the N subgroup of all lower-triangular matrices in Sp2n,and + the subgroup of Sp2n which consists of upper-triangular matrices with 1’s on the diagonal. The relation X X =1forX ∈N+ allows us to identify Fun N+ with the space of polynomials ∼ Fun N+ = C[yij | 1 i

Now apply Lemma 15.1.2 for A = Fii+1 with i ∈{1,...,n− 1}. The entries of the corresponding matrix X which diﬀer from the respective entries of Y are given by xii+1 = yii+1 − t and xis = yis − tyi+1 s for s = i +2,...,i. For the action of Fii+1 we have i Fii+1 →−∂ii+1 − yi+1 s ∂is. s=i+2 To express yi+1 i in terms of the admissible coordinates on N+,takethe(i +1,i) entries on both sides of YY =1toget

~~i−2 (15.4) yi+1 i − yii−1 + εs yisyi+1 s =0,i=1,...,n− 1. s=i+1 Therefore,~~

~~i−1 i−2 Fii+1 →−∂ii+1 − yi+1 s ∂is + yii−1 − εs yisyi+1 s ∂ii . s=i+2 s=i+1~~

−tA For A = Fnn+1 we have e =1− 2tEnn+1 which implies that the only entry of the corresponding matrix X which diﬀers from the respective entry of Y is xnn+1 = ynn+1 − 2t and so Fnn+1 →−2∂nn+1. 286 15. WAKIMOTO MODULES

The calculation of the entries of the matrix X for A = Fi+1 i with the values i ∈{1,...,n− 1} brings the same expressions as for o2n, complemented by the formulas yii xii = and 1+tyi−1 i x − = y − + t y − y − − y − − y i+1 i 1 i+1 i 1 i+1 i 1 i 1 i ii 1 i+1 i 2 + t yii − yii−1yi−1 i .

~~Hence using (15.4) and the relation yii+1 + yi−1 i =0weget~~

i Fi+1 i → − yri−1 yii+1 + yri ∂ri−1 + yri yii+1 ∂ri + yri+1 yii+1 ∂ri+1 r=1 i−2 i−1 − yis ∂i+1 s + yri+1 − yri yii+1 ∂ri s=i+2 r=1 i−2 − 2yii−1 − εs yis yi+1 s ∂i+1 i−1. s=i+2

−tA Finally, taking A = Fn+1 n we ﬁnd e =1− 2tEn+1 n. Therefore, for the entries of the matrix X we get from Lemma 15.1.2: xrn = yrn 1 − 2tynn+1 +2tyrn+1 for r =1,...,n− 1, and yrn+1 xrn+1 = for r =1,...,n. 1 − 2tynn+1

The action of Fn+1 n is then found by n−1 n Fn+1 n → 2 yrn+1 − yrnynn+1 ∂rn +2 yrn+1 ynn+1 ∂rn+1. r=1 r=1 To get the required formulas we twist the action by the automorphism changing sings of the simple root vectors and extend it to an arbitrary weight χ; cf. Propo- sition 15.2.2. Example . C 15.3.2 The representation of sp2 in the space of polynomials [y12] provided by Proposition 15.3.1 takes the form →− → →− 2 F11 2y12 ∂12 + χ1,F12 2∂12,F21 2y12 ∂12 +2χ1 y12.

It is isomorphic to the representation of sl2 obtained by the restriction from the − gl2-action of Example 15.1.3, and χ1 corresponds to χ1 χ2. Example . 15.3.3 TheactionoftheLiealgebrasp4 in the space of polynomials C[y12,y13,y14,y23]isgivenby

~~F11 →−y12 ∂12 − y13 ∂13 − 2y14 ∂14 + χ1,F22 → y12 ∂12 − y13 ∂13 − 2y23 ∂23 + χ2, together with~~

F12 → ∂12 + y23 ∂13 +(y13 − y12 y23)∂14,F23 → 2∂23, 15.4. WAKIMOTO MODULES IN TYPE A 287 and →− 2 − − F21 y12 ∂12 +(y14 + y12 y13)∂13 y12 y14 ∂14 +2y13 ∂23 +(χ1 χ2)y12, → − − − 2 F32 2 y12 y23 y13 ∂12 2y13 y23 ∂13 2y23 ∂23 +2χ2 y23.

15.4. Wakimoto modules in type A We will follow [46, Section 6.1] to reproduce the formulas defining the action of the affine Kac–Moody algebra glN in the Wakimoto modules of the critical level. A ∗ The Weyl algebra (glN ) is generated by elements aij[r]andaij[r], where 1 i0. ∗ Accordingly, the generators aij[r] with r 0andaij[r] with r>0 are called the ∗ annihilation operators, whereas the generators aij[r] with r<0andaij[r] with r 0 are called the creation operators. We can regard elements of M(glN )as polynomials in the creation operators applied to the vector |0. Introduce the formal Laurent series −r−1 ∗ ∗ −r (15.5) aij(z)= aij[r]z and aij(z)= aij[r]z r∈Z r∈Z A with coefficients in (glN ). Normally ordered products of such series are defined by applying the general rule (6.9). Equivalently, for any monomial P in the generators of the Weyl algebra, the corresponding normally ordered monomial : P : is obtained by moving all factors which are annihilation operators to the right of the factors which are creation operators. The normal ordering extends by linearity to arbitrary linear combinations of monomials. When applied to products of generating series (15.5), it will be understood that the normal ordering is applied to all coefficients of the series in z. Recall the formal Laurent series Eij(z) introduced in (7.22) and fix an N-tuple of series χ(z)= χ1(z),...,χN (z) with all χi(z) ∈ C((z)). The following formulas define the Wakimoto module Wχ(z) of the critical level. Proposition . 15.4.1 The following assignments define a representation of glN at the critical level on the vector space M(glN ): i−1 N → ∗ − ∗ Eii(z) : ari(z)ari(z): : ais(z)ais(z): +χi(z), r=1 s=i+1 for i =1,...,N, together with

N → ∗ Eii+1(z) aii+1(z)+ : ai+1 s(z)ais(z):, s=i+2 288 15. WAKIMOTO MODULES and i−1 → ∗ ∗ − ∗ Ei+1 i(z) : ari(z) aii+1(z) ari+1(z) ari(z): r=1 i N − ∗ ∗ ∗ : ari+1(z) aii+1(z) ari+1(z):+ : ais(z)ai+1 s(z): r=1 s=i+2 − ∗ − ∗ + χi(z) χi+1(z) aii+1(z) (i +1)∂z aii+1(z). for i =1,...,N − 1. Proof. The formulas follow from Proposition 15.1.1 providing a free field realization of glN and the general results of [46, Section 6.1]. The only additional calculation is needed to get the value ci = −i − 1 of the coefficient ci of the term ∗ ∂z aii+1(z) which is known to exist by the general results. To calculate the coefficient, note that since K = −N, by (7.3) we have (15.6) Eii+1[1],Ei+1 i[−1] = Eii[0] − Ei+1 i+1[0] − N. Write −r−1 (15.7) χi(z)= χi[r]z ,χi[r] ∈ C, r∈Z so that χi[r] = 0 for all sufficiently large positive values of r. Apply both sides of (15.6) to the vector |0 and note that Eii+1[1]|0 =0,whereasEii[0]|0 = χi[0]|0 for all i. Furthermore, taking the constant term in the formula for the action of Ei+1 i(z)weget i−1 − | ∗ ∗ − ∗ − | Ei+1 i[ 1] 0 = ari[0] aii+1[0] ari+1[0] ari[ 1] 0 r=1 i N − ∗ ∗ − | ∗ − | ari+1[0] aii+1[0] ari+1[ 1] 0 + ais[0]ai+1 s[ 1] 0 r=1 s=i+2 − − − ∗ − | ∗ − | + χi[p 1] χi+1[p 1] aii+1[ p] 0 + ci aii+1[ 1] 0 . p0 Now act on this vector by the operator N ∗ − Eii+1[1] = aii+1[1] + ai+1 s[ k +1]ais[k] s=i+2 k∈Z to obtain Eii+1[1] Ei+1 i[−1]|0 = −N + i +1+ci + χi[0] − χi+1[0] |0. Comparing this with the action of the right hand side of (15.6) on |0, we conclude that ci = −i − 1. Example . 15.4.2 The formulas for the Wakimoto module Wχ(z) over gl2 with χ(z)= χ1(z),χ2(z) have the form →− ∗ E11(z) : a12(z)a12(z): +χ1(z), → ∗ E22(z) : a12(z)a12(z): +χ2(z), 15.5. WAKIMOTO MODULES IN TYPES B AND D 289 together with

E12(z) → a12(z), →− ∗ 2 − ∗ − ∗ E21(z) : a12(z) a12(z): + χ1(z) χ2(z) a12(z) 2 ∂z a12(z). By the general results of [46, Section 8.3.3], elements of the center Z(glN )ofthe completed universal enveloping algebra U−N (glN ) act on the Wakimoto modules Wχ(z) by scalar multiplication. Moreover, as the next proposition shows, the eigenvalues of central elements essentially coincide with their Harish-Chandra images given in (13.47) and which were calculated in Section 13.3. Proposition . ∈ 15.4.3 The eigenvalue of any element S Z(glN ) acting in the Wakimoto module Wχ(z) coincides with the Harish-Chandra image f(S),wherethe −1 generators of the algebra Π=S h[t, t ] are respectively replaced by Eii[r] → χi[r].

Proof. We are interested in the eigenvalue of the vector |0∈Wχ(z) under the action of S. Write the element S as a (possibly inﬁnite) linear combination of ordered monomials with the ordering as deﬁned in Section 13.3. Consider a monomial occurring in this linear combination whose rightmost factor is an element of the form Eij[r] ∈ n+[t], so that i

Corollary 15.4.4. With the replacements μi(z) → χi(z) for i =1,...,N,the formulas of Proposition 13.3.3 give the eigenvalues for the action of the elements of Z(glN ) in the Wakimoto module Wχ(z).

~~15.5. Wakimoto modules in types B and D~~

~~Similar to type A, consider the Weyl algebra A(oN ) generated by elements ∗ ∈ Z aij[r]andaij[r], where 1 i~~

The Fock representation M(oN )ofA(oN ) is generated by a vector |0 such that | ∗ | aij[r] 0 =0 for r 0andaij[r] 0 =0 for r>0. ∗ The Laurent series aij(z)andaij(z) are introduced by the same formulas (15.5) as in type A. Their normally ordered products are deﬁned by the rule (6.9) which can also be stated in terms of the normally ordered monomials as in type A. 290 15. WAKIMOTO MODULES

~~Suppose now that N =2n and consider the formal Laurent series Fij(z)introduced in (8.50). Fix an n-tuple of series (15.8) χ(z)= χ1(z),...,χn(z) ,χi(z) ∈ C((z)).~~

~~The following formulas deﬁne the Wakimoto module Wχ(z) of the critical level.~~

~~Proposition 15.5.1. The following assignments deﬁne a representation of o2n at the critical level on the vector space M(o2n):~~

~~i−1 i−1 ∗ ∗ ∗ → − − Fii(z) : ari(z)ari(z): : ari (z)ari (z): : ais(z)ais(z): +χi(z) r=1 s=i+1 for i =1,...,n,~~

i−2 n ∗ ∗ ∗ → − Fii+1(z) aii+1(z)+ : ai+1 s(z)ais(z): : ai+1 s(z)ai+1 s (z)aii −1(z): s=i+2 s=i+2 for i =1,...,n− 1,andFn−1 n (z) → an−1 n (z), together with i−1 → ∗ ∗ − ∗ Fi+1 i(z) : ari(z) aii+1(z) ari+1(z) ari(z): r=1 ∗ ∗ ∗ ∗ ∗ − +: ari−1(z) aii+1(z)+ari (z) ari −1(z): : ari (z) aii+1(z) ari (z): i i−2 − ∗ ∗ ∗ : ari+1(z) aii+1(z) ari+1(z):+ : ais(z)ai+1 s(z): r=1 s=i+2 n ∗ ∗ ∗ ∗ − − − : ais(z)ais (z)aii −1(z): + χi(z) χi+1(z) aii+1(z) 2i∂z aii+1(z) s=i+2 for i =1,...,n− 1,and

~~n−2 ∗ ∗ ∗ → − Fn n−1(z) : arn−1(z) an−1 n (z) arn (z) arn−1(z): r=1 ∗ ∗ ∗ − ∗ ∗ +: arn(z) an−1 n (z)+arn+2(z) arn(z): : arn+2(z) an−1 n (z) arn+2(z):~~

n−1 ∗ ∗ − : arn (z) an−1 n (z) arn (z): r=1 ∗ − − ∗ + χn−1(z)+χn(z) an−1 n (z) (2n 2) ∂z an−1 n (z). Proof. The formulas follow from Proposition 15.2.2 providing a free field realization of o2n and the general results of [46, Section 6.1]. We only need to verify − ∗ − the values ci = 2i of the coefficients ci of the terms ∂z aii+1(z)andcn = 2n +2 ∗ of the coefficient cn of the term ∂z an−1 n (z). These values are known to exist by the general results. Since K = −2n + 2 at the critical level, by (8.2) we have Fii+1[1],Fi+1 i[−1] = Fii[0] − Fi+1 i+1[0] − 2n +2 15.5. WAKIMOTO MODULES IN TYPES B AND D 291 for i =1,...,n− 1. Writing χi(z) as in (15.7), and applying the right hand side to the vector |0 we get χi[0] − χi+1[0] − 2n +2 |0.Furthermore,

N ∗ Fii+1[1] = aii+1[1] + : ai+1 s[k]ais[l]: s=i+2 k+l=1 n ∗ ∗ − : ai+1 s[k]ai+1 s [l]aii −1[m]: s=i+2 k+l+m=1 so that Fii+1[1]|0 = 0, and all components of the vector Fi+1 i[−1]|0 are annihilated by the operator Fii+1[1] except for i−2 ∗ − | − ∗ − | ∗ − | ais[0]ai+1 s[ 1] 0 + χi[0] χi+1[0] aii+1[ 1] 0 + ci aii+1[ 1] 0 . s=i+2 By applying F [1] to this component we get ii+1 Fii+1[1] Fi+1 i[−1]|0 = −2n +2+2i + ci + χi[0] − χi+1[0] |0 and so ci = −2i. The coefficient cn is calculated in the same way by applying the operators on both sides of the relation Fn−1 n [1],Fn n−1[−1] = Fn−1 n−1[0] + Fnn[0] − 2n +2 to the vector |0 to get cn = −2n +2. Now let N =2n +1. Fixan n-tuple of series χ(z) as in (15.8) and use the series Fij(z) introduced in (8.50). The Wakimoto module Wχ(z) of the critical level is defined in the following proposition. Proposition 15.5.2. The following assignments define a representation of o2n+1 at the critical level on the vector space M(o2n+1): i−1 i−1 ∗ ∗ ∗ → − − Fii(z) : ari(z)ari(z): : ari (z)ari (z): : ais(z)ais(z): +χi(z) r=1 s=i+1 i−2 → ∗ Fii+1(z) aii+1(z)+ : ai+1 s(z)ais(z): s=i+2 i−2 1 ∗ ∗ − : a (z)a (z)a − (z): 2 i+1 s i+1 s ii 1 s=i+2 for i =1,...,n, together with i−1 → ∗ ∗ − ∗ Fi+1 i(z) : ari(z) aii+1(z) ari+1(z) ari(z): r=1 ∗ ∗ ∗ ∗ ∗ − +: ari−1(z) aii+1(z)+ari (z) ari −1(z): : ari (z) aii+1(z) ari (z): i i−2 − ∗ ∗ ∗ : ari+1(z) aii+1(z) ari+1(z):+ : ais(z)ai+1 s(z): r=1 s=i+2 i−2 1 ∗ ∗ ∗ ∗ − : a (z)a (z)a − (z): + χ (z) − χ (z) a (z) − 2i∂ a (z) 2 is is ii 1 i i+1 ii+1 z ii+1 s=i+2 292 15. WAKIMOTO MODULES for i =1,...,n− 1,and

~~n−1 → ∗ ∗ − ∗ Fn+1 n(z) : arn(z) ann+1(z) arn+1(z) arn(z): r=1 − ∗ ∗ ∗ : arn+2(z) ann+1(z) arn+2(z):+:arn+2(z) arn(z):~~

1 − : a∗ (z)2 a (z): +χ (z) a∗ (z) − (2n − 1) ∂ a∗ (z). 2 nn+1 nn+1 n nn+1 z nn+1 Proof. As with Proposition 15.5.1, we only need to calculate the coeﬃcients ∗ ∗ ci of the terms ∂z aii+1(z)andcn of the term ∂z ann+1(z). This time the critical level is K = −2n + 1 so that (8.2) gives Fii+1[1],Fi+1 i[−1] = Fii[0] − Fi+1 i+1[0] − 2n +1 for i =1,...,n. The same argument as for o2n yields the values ci = −2i for i =1,...,n− 1andcn = −2n +1.

The corresponding analogue of Proposition 15.4.3 holds in the same form for the orthogonal Lie algebras oN . Itisprovedbythesameargumentrelyingonthe formulas of Propositions 15.5.1 and 15.5.2. We use the homomorphism (13.47).

Proposition 15.5.3. The eigenvalue of any element S ∈ Z(oN ) acting in the Wakimoto module Wχ(z) coincides with the Harish-Chandra image f(S),wherethe generators of the algebra Π=Sh[t, t−1] are respectively replaced by the rule Fii[r] → χi[r] for i =1,...,n and r ∈ Z.

Corollary 15.5.4. With the replacements μi(z) → χi(z) for i =1,...,n, the formulas of Propositions 13.3.4, 13.3.5 and 13.3.6 give the eigenvalues for the action of the elements of Z(oN ) in the Wakimoto module Wχ(z).

~~15.6. Wakimoto modules in type C A ∗ The Weyl algebra (sp2n) is generated by elements aij[r]andaij[r], where 1 i0.~~

We use the same formulas (15.5) as in type A to introduce the Laurent series ∗ aij(z)andaij(z), and apply the rule (6.9) to deﬁne their normally ordered products. Fix an n-tuple of series χ(z) as in (15.8) and use the series Fij(z) introduced in (8.50). The following proposition deﬁnes the corresponding Wakimoto module Wχ(z) of the critical level. 15.6. WAKIMOTO MODULES IN TYPE C 293

~~Proposition . 15.6.1 The following assignments deﬁne a representation of sp2n at the critical level on the vector space M(sp2n):~~

~~i−1 ∗ ∗ → − Fii(z) : ari(z)ari(z): : ari (z)ari (z): r=1 i−1 ∗ ∗ − − : ais(z)ais(z): 2:aii (z)aii (z): +χi(z) s=i+1 for i =1,...,n,~~

i−1 → ∗ Fii+1(z) aii+1(z)+ : ai+1 s(z)ais(z): s=i+2 i−2 ∗ ∗ ∗ − +: aii−1(z) εs ais(z)ai+1 s (z) aii (z): s=i+1 for i =1,...,n− 1,andFnn+1(z) → 2ann+1(z), together with i ∗ ∗ ∗ → Fi+1 i(z) : ari−1(z) aii+1(z)+ari (z) ari −1(z): r=1 ∗ ∗ ∗ ∗ − − : ari (z) aii+1(z) ari (z): : ari+1(z) aii+1(z) ari+1(z):

i−2 i−1 ∗ ∗ ∗ − ∗ + : ais(z)ai+1 s(z):+ : ari(z) aii+1(z) ari+1(z) ari(z): s=i+2 r=1 i−2 ∗ ∗ ∗ − +: 2aii−1(z) εs ais(z)ai+1 s (z) ai+1 i −1(z): s=i+2 − ∗ − ∗ (2i +1)∂z aii+1(z)+ χi(z) χi+1(z) aii+1(z) for i =1,...,n− 1,and n−1 → ∗ ∗ − ∗ Fn+1 n(z) 2 : arn(z) ann+1(z) arn+1(z) arn(z): r=1 n − ∗ ∗ ∗ − ∗ 2 : arn+1(z) ann+1(z) arn+1(z):+2χn(z)ann+1(z) 2(n +1)∂z ann+1(z). r=1 Proof. We use again the general construction of the Wakimoto modules [46, Section 6.1], so that the formulas are implied by Proposition 15.3.1. We will only ∗ verify the values of the coeﬃcients ci of the terms ∂z aii+1(z)andthecoeﬃcientcn ∗ − − of the term ∂z ann+1(z). At the critical level we have K = n 1 and so (8.32) implies Fii+1[1],Fi+1 i[−1] = Fii[0] − Fi+1 i+1[0] − 2n − 2 for i =1,...,n−1. Write χi(z) as in (15.7) and apply the right hand side to the vector |0 to get χi[0] − χi+1[0] − 2n − 2 |0. Now observe that all components of 294 15. WAKIMOTO MODULES the vector Fi+1 i[−1]|0 are annihilated by the operator Fii+1[1] except for

i−2 ∗ ∗ ∗ − | − | − | aii [0]aii −1[ 1] 0 + ais[0]ai+1 s[ 1] 0 +2aii−1[0]ai+1 i −1[ 1] 0 s=i+2 − ∗ − | ∗ − | + χi[0] χi+1[0] aii+1[ 1] 0 + ci aii+1[ 1] 0 . By applying F [1] to this component we get ii+1 Fii+1[1] Fi+1 i[−1]|0 = −2n +1+2i + ci + χi[0] − χi+1[0] |0 and so ci = −2i − 1. The coefficient cn is calculated in the same way by applying the operators on both sides of the relation Fnn+1[1],Fn+1 n[−1] =4Fnn[0] − 4n − 4 to the vector |0 to get cn = −2n − 2. The following is an analogue of Proposition 15.4.3 which is verified by the same argument relying on the formulas of Proposition 15.6.1. We use the homomorphism f defined in (13.47). Proposition . ∈ 15.6.2 The eigenvalue of any element S Z(sp2n) acting in the Wakimoto module Wχ(z) coincides with the Harish-Chandra image f(S),where the generators of the algebra Π=S h[t, t−1] are respectively replaced by the rule Fii[r] → χi[r] for i =1,...,n and r ∈ Z.

Corollary 15.6.3. With the replacements μi(z) → χi(z) for i =1,...,n,the formulas of Proposition 13.3.7 give the eigenvalues for the action of the elements of Z(sp2n) in the Wakimoto module Wχ(z). 15.7. Bibliographical notes The Wakimoto modules were originally introduced by Wakimoto [151]for sl2 and by Feigin and Frenkel [38] for arbitrary aﬃne Kac–Moody algebras g. The construction and its generalizations are explained in detail in the book by Frenkel [46, Chapter 6]. Propositions 15.4.3, 15.5.3 and 15.6.2 are particular cases of the general results which hold for arbitrary simple Lie algebras g as shown in [46, Section 8.3.3]. Bibliography

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λ-bracket, 93, 213 creation operator, 287 n-th product, 92 critical level, 94, 182 q-character, 196, 201 crossing symmetry, 180 cyclic property of trace, 15, 19 affine Kac–Moody algebra, 93 affine Poisson vertex algebra, 214 degenerate affine Hecke algebra, 56 affine symmetric polynomial, 102 diagram, 1 annihilation operator, 287 differential algebra, 213 anti-symmetrizer distance in multiset, 198 in Brauer algebra, 14 dual Coxeter number, 93 in the symmetric group algebra, 5 elementary symmetric functions, 232 evaluation homomorphism, 170 Bethe ansatz equations, 270 evaluation module, 170 Bethe vector, 270 extremal projector, 39 Bolsinov’s criterion, 163 Bolsinov–Elashvili conjecture, 167 Feigin–Frenkel center, 94 box field, 91 addable, 1 Fock representation, 287, 289, 292 removable, 1 Fourier coefficient, 92 Brauer algebra, 7 free field realization, 277 Brauer diagram, 6 full trace, 15 breaking pair, 198 fusion procedure for symmetric group, 6 Capelli determinant, 64 Casimir element, 55 Gelfand invariant, 55, 66, 77 center of universal enveloping algebra, 55 Harish-Chandra of vertex algebra, 92 homomorphism, 55 character of representation, 3 isomorphism, 55 characteristic map, 27 harmonic polynomial, 194, 196, 198 Chevalley generators, 214 harmonic tensors, 194, 197 Chevalley involution, 263 highest vector, 57, 73 Chevalley isomorphism, 23 highest weight, 193 classical W-algebra, 215 highest weight representation, 57, 73 closure map, 14 hook length formula, 2 column-determinant, 47 Howe duality, 39 column-minor, 160 immanant, 26 comatrix, 48 quantum, 68 complete set of Segal–Sugawara vectors, 96 index of a Lie algebra, 163 complete symmetric functions, 232 completed universal enveloping algebra, 104 Jacobi identity, 149 content of a box, 1 Jucys–Murphy elements contents of updown tableau, 8 for Brauer algebra, 7 Coxeter number, 153 for symmetric group, 3

~~303 304 INDEX~~

Kirillov–Reshetikhin module, 196 Sugawara operators, 105 Kostant’s slice, 154 symmetric polynomial factorial complete, 58 Laplace operator, 194, 196 factorial elementary, 58 Leibniz rule, 149 symmetrizer length of diagram, 1 in Brauer algebra, 9 level of g-module, 93 in the symmetric group algebra, 4 Lie algebra general linear, 23 tableau, 1 orthogonal, 28 standard, 2 special linear, 23 trace, 18 symplectic, 28 transfer matrix, 176 Lie conformal algebra, 93 translation operator, 91 Lie–Poisson bracket, 149 lowest vector, 263, 265 unitarity property, 180 lowest weight representation, 263, 265 updown tableau, 8

MacMahon Master Theorem, 45 vacuum module, 93, 183 Manin matrix, 43 vacuum vector, 91 of type C,89 Verma module, 193 of types B and D,89 vertex algebra, 91 matrix presentation, 54 aﬃne, 94 Mishchenko–Fomenko subalgebra, 150 commutative, 92 Miura transformation, 228 holomorphic, 92 Vinberg’s quantization problem, 155 Nazarov–Wenzl algebra, 73 Newton identity, 50, 66 Wakimoto module, 287, 290–292 normal ordering, 94 Weyl algebra, 287, 289, 292 normalized Killing form, 93 Weyl group, 23 Yang R-matrix, 170 partial trace, 14 Yang–Baxter equation, 14, 170 partition, 1 Yangian, 169, 191 Pfaﬃan, 30, 82, 120, 145, 165 double, 180 plane partition, 103 dual, 177, 203 Poisson algebra, 149 extended, 191 Poisson bracket, 149 extended dual, 202 Poisson center, 149 Yangian character, 171 Poisson vertex algebra, 213 Young basis, 2 primitive idempotents, 2 Young diagram, 1 regular element, 151 reverse tableau, 179 right-quantum matrix, 51 Robinson hook dimension formula, 61 row-determinant, 48

Schur polynomial double, 58 factorial, 58 Schur–Weyl duality, 19, 60 screening operator, 187, 189, 207, 211, 228 Segal–Sugawara vector, 94 canonical, 95 shift of argument subalgebra, 150 skew diagram, 160 skew Howe duality, 41 skew Laplace operator, 198 smooth module, 104 state-ﬁeld correspondence, 91 subalgebra of vertex algebra, 92 Selected Published Titles in This Series

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For a complete list of titles in this series, visit the AMS Bookstore at www.ams.org/bookstore/survseries/. The celebrated Schur-Weyl duality gives rise to effective ways of constructing invariant polynomials on the classical Lie algebras. The emergence of the theory of quantum groups in the 1980s brought up special matrix techniques which allowed one to extend these constructions beyond polynomial invariants and produce new families of Casimir elements for finite-dimensional Lie algebras. Sugawara operators are analogs of Casimir elements for the affine Kac-Moody algebras. The goal of this book is to describe algebraic structures associated with the affine Lie algebras, including affine vertex algebras, Yangians, and classical W-algebras, which have numerous ties with many areas of mathematics and mathematical physics, including modular forms, conformal field theory, and soliton equations. An affine version of the matrix technique is developed and used to explain the elegant constructions of Sugawara operators, which appeared in the last decade. An affine analogue of the Harish-Chandra isomorphism connects the Sugawara operators with the classical W-algebras, which play the role of the Weyl group invariants in the finite-dimensional theory.

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