REPRESENTATIONS of SHIFTED YANGIANS 1. Introduction

REPRESENTATIONS OF SHIFTED YANGIANS

JONATHAN BRUNDAN AND ALEXANDER KLESHCHEV

Abstract. We study highest weight representations of shifted Yangians over an algebraically closed field of characteristic 0. In particular, we classify the finite dimensional irreducible representations and explain how to compute their Gelfand- Tsetlin characters in terms of known characters of standard modules and certain Kazhdan-Lusztig polynomials. Our approach exploits the relationship between shifted Yangians and the finite W -algebras associated to nilpotent orbits in general linear Lie algebras. At the end of the introduction, we formulate some conjectures. 1. Introduction 1 2. Shifted Yangians 10 3. Finite W -algebras 22 4. Dual canonical bases 35 5. Highest weight theory 46 6. Verma modules 52 7. Standard modules 64 8. Finite dimensional representations 79 References 92

1. Introduction Following work of Premet, there has been renewed interest recently in the representation theory of certain algebras that are associated to nilpotent orbits in complex semisimple Lie algebras. We refer to these algebras as finite W -algebras. They should be viewed as analogues of universal enveloping algebras for the Slodowy slice through the nilpotent orbit in question. Actually, in the special cases considered in this article, the definition of these algebras first appeared in 1979 in the Ph.D. thesis of Lynch [Ly], extending the celebrated work of Kostant [Ko2] treating regular nilpotent orbits. However, despite quite a lot of attention by a number of authors since then, see e.g. [Ka, M, Ma, BT, VD, GG, P1, P2], there is still surprisingly little concrete information about the representation theory of these algebras to be found in the literature. The goal in this article is to undertake a thorough study of finite dimensional representations of the finite W -algebras associated to nilpotent orbits in the Lie algebra glN (C). We are able to make progress in this case thanks largely to the relationship between finite W -algebras and shifted Yangians first noticed in [RS, BR] and developed in full generality in [BK5].

Fix for the remainder of the introduction a partition λ = (p1 ≤ · · · ≤ pn) of N. We draw the Young diagram of λ in a slightly unconventional way, so that there are pi boxes in the ith row, numbering rows 1, . . . , n from top to bottom in order of

2000 Mathematics Subject Classiﬁcation: 17B37. Research supported in part by NSF grant no. DMS-0139019.

1 2 JONATHAN BRUNDAN AND ALEXANDER KLESHCHEV increasing length. Also number the non-empty columns of this diagram by 1, . . . , l 0 from left to right, and let qi denote the number of boxes in the ith column, so λ = (q1 ≥ · · · ≥ ql) is the transpose partition to λ. For example, if (p1, p2, p3) = (2, 3, 4) then the Young diagram of λ is 1 4 2 5 7 3 6 8 9 and (q1, q2, q3, q4) = (3, 3, 2, 1). We number the boxes of the diagram by 1, 2,...,N down columns from left to right, and let row(i) and col(i) denote the row and column numbers of the ith box. Writing ei,j for the ij-matrix unit in the Lie algebra g = glN (C), let e denote P the matrix i,j ei,j summing over all 1 ≤ i, j ≤ N such that row(i) = row(j) and col(i) = col(j) − 1. This is a nilpotent matrix of Jordan type λ. For instance, if λ is L as above, then e = e1,4 +e2,5 +e5,7 +e3,6 +e6,8 +e8,9. Define a -grading g = gj Z j∈Z of the Lie algebra g by declaring that each ei,j is of degree (col(j) − col(i)). This is a good grading for e ∈ g1 in the sense of [KRW] (see also [EK] for the full classification). However, it is not the usual Dynkin grading arising from an sl2-triple unless all the parts of λ are equal. Actually, in the main body of the article, we work with more general good gradings than the one described here, replacing the Young diagram of λ with a more general diagram called a pyramid and denoted by the symbol π; see §3.1. When the pyramid π is left-justified, it coincides with the Young diagram of λ. We have chosen to focus just on this case in the introduction, since it plays a distinguished role in the theory. Now we give a formal definition of the finite W -algebra W (λ) associated to this L data. Let p denote the parabolic subalgebra j≥0 gj of g with Levi factor h = g0, and L let m denote the opposite nilradical j<0 gj. Taking the trace form with e defines a one dimensional representation χ : m → C. Let prχ : U(g) U(p) denote the projection along the direct sum decomposition U(g) = U(p) ⊕ U(g)Iχ, where U(g)Iχ is the left ideal of U(g) generated by the elements {x−χ(x)|x ∈ m}. Define a twisted action of x ∈ m on U(p) by setting x · u := prχ([x, u]) for all u ∈ U(p). Then, W (λ) is the subalgebra U(p)m of all twisted m-invariants in U(p); see §3.2. For example, if the Young diagram of λ consists of a single column and e is the zero matrix, W (λ) coincides with the entire universal enveloping algebra U(g). At the other extreme, if the Young diagram of λ consists of a single row and e is a regular nilpotent element, the work of Kostant [Ko2] shows that W (λ) is isomorphic to the center of U(g), in particular it is commutative. Let ξ : U(p) → U(h) be the algebra homomorphism induced by the natural projection p h. Let η : U(h) → U(h) be the automorphism mapping a matrix unit ei,j ∈ h to ei,j +δi,j(qcol(i)+1+···+ql); this simply provides a convenient renormalization of the generators. The restriction of the composition η ◦ ξ to W (λ) defines an injective algebra homomorphism µ : W (λ) ,→ U(h) which we call the Miura transform; see §3.6. To explain its importance in the theory, we note that h = gl ( ) ⊕ · · · ⊕ gl ( ), so U(h) q1 C ql C is naturally identified with the tensor product U(gl ( )) ⊗ · · · ⊗ U(gl ( )). Given q1 C ql C gl ( )-modules M for each k = 1, . . . , l, the outer tensor product M ··· M qk C k 1 l REPRESENTATIONS OF SHIFTED YANGIANS 3 is therefore a U(h)-module in the natural way. Hence, via the Miura transform, M1 ··· Ml is a W (λ)-module too. This construction plays the role of tensor product in the representation theory of W (λ). Next we want to recall the connection between W (λ) and shifted Yangians. Let σ be the upper triangular n×n matrix with ij-entry (pj −pi) for each 1 ≤ i ≤ j ≤ n. The shifted Yangian Yn(σ) associated to σ is the associative algebra over C with generators (r) (r) (r) Di (1 ≤ i ≤ n, r > 0),Ei (1 ≤ i < n, r > pi+1 − pi) and Fi (1 ≤ i < n, r > 0) subject to certain relations recorded explicitly in §2.1. In the case that σ is the zero matrix, i.e. all parts of λ are equal, Yn(σ) is precisely the usual Yangian Yn associated to the Lie algebra gln(C) and the defining relations are a variation on the Drinfeld presentation of [D]; see [BK4]. In general, the presentation of Yn(σ) is adapted to its natural triangular decomposition, allowing us to study representations in terms of (r) highest weight theory. In particular, the subalgebra generated by all the elements Di is a maximal commutative subalgebra which we call the Gelfand-Tsetlin subalgebra. We often work with the generating functions (1) −1 (2) −2 −1 Di(u) = 1 + Di u + Di u + · · · ∈ Yn(σ)[[u ]]. The main result of [BK5] shows that the finite W -algebra W (λ) is isomorphic to the (r) quotient of Yn(σ) by the two-sided ideal generated by all D1 (r > p1). A precise identification of W (λ) with this quotient is explained in §3.4, using explicit formulae for invariants in U(p)m. Under this identification, the tensor product construction defined using the Miura transform in the previous paragraph is induced by the comultiplication on the Hopf algebra Yn; see §2.5. We are ready to describe the first results about representation theory. We call a (r) vector v in a Yn(σ)-module M a highest weight vector if it is annihilated by all Ei (r) and each Di acts on v by a scalar. A critical point is that if v is a highest weight vector in a W (λ)-module, viewed as a Yn(σ)-module via the map Yn(σ) W (λ), (r) then in fact Di v = 0 for all r > pi. This is obvious for i = 1, since the image (r) of D1 in W (λ) is zero by the definition of the map for all r > p1. For i > 1, it follows from the following fundamental result proved in §3.7: for any i and r > pi, (r) the image of Di in W (λ) is congruent to zero modulo the left ideal generated by (s) all Ej . Hence, if v is a highest weight vector in a W (λ)-module, then there exist scalars (ai,j)1≤i≤n,1≤j≤pi such that p1 u D1(u)v = (u + a1,1)(u + a1,2) ··· (u + a1,p1 )v, p2 (u − 1) D2(u − 1)v = (u + a2,1)(u + a2,2) ··· (u + a2,p2 )v, . .

pn (u − n + 1) Dn(u − n + 1)v = (u + an,1)(u + an,2) ··· (u + an,pn )v.

Let A be the λ-tableau obtained by writing the scalars ai,1, . . . , ai,pi into the boxes on the ith row of the Young diagram of λ. In this way, the set of highest weights that can arise in W (λ)-modules are parametrized by the set Row(λ) of row symmetrized λ-tableaux, i.e. tableaux of shape λ with entries from C viewed up to row 4 JONATHAN BRUNDAN AND ALEXANDER KLESHCHEV equivalence. Conversely, given any row symmetrized λ-tableau A ∈ Row(λ), there exists a (non-zero) universal highest weight module M(A) ∈ W (λ) -mod generated by such a highest weight vector; see §6.1. We call M(A) the Verma module of type A. By familiar arguments, M(A) has a unique irreducible quotient L(A). The modules {L(A) | A ∈ Row(λ)} are all irreducible highest weight modules for W (λ) up to isomorphism. There is a natural abelian category M(λ) which is an analogue of the BGG category O for the algebra W (λ); see §7.5. (Actually, M(λ) is more like the category O∞ obtained by weakening the hypothesis that a Cartan subalgebra acts semisimply in the usual definition of O.) All objects in M(λ) are of finite length, and the simple objects are precisely the irreducible highest weight modules, hence the isomorphism classes {[L(A)] | A ∈ Row(λ)} give a canonical basis for the Grothendieck group [M(λ)] of the category M(λ). Verma modules belong to M(λ), and it is natural to consider the composition multiplicities [M(A): L(B)] for A, B ∈ Row(λ). We will formulate a precise combinatorial conjecture for these, in the spirit of the Kazhdan- Lusztig conjecture, later on in the introduction. For now, we just record the following basic result about the structure of Verma modules; see §6.3. For the statement, let ≤ denote the Bruhat ordering on row symmetrized λ-tableaux; see §4.1. Theorem A (Linkage principle). For A, B ∈ Row(λ), the composition multiplicity [M(A): L(A)] is equal to 1, and [M(A): L(B)] 6= 0 if and only if B ≤ A in the Bruhat ordering. Hence, {[M(A)] | A ∈ Row(λ)} is another natural basis for the Grothendieck group [M(λ)]. We want to say a few words about the proof of Theorem A, since it involves an important technique. Modules in the category M(λ) possess Gelfand-Tsetlin characters; see §5.2. This is a formal notion that keeps track of the dimensions of the generalized weight space decomposition of a module with respect to the Gelfand-Tsetlin subalgebra of Yn(σ), similar in spirit to the q-characters of Frenkel and Reshetikhin [FR]. The map sending a module to its Gelfand-Tsetlin character induces an embedding of the Grothendieck group [M(λ)] into a certain completion of the ring of ±1 Laurent polynomials Z[yi,a |i = 1, . . . , n, a ∈ C], for indeterminates yi,a. The key step in our proof of Theorem A is the computation of the Gelfand-Tsetlin character of the Verma module M(A) itself; see §6.2 for the precise statement. In general, ch M(A) is ±1 an infinite sum of monomials in the yi,a ’s involving both positive and negative powers, but the highest weight vector of M(A) contributes just the positive monomial y y ··· y y y ··· y ··· y y ··· y , 1,a1,1 1,a1,2 1,a1,p1 2,a2,1 2,a2,2 2,a2,p2 n,an,1 n,an,2 n,an,pn where ai,1, . . . , ai,pi are the entries in the ith row of A as above. The highest weight vector of any composition factor contributes a similar such positive monomial. So by analyzing the positive monomials appearing in the formula for ch M(A), we get information about the possible L(B)’s that can be composition factors of M(A). The Bruhat ordering on tableaux emerges naturally out of these considerations. Another important property of Verma modules has to do with tensor products. Let A ∈ Row(λ) be a row equivalence class of λ-tableaux. Pick a representative for A and let Ak denote its kth column with entries ak,1, . . . , ak,qk read from top to bottom, for REPRESENTATIONS OF SHIFTED YANGIANS 5 each k = 1, . . . , l. Let M(A ) denote the Verma module for the Lie algebra gl ( ) k qk C generated by a highest weight vector m+ which is annihilated by all strictly upper triangular matrices and on which ei,i acts as the scalar (ak,i +i−1) for each 1 ≤ i ≤ qk. Via the Miura transform, the tensor product M(A1)···M(Al) is then naturally a W (λ)-module as explained above, and the vector m+ ⊗· · ·⊗m+ is an obvious highest weight vector in this tensor product of highest weight A. In fact, our formula for the Gelfand-Tsetlin character of M(A) implies that

[M(A)] = [M(A1) ··· M(Al)], equality in the Grothendieck group [M(λ)]. Generically, M(A) is irreducible, hence it is actually isomorphic to M(A1) ··· M(Al) as a module. But there are exam- ples when, although equal in the Grothendieck group, M(A) is never isomorphic to M(A1) ··· M(Al), for any choice of representative for A. The ﬁrst part of next theorem, proved in §6.4, is a consequence of this equality in the Grothendieck group; the second part is an application of [FO].

Theorem B (Structure of center). The restriction of the linear map prχ : U(g) → U(p) defines an algebra isomorphism Z(U(g)) → Z(W (λ)) between the center of U(g) and the center of W (λ). Moreover, W (λ) is free as a module over its center. Now we switch our attention to finite dimensional W (λ)-modules. Let F(λ) denote the category of all finite dimensional W (λ)-module, viewed as a subcategory of the category M(λ). The problem of classifying all finite dimensional irreducible W (λ)- modules reduces to determining precisely which A ∈ Row(λ) have the property that L(A) is finite dimensional. To formulate the final result, we need one more definition. Call a λ-tableau A with entries in C column strict if in every column the entries belong to the same coset of C modulo Z and are strictly increasing from bottom to top. Let Col(λ) denote the set of all such column strict λ-tableaux. There is an obvious map R : Col(λ) → Row(λ) mapping a λ-tableau to its row equivalence class. Let Dom(λ) denote the image of this map, the set of all dominant row symmetrized λ-tableaux. Theorem C (Finite dimensional irreducible representations). For A ∈ Row(λ), the irreducible highest weight module L(A) is finite dimensional if and only if A is dominant. Hence, the modules {L(A) | A ∈ Dom(λ)} form a complete set of pairwise non-isomorphic finite dimensional irreducible W (λ)-modules. To prove this, there are two steps: one needs to show first that each L(A) with A ∈ Dom(λ) is finite dimensional, and second that all other L(A)’s are infinite dimensional. Let us explain the argument for the first step. Given A ∈ Col(λ) and k = 1, . . . , l, let Ak denote its kth column and let L(Ak) denote the irreducible highest weight module for the Lie algebra gl ( ) of the same highest weight as the Verma module M(A ) qk C k introduced above. Because A is column strict, each L(Ak) is finite dimensional. Hence we obtain a finite dimensional W (λ)-module

V (A) = L(A1) ··· L(Al), 6 JONATHAN BRUNDAN AND ALEXANDER KLESHCHEV which we call the standard module corresponding to A ∈ Col(λ). It contains an obvious highest weight vector of highest weight equal to the image of A in Row(λ). This simple construction is enough to finish the first step of the proof. The second step is actually much harder, and is an extension of the proof due to Tarasov [T1, T2] and Drinfeld [D] of the classification of finite dimensional irreducible representations of the Yangian Yn by Drinfeld polynomials. It is based on the remarkable fact that when n = 2, i.e. the Young diagram of λ has just two rows, every L(A)(A ∈ Row(λ)) can be expressed as an irreducible tensor product; see §7.1. Amongst all the standard modules, there are some special ones which are highest weight modules, and whose isomorphism classes form a basis for the Grothendieck group of the category F(λ). To formulate the key result here, we need one more combinatorial definition. Let A ∈ Col(λ) be a column strict λ-tableau with entries ai,1, . . . , ai,pi in its ith row read from left to right. We say that A is standard if ai,j ≤ ai,k for every 1 ≤ i ≤ n and 1 ≤ j < k ≤ pi such that ai,j and ai,k belong to the same coset of C modulo Z. If all entries of A are integers, this is the usual definition of a standard tableau: entries increase strictly up columns and weakly along rows. Our proof of the next theorem is based on an argument due to Chari [C2] in the context of quantum affine algebras; see §7.3. Theorem D (Highest weight standard modules). If A ∈ Col(λ) is standard, then the standard module V (A) is a highest weight module of highest weight equal to the row equivalence class of A. Most of the results so far are analogous to well known results in the representation theory of Yangians and quantum affine algebras, and do not exploit the finite W - algebra side of the picture in any significant way. To remedy this, we need to apply Skryabin’s theorem from [Sk]; see §8.1. This gives an explicit equivalence of categories between the category of all W (λ)-modules and the category W(λ) of all generalized Whittaker modules, namely, all g-modules on which (x−χ(x)) acts locally nilpotently for all x ∈ m. For any finite dimensional g-module V , it is obvious that the functor ? ⊗ V maps objects in W(λ) to objects in W(λ). Transporting through Skryabin’s equivalence of categories, we obtain a functor ? ~ V on W (λ) -mod itself; see §8.2. In this way, one can introduce translation functors on the categories M(λ) and F(λ). Actually, we just need some special translation functors, peculiar to the type A theory and denoted ei, fi for i ∈ C, which arise from ~’ing with the natural glN (C)-module and its dual. These functors fit into the axiomatic framework developed by Chuang and Rouquier [CR]; see §8.3. Now recall the parabolic subalgebra p of g with Levi factor h. We let O(λ) denote the corresponding parabolic category O, the category of all finitely generated g- modules on which p acts locally finitely and h acts semisimply. For A ∈ Col(λ) with entry ai in its ith box, we let N(A) ∈ O(λ) denote the parabolic Verma module generated by a highest weight vector m+ that is annihilated by all strictly upper triangular matrices in g and on which ei,i acts as the scalar (ai + i − 1) for each i = 1,...,N. Let K(A) denote the unique irreducible quotient of N(A). Both of the sets {[N(A)] | A ∈ Col(λ)} and {[K(A)] | A ∈ Col(λ)} form natural bases for the REPRESENTATIONS OF SHIFTED YANGIANS 7

Grothendieck group [O(λ)]. There is a remarkable functor F : O(λ) → F(λ) which is essentially the same as the Whittaker functor studied by Kostant and Lynch; see §8.4. It is an exact functor preserving central characters and commuting with translation functors. Moreover, it maps the parabolic Verma module N(A) to the standard module V (A) for every A ∈ Col(λ). The culmination of this article is the following theorem. Theorem E (Construction of irreducible modules). The functor F : O(λ) → F(λ) sends irreducible modules to irreducible modules or zero. More precisely, take any A ∈ Col(λ) and let B ∈ Row(λ) be its row equivalence class. Then, L(B) if A is standard, F (K(A)) =∼ 0 otherwise. Every finite dimensional irreducible W (λ)-module arises in this way. There are three main ingredients to the proof of this theorem. First, we need detailed information about the translation functors ei, fi, much of which is provided by [CR] as an application of the representation theory of degenerate affine Hecke algebras. Second, we need to know that the standard modules V (A) have simple cosocle whenever A is standard, which follows from Theorem D. Finally, we need to apply the Kazhdan-Lusztig conjecture for the Lie algebra glN (C) in order to determine exactly when F (K(A)) is non-zero. Let us discuss some of the combinatorial consequences of Theorem E in more detail. For this, we at last restrict our attention just to modules having integral central character. Let Col0(λ), Dom0(λ) and Row0(λ) denote the subsets of Col(λ), Dom(λ) and Row(λ) consisting of the tableaux all of whose entries are integers. Also let Std0(λ) denote the set of all standard tableaux in Col0(λ). So elements of Std0(λ) are tableaux of shape λ having entries from Z, such that entries are weakly increasing along rows from left to right and strictly increasing up columns from bottom to top. The map R actually gives a bijection between the set Std0(λ) and Dom0(λ). Let O0(λ), F0(λ) and M0(λ) denote the full subcategories of O(λ), F(λ) and M(λ) that consist of objects all of whose composition factors are of the form {K(A) | A ∈ Col0(λ)}, {L(A) | A ∈ Dom0(λ)} and {L(A) | A ∈ Row0(λ)}, respectively. The isomorphism classes of these three sets of objects give canonical bases for the Grothendieck groups [O0(λ)], [F0(λ)] and [M0(λ)], as do the isomorphism classes of the parabolic Verma modules {N(A) | A ∈ Col0(λ)}, the standard modules {V (A) | A ∈ Std0(λ)}, and the Verma modules {M(A) | A ∈ Row0(λ)}, respectively. The functor F above restricts to an exact functor F : O0(λ) → F0(λ), and we also have the natural embedding I of the category F0(λ) into M0(λ). At the level of Grothendieck groups, these functors induce maps F I [O0(λ)] [F0(λ)] ,→ [M0(λ)]. The translation functors ei, fi for i ∈ Z (and more generally their divided powers (r) (r) ei , fi defined as in [CR]) induce maps also denoted ei, fi on all these Grothendieck groups. The resulting maps satisfy the relations of the Chevalley generators (and 8 JONATHAN BRUNDAN AND ALEXANDER KLESHCHEV their divided powers) for the Kostant Z-form UZ of the universal enveloping algebra of the Lie algebra gl∞(C), that is, the Lie algebra of matrices with rows and columns labelled by Z all but finitely many of which are zero. The maps F and I are then UZ-module homomorphisms with respect to these actions. Now the point is that all of this categorifies a well known situation in linear algebra. Vλ0 Let VZ denote the natural UZ-module, on basis vi (i ∈ Z). We write (VZ) for the Vq1 Vql λ p1 tensor product (VZ) ⊗ · · · ⊗ (VZ) and S (VZ) for the tensor product S (VZ) ⊗ pn · · · ⊗ S (VZ). These free Z-modules have natural monomial bases denoted {NA | A ∈ Col0(λ)} and {MA | A ∈ Row0(λ)}, respectively; see §4.2. A well known consequence of the Littlewood-Richardson rule is that the space 0 Hom (Vλ (V ),Sλ(V )) UZ Z Z is a free Z-module of rank one; indeed, there is a canonical UZ-module homomorphism Vλ0 λ F : (VZ) → S (VZ) that generates the space of all such maps. The image of this λ map F is P (VZ), the familiar irreducible polynomial representation of UZ labelled by λ the partition λ. Recall P (VZ) also possesses a standard monomial basis {VA | A ∈ Vλ0 λ Std0(λ)}, defined from VA = F (NA). Finally, let i : (VZ) → [O0(λ)], j : P (VZ) → λ [F0(λ)] and k : S (VZ) → [M0(λ)] be the Z-module homomorphisms sending NA 7→ N(A),VA 7→ [V (A)] and MA 7→ [M(A)] for A ∈ Col0(λ), A ∈ Std0(λ) and A ∈ Row0(λ), respectively. Theorem F (Categorification of polynomial functors). The maps i, j, k are all isomorphisms of UZ-modules, and the following diagram commutes: Vλ0 F λ I λ (VZ) −−−−→ P (VZ) −−−−→ S (VZ)       iy yj yk

[O0(λ)] −−−−→ [F0(λ)] −−−−→ [M0(λ)]. F I −1 Moreover, setting LA = j ([L(A)]) for A ∈ Dom0(λ), the basis {LA | A ∈ Dom0(λ)} λ of P (VZ) coincides with Lusztig’s dual canonical basis.

Again, the Kazhdan-Lusztig conjecture for the Lie algebra glN (C) plays the central role in the proof of this theorem. Actually, we use the following increasingly well known reformulation of the Kazhdan-Lusztig conjecture in type A: setting KA = −1 i ([K(A)]), the basis {KA|A ∈ Col0(λ)} coincides with Lusztig’s dual canonical basis Vλ0 for the space (VZ). In particular, this implies that the decomposition numbers [V (A): L(B)] for A ∈ Std0(λ) and B ∈ Dom0(λ) can be computed in terms of certain Kazhdan-Lusztig polynomials associated to the symmetric group SN evaluated at q = 1. From a special case, one can also recover the analogous result for the Yangian Yn itself. We mention this, because it is interesting to compare the strategy followed here with that of Arakawa [A1], who also computes the decomposition matrices of the Yangian in terms of Kazhdan-Lusztig polynomials starting from the Kazhdan- Lusztig conjecture for the Lie algebra glN (C), via [AS]. There should also be a geometric approach to representation theory of shifted Yangians in the spirit of [V]. REPRESENTATIONS OF SHIFTED YANGIANS 9

As promised earlier in the introduction, let us now formulate a precise conjecture that explains how to compute the decomposition numbers [M(A): L(B)] of Verma modules for A, B ∈ Row0(λ), also in terms of Kazhdan-Lusztig polynomials associated −1 to the symmetric group SN . Setting LA = k ([L(A)]) for any A ∈ Row0(λ), we conjecture that {LA | A ∈ Row0(λ)} coincides with Lusztig’s dual canonical basis for λ the space S (VZ); see §7.5. This is a purely combinatorial reformulation in type A of the conjecture of de Vos and van Driel [VD] for arbitrary finite W -algebras, and is consistent with an idea of Premet that there should be an equivalence of categories between the category M(λ) here and a certain category N (λ) considered by Miliˇc´ıc and Soergel [MS]. Our conjecture is known to be true in the special case that the Young diagram of λ consists of a single column: in that case it is precisely the Kazhdan-Lusztig conjecture for the Lie algebra glN (C). It is also true if the Young diagram of λ has at most two rows, as can be verified by comparing the explicit construction of the simple highest weight modules in the two row case from §7.1 with the explicit description of the dual canonical basis in this case from [B, Theorem 20]. Finally, Theorem E would be an easy consequence of this conjecture. Despite the unreasonable length of this article, we seem to have only just broken the surface of this remarkably rich subject, and only in type A at that. There are a number of exciting directions for future research. In a subsequent paper [BK6], we will investigate two natural subcategories P(λ) and R(λ) of F(λ), consisting of polynomial and rational representations of W (λ), respectively. Let us make a few further, informal and in part conjectural, remarks about this forthcoming work that might help the reader to put this subject into context. L First, the category P(λ) is defined from P(λ) = d≥0 Pd(λ), where Pd(λ) is the category of all finite dimensional W (λ)-modules that are annihilated by the annihi- ⊗d lator of the tensor space Cρ ~ V . Here, V is the natural glN (C)-module and Cρ is the trivial W (λ)-module; see §3.6. Assuming at least d parts of λ are equal to l, we can prove a generalization of Schur-Weyl duality giving an equivalence of categories between the category Pd(λ) of polynomial representations of W (λ) of degree d and the cyclotomic quotients of the degenerate affine Hecke algebra Hd associated to the Pl weight i=1 Λn−qi for the affine Lie algebra glb ∞(C). In fact, one can recover the irreducible integrable highest weight representation of glb ∞(C) labelled by this level l dominant integral weight by considering the Grothendieck group of the category P(λ) and taking a natural limit n → ∞. This can be viewed as a categorification of the semi-infinite wedge construction for higher levels. Second, the category R(λ) of rational representations consists of all W (λ)-modules p M with the property that M ~ det is polynomial for some p 0. We expect, but have not yet been able to prove, that the functor F : O0(λ) → F0(λ) restricts to give an essentially surjective quotient functor F : O0(λ) → R(λ). Let us re- formulate this statement in a way which should be compared with [Ba, S]. Let P (λ) = L P (A), where P (A) is the projective cover of K(A) in the cat- A∈Std0(λ) egory O0(λ). These are precisely the self-dual PIMs in O0(λ); see [I, MSt]. Let op C(λ) = Endg(P (λ)) . Then, our statement is equivalent to saying that there is an equivalence of categories G : R(λ) → C(λ) -mof such that G ◦ F = Homg(P (λ), ?), 10 JONATHAN BRUNDAN AND ALEXANDER KLESHCHEV equality of functors from O0(λ) to C(λ) -mof. If true, this could have implications for vertex W -algebras, like in [A2]. Notation. Throughout we work over an algebraically closed field F of characteristic 0. Let ≥ denote the partial order on F defined by x ≥ y if (x − y) ∈ N, where N denotes {0, 1, 2,... } ⊂ F. We write simply gln for the Lie algebra gln(F). Acknowledgements. Part of this research was carried out during a stay by the first author at the (ex-) Institut Girard Desargues, UniversitéLyon I in Spring 2004. He would like to thank Meinolf Geck and the other members of the institute for their hospitality during this time.

2. Shifted Yangians In this preliminary chapter, we collect some basic deﬁnitions and results about shifted Yangians, most of which are taken from [BK5].

2.1. Generators and relations. By a shift matrix we mean a matrix σ = (si,j)1≤i,j≤n of non-negative integers such that

si,j + sj,k = si,k (2.1) whenever |i−j|+|j −k| = |i−k|. Note this means that s1,1 = ··· = sn,n = 0, and the matrix σ is completely determined by the upper diagonal entries s1,2, s2,3, . . . , sn−1,n and the lower diagonal entries s2,1, s3,2, . . . , sn,n−1. The shifted Yangian associated to the matrix σ is the algebra Yn(σ) over F deﬁned by generators (r) {Di | 1 ≤ i ≤ n, r > 0}, (2.2) (r) {Ei | 1 ≤ i < n, r > si,i+1}, (2.3) (r) {Fi | 1 ≤ i < n, r > si+1,i} (2.4) subject to certain relations. In order to write down these relations, let X (r) −r −1 Di(u) := Di u ∈ Yn(σ)[[u ]] (2.5) r≥0 (0) (r) where Di := 1, and then deﬁne some new elements Dei of Yn(σ) from the equation X (r) −r −1 Dei(u) = Dei u := −Di(u) . (2.6) r≥0 With this notation, the relations are as follows. (r) (s) [Di ,Dj ] = 0, (2.7) r+s−1 (r) (s) X (t) (r+s−1−t) [Ei ,Fj ] = δi,j Dei Di+1 , (2.8) t=0 REPRESENTATIONS OF SHIFTED YANGIANS 11

r−1 (r) (s) X (t) (r+s−1−t) [Di ,Ej ] = (δi,j − δi,j+1) Di Ej , (2.9) t=0 r−1 (r) (s) X (r+s−1−t) (t) [Di ,Fj ] = (δi,j+1 − δi,j) Fj Di , (2.10) t=0

(r) (s+1) (r+1) (s) (r) (s) (s) (r) [Ei ,Ei ] − [Ei ,Ei ] = Ei Ei + Ei Ei , (2.11) (r+1) (s) (r) (s+1) (r) (s) (s) (r) [Fi ,Fi ] − [Fi ,Fi ] = Fi Fi + Fi Fi , (2.12)

(r) (s+1) (r+1) (s) (r) (s) [Ei ,Ei+1 ] − [Ei ,Ei+1] = −Ei Ei+1, (2.13) (r+1) (s) (r) (s+1) (s) (r) [Fi ,Fi+1] − [Fi ,Fi+1 ] = −Fi+1Fi , (2.14)

(r) (s) [Ei ,Ej ] = 0 if |i − j| > 1, (2.15) (r) (s) [Fi ,Fj ] = 0 if |i − j| > 1, (2.16) (r) (s) (t) (s) (r) (t) [Ei , [Ei ,Ej ]] + [Ei , [Ei ,Ej ]] = 0 if |i − j| = 1, (2.17) (r) (s) (t) (s) (r) (t) [Fi , [Fi ,Fj ]] + [Fi , [Fi ,Fj ]] = 0 if |i − j| = 1, (2.18) for all meaningful r, s, t, i, j. (For example, the relation (2.13) should be understood to hold for all i = 1, . . . , n − 2, r > si,i+1 and s > si+1,i+2.) It is often helpful to view Yn(σ) as an algebra graded by the root lattice Qn associated to the Lie algebra gln. Let dn be the (abelian) Lie subalgebra of Yn(σ) spanned (1) (1) ∗ by the elements D1 ,...,Dn . Let ε1, . . . , εn be the basis for dn dual to the basis (1) (1) ∗ D1 ,...,Dn . We refer to elements of dn as weights and elements of n M ∗ Pn := Zεi ⊂ dn (2.19) i=1 as integral weights. The root lattice associated to the Lie algebra gln is then the Z-submodule Qn of Pn spanned by the simple roots εi − εi+1 for i = 1, . . . , n − 1. ∗ We have the usual dominance ordering on dn deﬁned by α ≥ β if (α − β) is a sum of simple roots. With this notation set up, the relations imply that we can deﬁne a Qn-grading M Yn(σ) = (Yn(σ))α (2.20)

α∈Qn (r) (r) (r) of the algebra Yn(σ) by declaring that the generators Di ,Ei and Fi are of degrees 0, εi − εi+1 and εi+1 − εi, respectively.

2.2. PBW theorem. For 1 ≤ i < j ≤ n and r > si,j resp. r > sj,i, we inductively (r) (r) deﬁne the higher root elements Ei,j resp. Fi,j of Yn(σ) from the formulae

(r) (r) (r) (r−sj−1,j ) (sj−1,j +1) Ei,i+1 := Ei ,Ei,j := [Ei,j−1 ,Ej−1 ], (2.21)

(r) (r) (r) (sj,j−1+1) (r−sj,j−1) Fi,i+1 := Fi ,Fi,j := [Fj−1 ,Fi,j−1 ]. (2.22) 12 JONATHAN BRUNDAN AND ALEXANDER KLESHCHEV

Introduce the canonical filtration F0Yn(σ) ⊆ F1Yn(σ) ⊆ · · · of Yn(σ) by declaring (r) (r) (r) that all Di ,Ei,j and Fi,j are of degree r, i.e. FdYn(σ) is the span of all monomials in these elements of total degree ≤ d. Then [BK5, Theorem 5.2] shows that the associated graded algebra gr Yn(σ) is free commutative on generators (r) {grr Di | 1 ≤ i ≤ n, si,i < r}, (2.23) (r) {grr Ei,j | 1 ≤ i < j ≤ n, si,j < r}, (2.24) (r) {grr Fi,j | 1 ≤ i < j ≤ n, sj,i < r}. (2.25) It follows immediately that the monomials in the elements (r) {Di | 1 ≤ i ≤ n, si,i < r}, (2.26) (r) {Ei,j | 1 ≤ i < j ≤ n, si,j < r}, (2.27) (r) {Fi,j | 1 ≤ i < j ≤ n, sj,i < r} (2.28) taken in some fixed order give a basis for the algebra Yn(σ). Moreover, letting Y(1n) + − (r) resp. Y(1n)(σ) resp. Y(1n)(σ) denote the subalgebra of Yn(σ) generated by the Di ’s (r) (r) resp. the Ei ’s resp. the Fi ’s, the monomials just in the elements (2.26) resp. (2.27) resp. (2.28) taken in some fixed order give bases for these subalgebras; see [BK5, Theorem 2.3]. These basis theorems imply in particular that multiplication defines a vector space isomorphism − + ∼ Y(1n)(σ) ⊗ Y(1n) ⊗ Y(1n)(σ) −→ Yn(σ), (2.29) giving us a triangular decomposition of the shifted Yangian. Also define the positive ] + [ − and negative Borel subalgebras Y(1n)(σ) := Y(1n)Y(1n)(σ) and Y(1n)(σ) := Y(1n)(σ)Y(1n). By the relations, these are indeed subalgebras of Yn(σ). Moreover, there are obvious surjective homomorphisms ] [ Y(1n)(σ) Y(1n),Y(1n)(σ) Y(1n) (2.30) ] [ (r) (r) with kernels K(1n)(σ) and K(1n)(σ) generated by all Ei,j and all Fi,j , respectively. We now introduce a new basis for Yn(σ), which will play a central role in this article. First, define the power series X (r) −r X (r) −r Ei,j(u) := Ei,j u ,Fi,j(u) := Fi,j u (2.31) r>si,j r>sj,i for 1 ≤ i < j ≤ n, and set Ei,i(u) = Fi,i(u) := 1 by convention. Recalling (2.5), let D(u) denote the n × n diagonal matrix with ii-entry Di(u) for 1 ≤ i ≤ n, let E(u) denote the n × n upper triangular matrix with ij-entry Ei,j(u) for 1 ≤ i ≤ j ≤ n, and let F (u) denote the n × n lower triangular matrix with ji-entry Fi,j(u) for 1 ≤ i ≤ j ≤ n. Consider the product T (u) = F (u)D(u)E(u) (2.32) REPRESENTATIONS OF SHIFTED YANGIANS 13

−1 of matrices with entries in Yn(σ)[[u ]]. The ij-entry of the matrix T (u) deﬁnes a power series min(i,j) X (r) −r X Ti,j(u) = Ti,j u := Fk,i(u)Dk(u)Ek,j(u) (2.33) r≥0 k=1 (r) (0) (r) for some new elements Ti,j ∈ FrYn(σ). Note that Ti,j = δi,j and Ti,j = 0 for 0 < r ≤ si,j.

Lemma 2.1. The associated graded algebra gr Yn(σ) is free commutative on gen- (r) erators {grr Ti,j | 1 ≤ i, j ≤ n, si,j < r}. Hence, the monomials in the elements (r) {Ti,j | 1 ≤ i, j ≤ n, si,j < r} taken in some ﬁxed order form a basis for Yn(σ). (r) Proof. Recall that Ti,j = 0 for 0 < r ≤ si,j. Given this, it is easy to see, e.g. by solving the equation (2.32) in terms of quasi-determinants as in [BK4, (5.2)–(5.4)], (r) (r) (r) that each of the elements Di ,Ei,j and Fi,j of Yn(σ) can be written as a linear (s) combination of monomials of total degree r in the elements {Ti,j | 1 ≤ i, j ≤ n, si,j < s}. Since we already know that gr Yn(σ) is free commutative on the generators (2.23)– (r) (2.25), it follows that the elements {grr Ti,j | 1 ≤ i, j ≤ n, si,j < r} also generate gr Yn(σ). Now the lemma follows by dimension considerations.

2.3. Some automorphisms. Letσ ˙ = (s ˙i,j)1≤i,j≤n be another shift matrix such that s˙i,i+1 +s ˙i+1,i = si,i+1 +si+1,i for all i = 1, . . . , n−1. Then the deﬁning relations imply that there is a unique algebra isomorphism ι : Yn(σ) → Yn(σ ˙ ) deﬁned on generators by the equations

(r) (r) (r) (r−si,i+1+s ˙i,i+1) (r) (r−si+1,i+s ˙i+1,i) ι(Di ) = Di , ι(Ei ) = Ei , ι(Fi ) = Fi . (2.34) t Another useful map is the anti-isomorphism τ : Yn(σ) → Yn(σ ) deﬁned by (r) (r) (r) (r) (r) (r) τ(Di ) = Di , τ(Ei ) = Fi , τ(Fi ) = Ei , (2.35) where σt denotes the transpose of the shift matrix σ. Note from (2.21)–(2.22) and (2.33) that (r) (r) (r) (r) (r) (r) τ(Ei,j ) = Fi,j , τ(Fi,j ) = Ei,j , τ(Ti,j ) = Tj,i . (2.36) −1 −1 Finally for any power series f(u) ∈ 1 + u F[[u ]], it is easy to check from the (r) (r) relations that there is an automorphism µf : Yn(σ) → Yn(σ) ﬁxing each Ei and Fi and mapping the power series Di(u) to the product f(u)Di(u), i.e. r (r) X (r−s) (r) (r) (r) (r) µf (Di ) = asDi , µf (Ei ) = Ei , µf (Fi ) = Fi , (2.37) s=0 P −s if f(u) = s≥0 asu . 14 JONATHAN BRUNDAN AND ALEXANDER KLESHCHEV

2.4. Parabolic generators. In this section, we recall some more complicated parabolic presentations of Yn(σ) from [BK5]. Actually the parabolic generators deﬁned here will be needed later on only in §3.7. By a shape we mean a tuple ν = (ν1, . . . , νm) of positive integers summing to n, which we think of as the shape of the standard

Levi subalgebra glν1 ⊕ · · · ⊕ glνm of gln. We say that a shape ν = (ν1, . . . , νm) is admissible (for σ) if si,j = 0 for all ν1 + ··· + νa−1 + 1 ≤ i, j ≤ ν1 + ··· + νa and a = 1, . . . , m, in which case we deﬁne

sa,b(ν) := sν1+···+νa,ν1+···+νb (2.38) for 1 ≤ a, b ≤ m. An important role is played by the minimal admissible shape (for σ), namely, the admissible shape whose length m is as small as possible. Suppose that we are given an admissible shape ν = (ν1, . . . , νm). Writing ei,j for the ij-matrix unit in the space Mr,s of r × s matrices over F, deﬁne ν X −1 Ta,b(u) := ei,j ⊗ Tν1+···+νa−1+i,ν1+···+νb−1+j(u) ∈ Mνa,νb ⊗ Yn(σ)[[u ]] (2.39) 1≤i≤νa 1≤j≤νb ν ν for each 1 ≤ a, b ≤ m. Let T (u) denote the m × m matrix with ab-entry Ta,b(u). Generalizing (2.32) (which is the special case ν = (1n) of the present deﬁnition), consider the Gauss factorization νT (u) = νF (u)νD(u)νE(u) (2.40) ν ν where D(u) is an m × m diagonal matrix with aa-entry denoted Da(u) ∈ Mνa,νa ⊗ −1 ν Yn(σ)[[u ]] for 1 ≤ a ≤ m, E(u) is an m × m upper unitriangular matrix with ν −1 ν ab-entry denoted Ea,b(u) ∈ Mνa,νb ⊗ Yn(σ)[[u ]] and F (u) is an m × m lower ν −1 unitriangular matrix with ba-entry denoted Fa,b(u) ∈ Mνb,νa ⊗ Yn(σ)[[u ]]. So, ν ν Ea,a(u) and Fa,a(u) are both the identity and min(a,b) ν X ν ν ν Ta,b(u) = Fc,a(u) Dc(u) Ec,b(u). (2.41) c=1 Also for 1 ≤ a ≤ m let ν ν −1 Dea(u) := − Da(u) , (2.42) −1 inverse computed in the algebra Mνa,νa ⊗ Yn(σ)[[u ]]. We expand ν X ν X ν (r) −r Da(u) = ei,j ⊗ Da;i,j(u) = ei,j ⊗ Da;i,ju , (2.43) 1≤i,j≤νa 1≤i,j≤νa r≥0

ν X ν X ν (r) −r Dea(u) = ei,j ⊗ Dea;i,j(u) = ei,j ⊗ Dea;i,ju , (2.44) 1≤i,j≤νa 1≤i,j≤νa r≥0

ν X ν X ν (r) −r Ea,b(u) = ei,j ⊗ Ea,b;i,j(u) = ei,j ⊗ Ea,b;i,ju , (2.45) 1≤i≤νa 1≤i≤νa 1≤j≤νb 1≤j≤νb r>sa,b(ν) REPRESENTATIONS OF SHIFTED YANGIANS 15

ν X ν X ν (r) −r Fa,b(u) = ei,j ⊗ Fa,b;i,j(u) = ei,j ⊗ Fa,b;i,ju , (2.46) 1≤i≤νb 1≤i≤νb 1≤j≤νa 1≤j≤νa r>sb,a(ν) ν ν ν ν −1 where Da;i,j(u), Dea;i,j(u), Ea,b;i,j(u) and Fa,b;i,j(u) are power series in Yn(σ)[[u ]], ν (r) ν (r) ν (r) ν (r) and Da;i,j, Dea;i,j, Ea,b;i,j and Fa,b;i,j are elements of Yn(σ). We will usually omit (r) (r) (r) (r) the superscript ν, writing simply Da;i,j, Dea;i,j, Ea,b;i,j and Fa,b;i,j, and also abbreviate (r) (r) (r) (r) Ea,a+1;i,j by Ea;i,j and Fa,a+1;i,j by Fa;i,j. Note finally that the anti-isomorphism τ from (2.35) satisfies (r) (r) (r) (r) (r) (r) τ(Da;i,j) = Da;j,i, τ(Ea,b;i,j) = Fa,b;j,i, τ(Fa,b;i,j) = Ea,b;j,i, (2.47) as follows from (2.41) and (2.36). In [BK5, §3], we proved that Yn(σ) is generated by the elements (r) {Da;i,j | a = 1, . . . , m, 1 ≤ i, j ≤ νa, r > 0}, (2.48) (r) {Ea;i,j | a = 1, . . . , m − 1, 1 ≤ i ≤ νa, 1 ≤ j ≤ νa+1, r > sa,a+1(ν)}, (2.49) (r) {Fa;i,j | a = 1, . . . , m − 1, 1 ≤ i ≤ νa+1, 1 ≤ j ≤ νa, r > sa+1,a(ν)} (2.50) subject to certain relations recorded explicitly in [BK5, (3.3)–(3.14)]. Moreover, the monomials in the elements (r) {Da;i,j | 1 ≤ a ≤ m, 1 ≤ i, j ≤ νa, sa,a(ν) < r}, (2.51) (r) {Ea,b;i,j | 1 ≤ a < b ≤ m, 1 ≤ i ≤ νa, 1 ≤ j ≤ νb, sa,b(ν) < r}, (2.52) (r) {Fa,b;i,j | 1 ≤ a < b ≤ m, 1 ≤ i ≤ νb, 1 ≤ j ≤ νa, sb,a(ν) < r} (2.53) taken in some fixed order form a basis for Yn(σ). Actually the definition of the higher (r) (r) root elements Ea,b;i,j and Fa,b;i,j given here is different from the definition given in [BK5]. The equivalence of the two definitions is verified by the following lemma.

Lemma 2.2. For 1 ≤ a sa,b(ν), we have that (r) (r−sb−1,b(ν)) (sb−1,b(ν)+1) Ea,b;i,j = [Ea,b−1;i,k ,Eb−1;k,j ] for any 1 ≤ k ≤ νb−1. Similarly, for 1 ≤ a sb,a(ν), we have that

(r) (sb,b−1(ν)+1) (r−sb,b−1(ν)) Fa,b;i,j = [Fb−1;i,k ,Fa,b−1;k,j ] for any 1 ≤ k ≤ νb−1. Proof. We just prove the statement about the E’s; the statement about the F ’s then follows on applying the anti-isomorphism τ. Proceed by downward induction on the length of the admissible shape ν = (ν1, . . . , νm). The base case m = n is the deﬁnition (2.21), so suppose m < n. Pick 1 ≤ p ≤ m and x, y > 0 such that νp = x+y, then let µ = (ν1, . . . , νp−1, x, y, νp+1, . . . , νm), an admissible shape of strictly longer 16 JONATHAN BRUNDAN AND ALEXANDER KLESHCHEV length. A matrix calculation from the deﬁnitions shows for each 1 ≤ a x; ν  µ Ea,b;i,j(u) = Ea,b+1;i,j(u) if a < p, b > p;  µ Py µ µ  Ea,b+1;i,j(u) − h=1 Ea,a+1;i,h(u) Ea+1,b+1;h,j(u) if a = p, i ≤ x;  µ  Ea+1,b+1;i−x,j(u) if a = p, i > x;  µ  Ea+1,b+1;i,j(u) if a > p. Now suppose that b > a + 1. We need to prove that

ν ν ν (sb−1,b(ν)+1) −sb−1,b(ν) Ea,b;i,j(u) = [ Ea,b−1;i,k(u), Eb−1;k,j ]u for each 1 ≤ k ≤ νb−1. The strategy is as follows: rewrite both sides of the identity we are trying to prove in terms of the µE’s and then use the induction hypothesis, which asserts that

µ µ µ (sb−1,b(µ)+1) −sb−1,b(µ) Ea,b;i,j(u) = [ Ea,b−1;i,k(u), Eb−1;k,j ]u for each 1 ≤ a < b − 1 ≤ m, 1 ≤ i ≤ µa, 1 ≤ j ≤ µb and 1 ≤ k ≤ µb−1. Most of the cases follow at once on doing this; we just discuss the more diﬃcult ones in detail below. Case one: b < p. Easy. Case two: b = p, j ≤ x. Easy. Case three: b = p, j > x. We have by induction that

µ µ µ (sb,b+1(µ)+1) −sb,b+1(µ) Ea,b+1;i,j−x(u) = [ Ea,b;i,h(u), Eb;h,j−x ]u µ (1) ν (1) for 1 ≤ h ≤ x. Noting that sb,b+1(µ) = 0 and that Eb;h,j−x = Db;h,j, this shows ν ν ν (1) that Ea,b;i,j(u) = [ Ea,b;i,h(u), Db;h,j]. Using the cases already considered and the relations, we get that

ν ν (1) ν ν (sb−1,b(ν)+1) ν (1) −sb−1,b(ν) [ Ea,b;i,h(u), Db;h,j] = [[ Ea,b−1;i,k(u), Eb−1;k,h ], Db;h,j]u

ν ν (sb−1,b(ν)+1) −sb−1,b(ν) = [ Ea,b−1;i,k(u), Eb−1;k,j ]u for any 1 ≤ k ≤ νb−1. Case four: a < p, b > p. Easy if b > p + 1 or if b = p + 1 and k > x. Now suppose that b = p + 1 and k ≤ x. We know already that

ν ν ν (sb−1,b(ν)+1) −sb−1,b(ν) Ea,b;i,j(u) = [ Ea,b−1;i,x+1(u), Eb−1;x+1,j ]u . ν (r) Using the cases already considered to express Ea,b−1;i,k as a commutator then using ν (r) ν (s) the relation [BK5, (3.11)], we have that [ Ea,b−1;i,k, Eb−1;x+1,j] = 0. Bracketing with ν (1) ν (r) ν (s) Db−1;k,x+1 and using the relations one deduces that [ Ea,b−1;i,x+1, Eb−1;x+1,j] = ν (r) ν (s) [ Ea,b−1;i,k, Eb−1;k,j]. Hence,

ν ν (sb−1,b(ν)+1) ν ν (sb−1,b(ν)+1) [ Ea,b−1;i,x+1(u), Eb−1;x+1,j ] = [ Ea,b−1;i,k(u), Eb−1;k,j ]. REPRESENTATIONS OF SHIFTED YANGIANS 17

(s (ν)+1) ν ν ν b−1,b −sb−1,b(ν) Using this we do indeed get that Ea,b;i,j(u) = [ Ea,b−1;i,k(u), Eb−1;k,j ]u . Case ﬁve: a = p, i ≤ x. The left hand side of the identity we are trying to prove is equal to y µ X µ µ Ea,b+1;i,j(u) − Ea,a+1;i,h(u) Ea+1,b+1;h,j(u). h=1 The right hand side equals y µ X µ µ µ (sb,b+1(µ)+1) −sb,b+1(µ) [ Ea,b;i,k(u) − Ea,a+1;i,h(u) Ea+1,b;h,k(u), Eb;k,j ]u . h=1 Now apply the induction hypothesis together with the fact from the relations that µ µ (sb,b+1(µ)+1) Ea,a+1;i,h(u) and Eb;k,j commute. Case six: a = p, i > x. Easy. Case seven: a > p. Easy.

We also introduce here one more family of elements of Yn(σ) needed in §3.7. Con- tinue with ν = (ν1, . . . , νm) being a ﬁxed admissible shape for σ. Recalling that ν ν Ea,a(u) and Fa,a(u) are both the identity, we deﬁne b−1 (s (ν)+1) ν ¯ ν X ν ν c,b −sc,b(ν)−1 Ea,b(u) := Ea,b(u) − Ea,c(u) Ec,b u , (2.54) c=a b−1 (s (ν)+1) ν ¯ ν X ν b,c ν −sb,c(ν)−1 Fa,b(u) := Fa,b(u) − Fc,b Fa,c(u)u , (2.55) c=a for 1 ≤ a ≤ b ≤ m. As in (2.45)–(2.46), we expand ν ¯ X ν ¯ X ν ¯(r) −r Ea,b(u) = ei,j ⊗ Ea,b;i,j(u) = ei,j ⊗ Ea,b;i,ju , (2.56) 1≤i≤νa 1≤i≤νa 1≤j≤νb 1≤j≤νb r>sa,b(ν)+1 ν ¯ X ν ¯ X ν ¯(r) −r Fa,b(u) = ei,j ⊗ Fa,b;i,j(u) = ei,j ⊗ Fa,b;i,ju , (2.57) 1≤i≤νb 1≤i≤νb 1≤j≤νa 1≤j≤νa r>sb,a(ν)+1 ν ¯ ν ¯ −1 ν ¯(r) where Ea,b;i,j(u) and Fa,b;i,j(u) are power series in Yn(σ)[[u ]], and Ea,b;i,j and ν ¯(r) Fa,b;i,j are elements of Yn(σ). We usually drop the superscript ν from this notation.

Lemma 2.3. For 1 ≤ a sa,b(ν) + 1, we have that ¯(r) (r−sb−1,b(ν)−1) (sb−1,b(ν)+2) Ea,b;i,j = [Ea,b−1;i,k ,Eb−1;k,j ] for any 1 ≤ k ≤ νb−1. Similarly, for 1 ≤ a sb,a(ν) + 1, we have that ¯(r) (sb,b−1(ν)+2) (r−sb,b−1(ν)−1) Fa,b;i,j = [Fb−1;i,k ,Fa,b−1;k,j ] for any 1 ≤ k ≤ νb−1. 18 JONATHAN BRUNDAN AND ALEXANDER KLESHCHEV

Proof. We just prove the statement about the E’s; the statement for the F ’s then follows on applying the anti-isomorphism τ. We need to prove that (s (ν)+2) ¯ b−1,b −sb−1,b(ν)−1 Ea,b;i,j(u) = [Ea,b−1;i,k(u),Eb−1;k,j ]u . Proceed by induction on b = a + 2, . . . , m. For the base case b = a + 2, we have by the relation [BK5, (3.9)] that

(sb−1,b(ν)+2) (sb−1,b(ν)+1) [Ea,b−1;i,k(u),Eb−1;k,j ] − [Ea,b−1;i,k(u),Eb−1;k,j ]u = νb−1 (sa,b−1(ν)+1) (sb−1,b(ν)+1) −sa,b−1(ν) X (sb−1,b(ν)+1) − [Ea,b−1;i,k ,Eb−1;k,j ]u − Ea,b−1;i,h(u)Eb−1;h,j . h=1 Multiplying by u−sb−1,b(ν)−1 and using Lemma 2.2, this shows that

(sb−1,b(ν)+2) −sb−1,b(ν)−1 [Ea,b−1;i,k(u),Eb−1;k,j ]u = Ea,b;i,j(u) νb−1 (sa,b(ν)+1) −sa,b(ν)−1 X (sb−1,b(ν)+1) −sb−1,b(ν)−1 − Ea,b;i,j u − Ea,b−1;i,h(u)Eb−1;h,j u . h=1

The right hand side is exactly the deﬁnition (2.54) of E¯a,b;i,j(u) in this case. Now assume that b > a + 2 and calculate using Lemma 2.2, relations [BK5, (3.9)] and [BK5, (3.11)] and the induction hypothesis:

(sb−1,b(ν)+2) −1 [Ea,b−1;i,k(u),Eb−1;k,j ]u

(sb−2,b−1(ν)+1) (sb−1,b(ν)+2) −sb−2,b−1(ν)−1 =[[Ea,b−2;i,1(u),Eb−2;1,k ],Eb−1;k,j ]u

(sb−2,b−1(ν)+1) (sb−1,b(ν)+2) −sb−2,b−1(ν)−1 =[Ea,b−2;i,1(u), [Eb−2;1,k ,Eb−1;k,j ]]u

(sb−2,b−1(ν)+2) (sb−1,b(ν)+1) −sb−2,b−1(ν)−1 =[Ea,b−2;i,1(u), [Eb−2;1,k ,Eb−1;k,j ]]u νb−1 X (sb−2,b−1(ν)+1) (sb−1,b(ν)+1) −sb−2,b−1(ν)−1 − [Ea,b−2;i,1(u),Eb−2;1,h Eb−1;h,j ]u h=1

(sb−2,b−1(ν)+2) (sb−1,b(ν)+1) −sb−2,b−1(ν)−1 =[[Ea,b−2;i,1(u),Eb−2;1,k ],Eb−1;k,j ]u νb−1 X (sb−2,b−1(ν)+1) (sb−1,b(ν)+1) −sb−2,b−1(ν)−1 − [Ea,b−2;i,1(u),Eb−2;1,h ]Eb−1;h,j u h=1 νb−1 ¯ (sb−1,b(ν)+1) X (sb−1,b(ν)+1) −1 =[Ea,b−1;i,k(u),Eb−1;k,j ] − Ea,b−1;i,h(u)Eb−1;h,j u . h=1 Multiplying both sides by u−sb−1,b(ν) and using the deﬁnition (2.54) together with Lemma 2.2 once more gives the conclusion. REPRESENTATIONS OF SHIFTED YANGIANS 19

2.5. Hopf algebra structure. In the special case that the shift matrix σ is the zero matrix, we denote Yn(σ) simply by Yn. Observe that the parabolic generators (r) D1;i,j of Yn deﬁned from (2.40) relative to the admissible shape ν = (n) are simply (r) equal to the elements Ti,j from (2.33). Hence the parabolic presentation from [BK5, (r) (3.3)–(3.14)] asserts in this case that the elements {Ti,j | 1 ≤ i, j ≤ n, r > 0} generate Yn subject only to the relations min(r,s)−1 (r) (s) X (r+s−1−t) (t) (t) (r+s−1−t) [Ti,j ,Th,k] = Ti,k Th,j − Ti,k Th,j (2.58) t=0 (0) for every 1 ≤ h, i, j, k ≤ n and r, s > 0, where Ti,j = δi,j. This is precisely the RTT presentation for the Yangian associated to the Lie algebra gln originating in the work of Faddeev, Reshetikhin and Takhtadzhyan [FRT]; see also [D] and [MNO, §1]. It is well known that the Yangian Yn is actually a Hopf algebra with comultiplication ∆ : Yn → Yn ⊗ Yn and counit ε : Yn → F deﬁned in terms of the generating function (2.33) by n X ∆(Ti,j(u)) = Ti,k(u) ⊗ Tk,j(u), (2.59) k=1 ε(Ti,j(u)) = δi,j. (2.60)

Note also that the algebra anti-automorphism τ : Yn → Yn from (2.36) is a coalgebra anti-automorphism, i.e. we have that ∆ ◦ τ = P ◦ (τ ⊗ τ) ◦ ∆ (2.61) where P denotes the permutation operator x ⊗ y 7→ y ⊗ x. It is usually diﬃcult to compute the comultiplication ∆ : Yn → Yn ⊗ Yn in terms (r) (r) (r) of the generators Di ,Ei and Fi . At least the case n = 2 can be worked out explicitly like in [M1, Deﬁnition 2.24]: we have that

∆(D1(u)) = D1(u) ⊗ D1(u) + D1(u)E1(u) ⊗ F1(u)D1(u), (2.62) X k k k ∆(D2(u)) = D2(u) ⊗ D2(u) + (−1) D2(u)E1(u) ⊗ F1(u) D2(u), (2.63) k≥1 X k k k−1 ∆(E1(u)) = 1 ⊗ E1(u) + (−1) E1(u) ⊗ De1(u)F1(u) D2(u), (2.64) k≥1 X k k−1 k ∆(F1(u)) = F1(u) ⊗ 1 + (−1) D2(u)E1(u) De1(u) ⊗ F1(u) , (2.65) k≥1 as can be checked directly from (2.59) and (2.32). The next lemma gives some further information about ∆ for n > 2; cf. [CP2, Lemma 2.1]. To formulate the lemma precisely, recall from (2.20) how Yn is viewed as a Qn-graded algebra; the elements (r) Ti,j are of degree (εi − εj) for this grading. For any s ≥ 0 and m ≥ 1 with m + s ≤ n there is an algebra embedding (r) (r) (r) (r) (r) (r) ψs : Ym ,→ Yn,Di 7→ Di+s,Ei 7→ Ei+s,Fi 7→ Fi+s. (2.66) 20 JONATHAN BRUNDAN AND ALEXANDER KLESHCHEV

(r) A different description of this map in terms of the generators Ti,j of Yn is given in [BK4, (4.2)]. The map ψs is not a Hopf algebra embedding; in particular, the maps ∆ ◦ ψs and (ψs ⊗ ψs) ◦ ∆ from Ym to Yn ⊗ Yn are definitely different if m < n.

Lemma 2.4. For any x ∈ Ym such that ψs(x) ∈ (Yn)α for some α ∈ Qn, we have that X ∆(ψs(x)) − (ψs ⊗ ψs)(∆(x)) ∈ (Yn)β ⊗ (Yn)α−β + 06=β∈Qn + Pn−1 where Qn here denotes the set of all elements i=1 ci(εi − εi+1) of the root lattice Qn such that ci ≥ 0 for all i ∈ {1, . . . , s} ∪ {m + s, . . . , n − 1}. Proof. It suffices to prove the lemma in the two special cases s = 0 and m + s = n. Consider first the case that s = 0. Then ψs : Ym ,→ Yn is just the map sending (r) (r) Ti,j ∈ Ym to Ti,j ∈ Yn for 1 ≤ i, j ≤ m and r > 0. For these elements the statement of the lemma is clear from the explicit formula for ∆ from (2.59). It follows in general + since Ym is generated by these elements and Qn is closed under addition. (r) (r) Instead suppose that m + s = n. Let Tei,j := −S(Ti,j ) where S is the antipode. (r) (r) Then by [BK4, (4.2)], ψs : Ym ,→ Yn is the map sending Tei,j ∈ Ym to Tei+s,j+s ∈ Yn for 1 ≤ i, j ≤ m, r > 0. Since (2.59) implies that n r (r) X X (t) (r−t) ∆(Tei,j ) = − Tek,j ⊗ Tei,k , k=1 t=0 the proof can now be completed as in the previous paragraph. Now we can formulate a very useful result describing the effect of ∆ on the gener- ] [ ators of Yn in general. Recall from (2.30) that K(1n)(σ) resp. K(1n)(σ) denotes the ] [ (r) two-sided ideal of the Borel subalgebra Y(1n)(σ) resp. Y(1n)(σ) generated by the Ei (r) ] resp. the Fi ; in the case σ is the zero matrix, we denote these simply by K(1n) and [ K(1n). Also define X (r) −r Hi(u) = Hi u := Dei(u)Di+1(u) (2.67) r≥0 −1 (0) for each i = 1, . . . , n − 1. Since Dei(u) = −Di(u) , we have that Hi = −1.

Theorem 2.5. The comultiplication ∆ : Yn → Yn ⊗ Yn has the following properties: (r) Pr (s) (r−s) ] [ (i) ∆(Di ) ≡ s=0 Di ⊗ Di (mod K(1n) ⊗ K(1n)); (r) (r) Pr (s) (r−s) ] 2 [ (ii) ∆(Ei ) ≡ 1 ⊗ Ei − s=1 Ei ⊗ Hi (mod (K(1n)) ⊗ K(1n)); (r) (r) Pr (r−s) (s) ] [ 2 (iii) ∆(Fi ) ≡ Fi ⊗ 1 − s=1 Hi ⊗ Fi (mod K(1n) ⊗ (K(1n)) ). Proof. This follows from Lemma 2.4, (2.62)–(2.65) and [BK5, Corollary 11.11].

Returning to the general case, there is for any shift matrix σ = (si,j)1≤i,j≤n a (r) (r) (r) canonical embedding Yn(σ) ,→ Yn such that the generators Di ,Ei and Fi of Yn(σ) from (2.2)–(2.4) map to the elements of Yn with the same name. However, the REPRESENTATIONS OF SHIFTED YANGIANS 21

(r) (r) higher root elements Ei,j and Fi,j of Yn(σ) do not in general map to the elements of (r) Yn with the same name under this embedding, and the elements Ti,j of Yn(σ) do not (r) in general map to the elements Ti,j of Yn. In particular, if σ 6= 0 we do not know a (r) full set of relations for the generators Ti,j of Yn(σ). Write σ = σ0 + σ00 where σ0 is strictly lower triangular and σ00 is strictly upper 0 00 triangular. Embedding the shifted Yangians Yn(σ), Yn(σ ) and Yn(σ ) into Yn in the canonical way, the ﬁrst part of [BK5, Theorem 11.9] asserts that the comultiplication ∆ : Yn → Yn ⊗ Yn restricts to a map 0 00 ∆ : Yn(σ) → Yn(σ ) ⊗ Yn(σ ). (2.68)

Also the restriction of the counit ε : Yn → F gives us the trivial representation ε : Yn(σ) → F (2.69) of the shifted Yangian, with ε(Di(u)) = 1 and ε(Ei(u)) = ε(Fi(u)) = 0.

2.6. The center of Yn(σ). Let us ﬁnally describe the center Z(Yn(σ)) of Yn(σ). Recalling the notation (2.5), let X (r) −r −1 Cn(u) = Cn u := D1(u)D2(u − 1) ··· Dn(u − n + 1) ∈ Yn(σ)[[u ]]. (2.70) r≥0

In the case of the Yangian Yn itself, there is a well known alternative description of the power series Cn(u) in terms of quantum determinants due to Drinfeld [D] (see also [BK4, Theorem 8.6]). To recall this, given an n×n matrix A = (ai,j)1≤i,j≤n with entries in some (not necessarily commutative) ring, set X rdet A := sgn(w)a1,w1a2,w2 ··· an,wn, (2.71)

w∈Sn X cdet A := sgn(w)aw1,1aw2,2 ··· awn,n, (2.72)

w∈Sn −1 where Sn is the symmetric group. Then, working in Yn[[u ]], we have that   T1,1(u − n + 1) T1,2(u − n + 1) ··· T1,n(u − n + 1)  . . .. .   . . . .  Cn(u) = rdet   (2.73)  Tn−1,1(u − 1) Tn−1,2(u − 1) ··· Tn−1,n(u − 1)  Tn,1(u) Tn,2(u) ··· Tn,n(u)   T1,1(u) T1,2(u − 1) ··· T1,n(u − n + 1)  . . .. .   . . . .  = cdet   . (2.74)  Tn−1,1(u) Tn−1,2(u − 1) ··· Tn−1,n(u − n + 1)  Tn,1(u) Tn,2(u − 1) ··· Tn,n(u − n + 1) In particular, in view of this alternative description, [MNO, Proposition 2.19] shows that ∆(Cn(u)) = Cn(u) ⊗ Cn(u). (2.75) 22 JONATHAN BRUNDAN AND ALEXANDER KLESHCHEV

(1) (2) Theorem 2.6. The elements Cn ,Cn ,... are algebraically independent and generate Z(Yn(σ)).

Proof. Exploiting the embedding Yn(σ) ,→ Yn, it is known by [MNO, Theorem 2.13] (1) (2) that the elements Cn ,Cn ,... are algebraically independent and generate Z(Yn) (see also [BK4, Theorem 7.2] for a slight variation on this argument). So they certainly belong to Z(Yn(σ)). The fact that Z(Yn(σ)) is no larger than Z(Yn) may be proved L by passing to the associated graded algebra gr Yn(σ) from [BK5, Theorem 2.1] and using a variation on the trick from the proof of [MNO, Theorem 2.13]. We omit the details since we give an alternative argument in Corollary 6.11 below.

Recall the automorphisms µf : Yn(σ) → Yn(σ) from (2.37). Define −1 −1 SYn(σ) := {x ∈ Yn(σ) | µf (x) = x for all f(u) ∈ 1 + u F[[u ]]}. (2.76) Like in [MNO, Proposition 2.16], one can show that multiplication defines an algebra isomorphism ∼ Z(Yn(σ)) ⊗ SYn(σ) → Yn(σ). (2.77) (r) Recalling (2.67), ordered monomials in the elements {Hi | i = 1, . . . , n − 1, r > 0}, (r) (r) {Ei,j | 1 ≤ i < j ≤ n, r > si,j} and {Fi,j | 1 ≤ i < j ≤ n, r > sj,i} form a basis for SYn(σ). 3. Finite W -algebras In this chapter we review the definition of the finite W -algebras associated to nilpotent orbits in the Lie algebra glN , then explain their connection to the shifted Yangians. Again, much of this material is taken from [BK5], though there are some important new results too.

3.1. Pyramids. By a pyramid we mean a sequence π = (q1, . . . , ql) of integers such that 0 < q1 ≤ · · · ≤ qk, qk+1 ≥ · · · ≥ ql > 0 (3.1) for some 0 ≤ k ≤ l. We visualize π by means of a diagram consisting of q1 bricks stacked in the first column, q2 bricks stacked in the second column, . . . , ql bricks stacked in the lth column, where columns are numbered 1, 2, . . . , l from left to right. The level of the pyramid π is then the number l of non-empty columns and the height is the number max(q1, . . . , ql) of non-empty rows in the diagram. For example, the pyramid π = (1, 2, 4, 3, 1) is of level 5 and height 4, and its diagram is 4 5 8 2 6 9 (3.2) 1 3 7 10 11 . As in this example, we always number the bricks of the diagram 1, 2,...,N down columns starting with the first column. Let col(i) denote the number of the column containing the entry i in the diagram. We say that the pyramid is left-justified if q1 ≥ · · · ≥ ql and right-justified if q1 ≤ · · · ≤ ql. REPRESENTATIONS OF SHIFTED YANGIANS 23

We often need to make the following two additional choices: (a) an integer n greater than or equal to the height of π; (b) an integer k with 0 ≤ k ≤ l such that q1 ≤ · · · ≤ qk, qk+1 ≥ · · · ≥ ql as in (3.1). Having made such choices, we read oﬀ various further pieces of data from π. First, numbering the rows of the diagram of π by 1, 2, . . . , n from top to bottom so that the nth row is the last row containing l bricks, let pi denote the number of bricks on the ith row. This deﬁnes the tuple (p1, . . . , pn) of row lengths, with

0 ≤ p1 ≤ · · · ≤ pn = l. (3.3) Write row(i) for the number of the row containing the entry i in the diagram of π. Next, deﬁne a shift matrix σ = (si,j)1≤i,j≤n from the equation #{c = 1, . . . , k | i > n − qc ≥ j} if i ≥ j, si,j := (3.4) #{c = k + 1, . . . , l | i ≤ n − qc < j} if i ≤ j. To make sense of this formula, we just point out that the pyramid π (hence all the other data) can easily be recovered given just this shift matrix σ and the level l, since its diagram consists of pi = l − sn,i − si,n bricks on the ith row indented sn,i columns from the left edge and si,n columns from the right edge. Finally, let

Si,j := si,j + pmin(i,j). (3.5)

3.2. Finite W -algebras. Let g denote the Lie algebra glN , equipped with the trace form (., .). Given a pyramid π = (q1, . . . , ql) with N bricks, there is a corresponding L Z-grading g = j∈ gj defined by declaring that the ij-matrix unit ei,j is of degree Z L L (col(j) − col(i)) for each 1 ≤ i, j ≤ N. Let h := g0, p := j≥0 gj and m := j<0 gj. Thus p is a standard parabolic subalgebra of g with Levi factor h ∼ gl ⊕ · · · ⊕ gl , = q1 ql and m is the opposite nilradical. Let e ∈ p denote the nilpotent matrix X e = ei,j (3.6) i,j summing over all pairs i j of adjacent entries in the diagram; for example if π is as in L (3.2) then e = e5,8 +e2,6 +e6,9 +e1,3 +e3,7 +e7,10 +e10,11. The -grading g = gj Z j∈Z is then a good grading for e ∈ g1 in the sense of [KRW, EK]. The map χ : m → F, x 7→ (x, e) is a Lie algebra homomorphism. Let Iχ denote the kernel of the associated homomorphism U(m) → F (here U(.) denotes universal enveloping algebra). We will work with the twisted action of m on U(p) defined by setting x · u := prχ([x, u]) (3.7) for x ∈ m, u ∈ U(p), where prχ : U(g) → U(p) denotes the projection along the direct sum decomposition U(g) = U(p) ⊕ U(g)Iχ. Following [BK5, §8], we define the finite W -algebra corresponding to the pyramid π to be the subalgebra m W (π) := U(p) = {u ∈ U(p) | [x, u] ∈ U(g)Iχ for all x ∈ m} (3.8) of U(p). In this form, the definition of the algebra W (π) goes back to Kostant [Ko2] and Lynch [Ly]; it is a special case of the construction due to Premet [P1] then 24 JONATHAN BRUNDAN AND ALEXANDER KLESHCHEV

Gan and Ginzburg [GG] of non-commutative filtered deformations of the coordinate algebra of the Slodowy slice associated to the nilpotent orbit containing e. To make the last statement precise, we need to introduce the Kazhdan filtration F0U(p) ⊆ F1U(p) ⊆ · · · of U(p). This can be defined simply by declaring that each matrix unit ei,j ∈ p is of degree

deg(ei,j) := (1 + col(j) − col(i)), (3.9) i.e. FdU(p) is the span of all monomials ei1,j1 ··· eir,jr in U(p) with deg(ei1,j1 ) + ··· + deg(eir,jr ) ≤ d. The associated graded algebra gr U(p) is obviously identified with the symmetric algebra S(p) viewed as a graded algebra via (3.9) once more. We get induced a filtration F0W (π) ⊆ F1W (π) ⊆ · · · of W (π), also called the Kazhdan filtration, by setting FdW (π) := W (π) ∩ FdU(p); so gr W (π) is naturally a graded ⊥ subalgebra of gr U(p) = S(p). Let cg(e) denote the centralizer of e in g and p denote the nilradical of p. Also define elements h ∈ g0 and f ∈ g−1 so that (e, h, f) is an sl2-triple in g (taking h = f = 0 in the degenerate case e = 0). By [BK5, Lemma 8.1(ii)], we have that ⊥ p = cg(e) ⊕ [p , f]. (3.10) The projection p cg(e) along this direct sum decomposition induces a homomorphism S(p) S(cg(e)). Now the precise statement is that the restriction of this ∼ homomorphism to gr W (π) is an isomorphism gr W (π) → S(cg(e)) of graded algebras; see [Ly, Theorem 2.3].

3.3. Invariants. Given a pyramid π = (q1, . . . , ql), choose integers n and k and use these choices to read oﬀ the extra data (p1, . . . , pn) and σ = (si,j)1≤i,j≤n as in §3.1. Let ρ : U(p) → F be the homomorphism with

ρ(ei,j) = δi,j(qcol(j) + qcol(j)+1 + ··· + ql − n). (3.11) The corresponding one dimensional p-module will be denoted Fρ. We stress that this depends on our fixed choice of n, as do many subsequent definitions. For 1 ≤ i, j ≤ N, introduce the shorthand col(j)−col(i) e˜i,j := (−1) ei,j + δi,j(n − qcol(i) − qcol(i)+1 − · · · − ql). (3.12) So,e ˜i,j acts as zero on the module Fρ. For 1 ≤ i, j ≤ n, 0 ≤ x ≤ n and r ≥ 1 define r (r) X X #{t=1,...,s−1 | row(jt)≤x} Ti,j;x := (−1) e˜i1,j1 ··· e˜is,js (3.13) s=1 i1,...,is j1,...,js where the second sum is over all 1 ≤ i1, . . . , is, j1, . . . , js ≤ N such that

(a) deg(ei1,j1 ) + ··· + deg(eis,js ) = r (recall (3.9)); (b) col(it) ≤ col(jt) for each t = 1, . . . , s; (c) if row(jt) > x then col(jt) < col(it+1) for each t = 1, . . . , s − 1; (d) if row(jt) ≤ x then col(jt) ≥ col(it+1) for each t = 1, . . . , s − 1; (e) row(i1) = i, row(js) = j; (f) row(jt) = row(it+1) for each t = 1, . . . , s − 1. REPRESENTATIONS OF SHIFTED YANGIANS 25

Also set  1 if i = j > x, (0)  Ti,j;x := −1 if i = j ≤ x, (3.14)  0 if i 6= j, and introduce the generating function X (r) −r −1 Ti,j;x(u) := Ti,j;xu ∈ U(p)[[u ]]. (3.15) r≥0 These remarkable elements were ﬁrst introduced in [BK5, (9.6)]. As we will explain (r) m in the next section, certain Ti,j;x in fact generate the subalgebra W (π) = U(p) of twisted m-invariants. (r) Here is a quite diﬀerent description of the elements Ti,j;0 in the spirit of [BK5, (12.6)]. If either the ith or the jth row of the diagram is empty then we have simply that Ti,j;0(u) = δi,j. Otherwise, let a ∈ {1, . . . , l} be minimal such that i > n − qa and let b ∈ {2, . . . , l + 1} be maximal such that j > n − qb−1. Using the shorthand π(r, c) for the entry (q1 + ··· + qc + r − n) in the rth row and the cth column of the diagram of π (which makes sense only if r > n − qc), we have that

Si,j m −Si,j X X Y Ti,j;0(u) = u e˜π(rt−1,ct−1),π(rt,ct−1) + δπ(rt−1,ct−1),π(rt,ct−1)u m=1 r0,...,rm t=1 c0,...,cm (3.16) where the second summation is over all rows r0, . . . , rm and columns c0, . . . , cm such that a = c0 < ··· < cm = b, r0 = i and rm = j, and max(n − qct−1, n − qct ) < rt ≤ n for each t = 1, . . . , m − 1. This identity is proved by multiplying out the parentheses and comparing with the original deﬁnition (3.13). 3.4. Finite W -algebras are quotients of shifted Yangians. Now we can formulate the main theorem from [BK5] precisely. Continue with the notation from §3.3; in particular, we have made a ﬁxed choice σ = (si,j)1≤i,j≤n of shift matrix corresponding to the pyramid π. Then, [BK5, Theorem 10.1] asserts that the elements (r) {Ti,i;i−1 | i = 1, . . . , n, r > si,i}, (3.17) (r) {Ti,i+1;i | i = 1, . . . , n − 1, r > si,i+1}, (3.18) (r) {Ti+1,i;i | i = 1, . . . , n − 1, r > si+1,1} (3.19) of U(p) from (3.13) generate the subalgebra W (π) = U(p)m. Moreover, there is a unique surjective homomorphism

κ : Yn(σ) W (π) (3.20) under which the generators (2.2)–(2.4) of Yn(σ) map to the corresponding generators (r) (r) (r) (r) (r) (r) (3.17)–(3.19) of W (π), i.e. κ(Di ) = Ti,i;i−1, κ(Ei ) = Ti,i+1;i and κ(Fi ) = Ti+1,i;i. (r) The kernel of κ is the two-sided ideal of Yn(σ) generated by the elements {D1 | r > p1}. Finally, viewing Yn(σ) as a filtered algebra via the canonical filtration and W (π) as a filtered algebra via the Kazhdan filtration, we have that κ(FdYn(σ)) = FdW (π). 26 JONATHAN BRUNDAN AND ALEXANDER KLESHCHEV

From now onwards we will abuse notation by using exactly the same notation for −1 the elements of Yn(σ) (or Yn(σ)[[u ]]) introduced in chapter 2 as for their images in W (π) (or W (π)[[u−1]]) under the map κ, relying on context to decide which we mean. So in particular we will denote the invariants (3.17)–(3.19) from now on just (r) (r) (r) by Di ,Ei and Fi . Thus, W (π) is generated by these elements subject only to the relations (2.7)–(2.18) together with the one additional relation (r) D1 = 0 for r > p1. (3.21)

More generally, given an admissible shape ν = (ν1, . . . , νm) for σ, W (π) is generated by the parabolic generators (2.48)–(2.50) subject only to the relations from [BK5, (3.3)–(3.14)] together with the one additional relation

(r) D1;i,j = 0 for 1 ≤ i, j ≤ ν1 and r > p1. (3.22) (r) These parabolic generators of W (π) are also equal to certain of the Ti,j;x’s; see [BK5, Theorem 9.3] for the precise statement here. We should also mention the special case that the pyramid π is an n × l rectangle, when the nilpotent matrix e consists of n Jordan blocks all of the same size l and the shift matrix σ is the zero matrix. In this case, the relation (3.22) implies that W (π) is the quotient of the usual Yangian Yn from §2.5 by the two-sided ideal generated (r) by {Ti,j | 1 ≤ i, j ≤ n, r > l}. Hence W (π) is isomorphic to the Yangian of level l introduced by Cherednik [C1, C2], as was ﬁrst noticed by Ragoucy and Sorba [RS]. 3.5. More automorphisms. The isomorphism type of the algebra W (π) only actually depends on the conjugacy class of the nilpotent matrix e ∈ glN , i.e. on the row lengths (p1, . . . , pn) of π, not on the pyramid π itself. To be precise, suppose thatπ ˙ is another pyramid with the same row lengths as π, and choose a shift matrix σ˙ = (s ˙i,j)1≤i,j≤n corresponding toπ ˙ . Then, viewing W (π) as a quotient of Yn(σ) and W (π ˙ ) as a quotient of Yn(σ ˙ ), the automorphism ι from (2.34) obviously factors through the quotients to induce an isomorphism ι : W (π) → W (π ˙ ). (3.23) In a similar fashion, the map τ from (2.35) induces an anti-isomorphism τ :W (π) → W (πt), (3.24) t where here π denotes the transpose pyramid (ql, . . . , q1) obtained by reversing the order of the columns of π. Here is a useful invariant deﬁnition of τ. Recalling (3.11), letρ ¯ : U(p) → F be the homomorphism with

ρ¯(ei,j) = δi,j(q1 + q2 + ··· + qcol(j) − n). (3.25) The corresponding one dimensional p-module will be denoted Fρ¯. Also let wπ ∈ SN denote the permutation which when applied to the entries of the diagram π numbered in the standard way down columns from left to right gives the numbering down columns from right to left. For example, if π is as in (3.2) then REPRESENTATIONS OF SHIFTED YANGIANS 27 wπ = (1 11)(2 9 3 10 4 5 6 7 8): 4 5 5 8 6 2 2 6 9 −→wπ 9 7 3 1 3 7 10 11 11 10 8 4 1 . τ t Let p denote the parabolic subalgebra of glN associated to the pyramid π . This is t τ t −1 related to the usual transpose p of the subalgebra p by p = wπp wπ , viewing wπ here as a permutation matrix in the usual way. Define an algebra anti-isomorphism τ : U(p) → U(pτ ) (3.26) t −1 τ mapping x ∈ p to wπx wπ + (ρ − ρ¯)(x) ∈ U(p ). In other words, the elements τ e˜i,j ∈ U(p) from (3.12) map to the analogously defined elementse ˜wπ(j),wπ(i) ∈ U(p ). Considering the form of the definition (3.13) explicitly, one deduces from this that (r) (r) τ(Ti,j;x) = Tj,i;x (3.27) for all 1 ≤ i, j ≤ n, 0 ≤ x ≤ n and r ≥ 0. Combining this with the results of §3.4, it follows that τ : U(p) → U(pτ ) maps the subalgebra W (π) of U(p) to the subalgebra W (πt) of U(pτ ), and moreover its restriction to W (π) coincides with the map (3.24). There is one more useful automorphism of W (π). For a scalar c ∈ F, let ηc : U(glN ) → U(glN ) be the algebra automorphism mapping ei,j 7→ ei,j + δi,jc for each 1 ≤ i, j ≤ N. It is obvious from the definitions in §3.2 that this leaves the subalgebra W (π) of U(glN ) invariant, hence it restricts to an algebra automorphism

ηc : W (π) → W (π) (3.28)

The following lemma gives a description of ηc in terms of the generators of W (π). Lemma 3.1. For any c ∈ F, the following equations hold: p p (i) ηc(u i Di(u)) = (u + c) i Di(u + c) for 1 ≤ i ≤ n; s s (ii) ηc(u i,i+1 Ei(u)) = (u + c) i,i+1 Ei(u + c) for 1 ≤ i < n; s s (iii) ηc(u i+1,i Fi(u)) = (u + c) i+1,i Fi(u + c) for 1 ≤ i < n. Proof. It is immediate from (3.16) that

Si,j Si,j ηc(u Ti,j;0(u)) = (u + c) Ti,j;0(u + c). We will deduce the lemma from this formula. To do so, let Tb(u) denote the n×n matrix with ij-entry Ti,j;0(u). Consider the Gauss factorization Tb(u) = Fb(u)Db(u)Eb(u) −1 where Db(u) is a diagonal matrix with ii-entry Dbi(u) ∈ U(p)[[u ]], Eb(u) is an upper −1 unitriangular matrix with ij-entry Ebi,j(u) ∈ U(p)[[u ]] and Fb(u) is a lower unitri- −1 angular matrix with ji-entry Fbi,j(u) ∈ U(p)[[u ]]. Thus, min(i,j) X Ti,j;0(u) = Fbk,i(u)Dbk(u)Ebk,j(u). k=1 28 JONATHAN BRUNDAN AND ALEXANDER KLESHCHEV

Since Si,j = si,k + pk + sk,j, it follows that min(i,j) X −1 si,k −1 pk −1 sk,j ηc(Ti,j;0(u)) = (1 + cu ) Fbk,i(u)(1 + cu ) Dbk(u)(1 + cu ) Ebk,j(u). k=1 From this equation we can read oﬀ immediately the Gauss factorization of the matrix ηc(Tb(u)), hence the matrices ηc(Db(u)), ηc(Eb(u)) and ηc(Fb(u)), to get that

pi pi ηc(u Dbi(u)) = (u + c) Dbi(u + c),

si,j si,j ηc(u Ebi,j(u)) = (u + c) Ebi,j(u + c),

sj,i sj,i ηc(u Fbi,j(u)) = (u + c) Fbi,j(u + c). The ﬁrst of these equations gives (i), since by [BK5, Corollary 9.4] we have that −1 Dbi(u) = Di(u) in U(p)[[u ]]. Similarly, (ii) and (iii) follow from the second and third equations with j = i + 1, looking just at the negative powers of u and using [BK5, Corollary 9.4] again.

The quotient map κ : Yn(σ) W (π) from (3.20) of course depends on the choice of (r) (r) the shift matrix σ = (si,j)1≤i,≤n. Hence the particular choice of generators Di ,Ei (r) and Fi in our presentation of the algebra W (π) also depends ultimately on the choice of the integers n and k made in §3.1. Using Lemma 3.1, we can explain what happens when we switch to a different choiceσ ˙ = (s ˙i,j)1≤i,j≤n˙ of shift matrix coming from different choicesn ˙ and k˙ of these integers. Assume without loss of generality that n ≥ n˙ and set c := n − n˙ . When working withσ ˙ , we are numbering the rows of π by 1, 2,..., n˙ rather than by 1, 2, . . . , n; in particular, the row lengths (p ˙1,..., p˙n˙ ) defined fromσ ˙ are related to the original row lengths (p1, . . . , pn) by the formula p˙i = pi+c. ˙ (r) ˙ (r) ˙ (r) ˙ (r) Lemma 3.2. With the above notation, the generators Di , Ei,j , Fi,j and Ti,j of W (π) defined relative to σ˙ are related to the original generators by the equations

p˙i pi+c (u + c) D˙ i(u + c) = u Di+c(u),

s˙i,j si+c,j+c (u + c) E˙ i,j(u + c) = u Ei+c,j+c(u), s˙j,i sj+c,i+c (u + c) F˙i,j(u + c) = u Fi+c,j+c(u),

S˙i,j Si+c,j+c (u + c) T˙i,j(u + c) = u Ti+c,j+c(u). Moreover, for 1 ≤ i ≤ c, we have that Di(u) = 1 and Ei,j(u) = Fi,j(u) = 0 for all j > i. ˙ (r) Proof. By the deﬁnition (3.13), the elements Ti,j;x of U(p) deﬁned relative toσ ˙ (r) ˙ (r) (r) are related to the original elements Ti,j;x by the formula ηc(Ti,j;x) = Ti+c,j+c;x+c; the automorphism ηc appears here because the elementse ˜i,j from (3.12) involve n. Given REPRESENTATIONS OF SHIFTED YANGIANS 29 this, it follows by Lemma 3.1 that

p˙i pi+c (u + c) D˙ i(u + c) = u Di+c(u),

s˙i,i+1 si+c,i+1+c (u + c) E˙ i(u + c) = u Ei+c(u),

s˙i+1,i si+1+c,i+c (u + c) F˙i(u + c) = u Fi+c(u). The remaining equations follow directly from these using (2.21)–(2.22) and (2.33).

3.6. Miura transform. Recall from the definition that W (π) is a subalgebra of L U(p), where p is the parabolic subalgebra j≥0 gj of g = glN with Levi factor h = g ∼ gl ⊕ · · · ⊕ gl . The elements {e[c] | 1 ≤ c ≤ l, 1 ≤ i, j ≤ q } defined from 0 = q1 ql i,j c [c] ei,j := eq1+···+qc−1+i,q1+···+qc−1+j form a basis for h. We will often identify U(h) with U(gl ) ⊗ · · · ⊗ U(gl ) so that e[c] is identified with 1⊗(c−1) ⊗ e ⊗ 1⊗(l−c). Following q1 ql i,j i,j [BK5, (11.5)], we define an algebra automorphism [c] [c] η : U(h) → U(h), ei,j 7→ ei,j + δi,j(qc+1 + ··· + ql). (3.29) Also let ξ : U(p) U(h) be the algebra homomorphism induced by the natural projection p h. The composite µ := η ◦ ξ : W (π) → U(h) (3.30) is called the Miura transform. By [BK5, Theorem 11.4] or [Ly, Corollary 2.3.2], µ is an injective algebra homomorphism, allowing us to view W (π) as a subalgebra of U(h). 0 00 0 00 0 Now suppose that l = l +l for non-negative integers l , l , and let π := (q1, . . . , ql0 ) 00 0 00 and π := (ql0+1, . . . , ql). We write π = π ⊗ π whenever a pyramid is cut into two in this way. Letting h0 := gl ⊕· · ·⊕gl and h00 := gl ⊕· · ·⊕gl , the Miura transform q1 ql0 ql0+1 ql allows us to view the algebras W (π0) and W (π00) as subalgebras of U(h0) and U(h00), respectively. Moreover, identifying h with h0 ⊕ h00 hence U(h) with U(h0) ⊗ U(h00), it follows from the definition [BK5, (11.2)] and injectivity of the Miura transforms that the subalgebra W (π) of U(h) is contained in the subalgebra W (π0) ⊗ W (π00) of U(h0) ⊗ U(h00). We denote the resulting inclusion homomorphism by 0 00 ∆l0,l00 : W (π) → W (π ) ⊗ W (π ). (3.31) This is precisely the comultiplication from [BK5, §11]. It is coassociative in an obvious sense; see [BK5, Lemma 11.2]. Observing in the case π consists of a single column of height t that W (π) is simply equal to U(glt), the Miura transform µ for general π may be recovered by iterating the comultiplication a total of (l − 1) times to split π into its individual columns. Let us explain the relationship between ∆l0,l00 and the comultiplication ∆ from (2.68). Letπ ˙ 0 be the right-justified pyramid with the same row lengths as π0, and let π˙ 00 be the left-justified pyramid with the same row lengths as π00. Soπ ˙ :=π ˙ 0 ⊗ π˙ 00 is a pyramid with the same row lengths as π. Fixing an integer n greater than or equal to the height of π, read off a shift matrix σ = (si,j)1≤i,j≤n from the pyramidπ ˙ by choosing the integer k in (3.4) to equal the integer l0. Finally define σ0 resp. σ00 to 30 JONATHAN BRUNDAN AND ALEXANDER KLESHCHEV be the strictly lower resp. upper triangular matrices with σ = σ0 + σ00. Then, W (π ˙ ) 0 00 is naturally a quotient of the shifted Yangian Yn(σ) and similarly W (π ˙ ) ⊗ W (π ˙ ) is 0 00 a quotient of Yn(σ ) ⊗ Yn(σ ). Composing these quotient maps with the canonical isomorphisms W (π ˙ ) →∼ W (π) and W (π ˙ 0)⊗W (π ˙ 00) →∼ W (π0)⊗W (π00) defined by (3.23), 0 00 0 00 we obtain epimorphisms Yn(σ) W (π) and Yn(σ )⊗Yn(σ ) W (π )⊗W (π ). Now the second part of [BK5, Theorem 11.9] together with [BK5, Remark 11.10] asserts that the following diagram commutes: ∆ 0 00 Yn(σ) −−−−→ Yn(σ ) ⊗ Yn(σ )     y y (3.32) ∆ 0 00 W (π) −−−−→l ,l W (π0) ⊗ W (π00). Using this diagram, the results about ∆ obtained in §2.5 easily lead to analogous 0 00 statements for the maps ∆l0,l00 : W (π) → W (π ) ⊗ W (π ) in general. For example, (2.61) implies that ∆l00,l0 ◦ τ = P ◦ τ ⊗ τ ◦ ∆l0,l00 , (3.33) equality of maps from W (π) to W ((π00)t) ⊗ W ((π0)t). This can also be seen directly from the alternative description of τ as the restriction of the map (3.26). Note finally that the trivial Yn(σ)-module from (2.69) factors through the quotient (r) (r) (r) map κ to induce a one dimensional W (π)-module on which Di ,Ei and Fi act as zero for all meaningful i and r > 0. We call this the trivial W (π)-module. Recalling (3.12)–(3.13), the trivial W (π)-module is precisely the restriction of the U(p)-module Fρ from (3.11) to the subalgebra W (π). (r) 3.7. Vanishing of higher Ti,j ’s. Continue with the notation of §3.3; in particular, we have chosen n, k and the shift matrix σ = (si,j)1≤i,j≤n as there. We wish to show (r) (r) that Ti,j ∈ W (π), i.e. the image of the element Ti,j ∈ Yn(σ) from (2.33) under the homomorphism κ : Yn(σ) → W (π), is zero whenever r > Si,j. In order to prove this, (r) we first derive a recursive formula for Ti,j as an element of U(p). Given 1 ≤ x ≤ k+1 and k ≤ y ≤ l, let πx,y denote the subpyramid (qx, qx+1, . . . , qy) of level (y − x + 1) of our fixed pyramid π = (q1, . . . , ql) of level l. Fix the choice σx,y of shift matrix for πx,y defined according to the formula (3.4) but replacing k there by (k − x + 1). Writing px,y for the parabolic associated to πx,y, define an algebra embedding Jx,y : U(px,y) → U(p) (3.34) mapping ei,j ∈ U(px,y) to eq1+···+qx−1+i,q1+···+qx−1+j − δi,j(qy+1 + ··· + ql) ∈ U(p).

In other words,e ˜i,j ∈ U(px,y) maps toe ˜q1+···+qx−1+i,q1+···+qx−1+j ∈ U(p). In the remainder of the section, we need to consider elements both of W (π) ⊆ U(p) and of W (πx,y) ⊆ U(px,y), relying on the context to discern which we mean. For instance, ¯(r) recalling the deﬁnitions from §2.4, the notation J1,l−1(Ea,b;i,j) in the following lemma ¯(r) means the image of the element Ea,b;i,j of W (π1,l−1) under the map J1,l−1. Lemma 3.3. Assume that l ≥ 2 and work relative to the minimal admissible shape ν = (ν1, . . . , νm) for σ. REPRESENTATIONS OF SHIFTED YANGIANS 31

(i) If q1 ≥ ql and k ≤ l − 1, then the following equalities hold in U(p) for all meaningful a, b, i, j, r:

νm (r) (r) X (r−1) Da;i,j = J1,l−1(Da;i,j) + δa,m J1,l−1(Dm;i,h )˜eq1+···+ql−1+h,q1+···+ql−1+j h=1 h (r−1) i + δa,m J1,l−1(Dm;i,j ), e˜q1+···+ql−1+j−ql,q1+···+ql−1+j ,  (r)  J1,l−1(Ea,b;i,j) if b < m,  (r)  J1,l−1(E¯ ) E(r) = a,m;i,j a,b;i,j + Pνm J (E(r−1) )˜e  h=1 1,l−1 a,m;i,h q1+···+ql−1+h,q1+···+ql−1+j  h (r−1) i  + J1,l−1(Ea,m;i,j), e˜q1+···+ql−1+j−ql,q1+···+ql−1+j if b = m, (r) (r) Fa,b;i,j = J1,l−1(Fa,b;i,j).

(ii) If q1 ≤ ql and k ≥ 1, then the following equalities hold in U(p) for all meaningful a, b, i, j, r:

νm (r) (r) X (r−1) h (r−1) i Da;i,j = J2,l(Da;i,j) + δa,m e˜i,hJ2,l(Dm;h,j ) + δa,m e˜i,q2+i,J2,l(Dm;i,j ) , h=1 (r) (r) Ea,b;i,j = J2,l(Ea,b;i,j),  J (F (r) ) if b < m,  2,l a,b;i,j (r)  ¯(r) F = J2,l(Fa,m;i,j) a,b;i,j h i  Pνm (r−1) (r−1)  + h=1 e˜i,hJ2,l(Fa,m;h,j) + e˜i,q2+i,J2,l(Fa,m;i,j) if b = m.

(r) Proof. (i) The ﬁrst equation involving Da;i,j and the second two equations in the case b = a+1 follow immediately from [BK5, Lemma 10.4]. The second two equations for b > a + 1 may then be deduced in exactly the same way as [BK5, Lemma 4.2]. In the diﬃcult case when b = m, one needs to use Lemma 2.3 and also the observation that h (r−sm−1,m(ν)) i J1,l−1(Ea,m−1;i,h ), e˜q1+···+ql−1+j−ql,q1+···+ql−1+j = 0 for any 1 ≤ h ≤ νm along the way. The latter fact is checked by considering the (r−sm−1,m(ν)) expansion of Ea,m−1;i,h using [BK5, Theorem 9.3] and Lemma 2.2. (ii) Argue similarly using [BK5, Lemma 10.11] instead of [BK5, Lemma 10.4], or alternatively apply the map τ from (3.26) to (i).

Lemma 3.4. Assume that l ≥ 2, 1 ≤ i, j ≤ n and r > 0. 32 JONATHAN BRUNDAN AND ALEXANDER KLESHCHEV

(i) If q1 ≥ ql and k ≤ l − 1 then

(r) (r) X (r−sh,j ) (sh,j ) Ti,j = J1,l−1(Ti,j ) − J1,l−1(Ti,h )J1,l−1(Th,j ) 1≤h≤n−ql sh,j ≤r X (r−1) + J1,l−1(Ti,h )˜eq1+···+ql+h−n,q1+···+ql+j−n n−ql

omitting the last three terms on the right hand side if j ≤ n − ql. (ii) If q1 ≤ ql and k ≥ 1 then

(r) (r) X (si,h) (r−si,h) Ti,j = J2,l(Ti,j ) − J2,l(Ti,h )J2,l(Th,j ) 1≤h≤n−q1 si,h≤r X (r−1) h (r−1) i + e˜q1+i−n,q1+h−nJ2,l(Th,j ) + e˜q1+i−n,q1+q2+i−n,J2,l(Ti,j ) , n−q1

omitting the last three terms on the right hand side if j ≤ n − q1.

Proof. (i) Let ν = (ν1, . . . , νm) be the minimal admissible shape for σ. Take 1 ≤ a, b ≤ m, 1 ≤ i ≤ νa, 1 ≤ j ≤ νb and r > 0. By deﬁnition, min(a,b) ν X Xc Tν1+···+νa−1+i,ν1+···+νb−1+j(u) = Fc,a;i,s(u)Dc;s,t(u)Ec,b;t,j(u). c=1 s,t=1 Now apply Lemma 3.3(i) to rewrite the terms on the right hand side then simplify using the deﬁnition (2.54). (ii) Similar, or apply τ to (i).

(r) Theorem 3.5. The generators Ti,j of W (π) are zero for all 1 ≤ i, j ≤ n and r > Si,j. Proof. Proceed by induction on the level l. The base case l = 1 is easy to verify directly from the definitions. For l > 1, assume that q1 ≥ ql, the argument in the case q1 ≤ ql being entirely similar. In view of Lemma 3.2 we just need to prove the result for one particular choice of the shift matrix σ, so we may assume moreover that k ≤ l − 1. Noting that Si,j − sh,j = Si,h for i, h ≤ j, the induction hypothesis implies that all the terms on the right hand side of Lemma 3.4(i) are zero if r > Si,j. (r) Hence Ti,j = 0 as required. Finally we describe some PBW bases for the algebra W (π). Recalling the definition of the Kazhdan filtration on W (π) from §3.2, [BK5, Theorem 6.2] shows that the associated graded algebra gr W (π) is free commutative on generators (r) {grr Di | 1 ≤ i ≤ n, si,i < r ≤ Si,i}, (3.35) (r) {grr Ei,j | 1 ≤ i < j ≤ n, si,j < r ≤ Si,j}, (3.36) (r) {grr Fi,j | 1 ≤ i < j ≤ n, sj,i < r ≤ Sj,i}. (3.37) REPRESENTATIONS OF SHIFTED YANGIANS 33

Hence, as in [BK5, Corollary 6.3], the monomials in the elements (r) {Di | 1 ≤ i ≤ n, si,i < r ≤ Si,i}, (3.38) (r) {Ei,j | 1 ≤ i < j ≤ n, si,j < r ≤ Si,j}, (3.39) (r) {Fi,j | 1 ≤ i < j ≤ n, sj,i < r ≤ Sj,i} (3.40) taken in some ﬁxed order give a basis for the algebra W (π). Lemma 3.6. The associated graded algebra gr W (π) is free commutative on genera- (r) tors {grr Ti,j | 1 ≤ i, j ≤ n, si,j < r ≤ Si,j}. Hence, the monomials in the elements (r) {Ti,j | 1 ≤ i, j ≤ n, si,j < r ≤ Si,j} taken in some ﬁxed order form a basis for W (π). Proof. Similar to the proof of Lemma 2.1, but using Theorem 3.5 too.

3.8. Harish-Chandra homomorphisms. Finally in this chapter we review the classical description of the center Z(U(glN )). Recalling the notation (2.71)–(2.72), define a monic polynomial N X (r) N−r ZN (u) = ZN u ∈ U(glN )[u] (3.41) r=0 by setting   e1,1 + u − N + 1 ··· e1,N−1 e1,N  . .. . .   . . . .  ZN (u) := rdet   (3.42)  eN−1,1 ··· eN−1,N−1 + u − 1 eN−1,N  eN,1 ··· eN,N−1 eN,N + u   e1,1 + u e1,2 ··· e1,N  e2,1 e2,2 + u − 1 ··· e2,N    = cdet  . . .. .  . (3.43)  . . . .  eN,1 eN,2 ··· eN,N + u − N + 1 (1) (N) Then, the coefficients ZN ,...,ZN of this polynomial are algebraically independent and generate the center Z(U(glN )). For a proof, see [CL, §2.2] where this is deduced from the classical Capelli identity or [MNO, Remark 2.11] where it is deduced from (2.73)–(2.74). So it is natural to parametrize the central characters of U(glN ) by monic polynomials in F[u] of degree N, the polynomial f(u) corresponding to the central character Z(U(glN )) → F,ZN (u) 7→ f(u). Let P denote the free abelian group M P = Zεa. (3.44) a∈F Given a monic f(u) ∈ F[u] of degree N, we associate the element X θ = caεa ∈ P (3.45) a∈F 34 JONATHAN BRUNDAN AND ALEXANDER KLESHCHEV whose coefficients {ca | a ∈ F} are defined from the factorization Y f(u) = (u + a)ca . (3.46) a∈F This defines a bijection between the set of monic polynomials of degree N and the set of elements θ ∈ P whose coefficients are non-negative integers summing to N. We will from now on always use this latter set to parametrize central characters. (1) (N) Let us compute the images of the elements ZN ,...,ZN under the Harish-Chandra homomorphism. Let dN denote the standard Cartan subalgebra of glN with basis ∗ e1,1, . . . , eN,N and let ε1, . . . , εN be the dual basis for dN . We often represent an ∗ element α ∈ dN simply as an N-tuple α = (a1, . . . , aN ) of elements of the field F, PN defined from α = i=1 aiεi. Also let bN be the standard Borel subalgebra consisting of upper triangular matrices. We will parametrize highest weight modules already ∗ in “ρ-shifted notation”: for α ∈ dN , let M(α) denote the Verma module of highest weight (α − ρ), namely, the module

M(α) := U(glN ) ⊗U(bN ) Fα−ρ (3.47) induced from the one dimensional bN -module of weight (α − ρ), where ρ in this place only means the weight −ε2 − 2ε3 − · · · − (N − 1)εN . Thus, if α = (a1, . . . , aN ), the diagonal matrix ei,i acts on the highest weight space of M(α) by the scalar ∗ (ai + i − 1). Viewing the symmetric algebra S(dN ) as an algebra of functions on dN , with the symmetric group SN acting by wei,i := ewi,wi as usual, the Harish-Chandra homomorphism ∼ SN ΨN : Z(U(glN )) → S(dN ) (3.48) may be deﬁned as the map sending z ∈ Z(U(glN )) to the unique element of S(dN ) ∗ with the property that z acts on M(α) by the scalar (ΨN (z))(α) for each α ∈ dN . Using (3.43) it is easy to see directly from this deﬁnition that

ΨN (ZN (u)) = (u + e1,1)(u + e2,2) ··· (u + eN,N ). (3.49) The coeﬃcients on the right hand side are the elementary symmetric functions. Deﬁne ∗ the content θ(α) of the weight α = (a1, . . . , aN ) ∈ dN by setting

θ(α) := εa1 + ··· + εaN ∈ P. (3.50) By (3.49), the central character of the Verma module M(α) is precisely the central character parametrized by θ(α). Now return to the setup of §3.2. The restriction of the projection prχ : U(g) → U(p) deﬁned after (3.7) deﬁnes an algebra homomorphism

ψ : Z(U(glN )) → Z(W (π)). (3.51) (1) (N) Applying this to the polynomial ZN (u) we obtain elements ψ(ZN ), . . . , ψ(ZN ) of Z(W (π)). The following lemma explains the relationship between these elements and (1) (2) the elements Cn ,Cn ,... of Z(W (π)) deﬁned by the formula (2.70).

p p pn Lemma 3.7. ψ(ZN (u)) = u 1 (u − 1) 2 ··· (u − n + 1) Cn(u). REPRESENTATIONS OF SHIFTED YANGIANS 35

Proof. Using (3.42), it is easy to see that the image of ψ(ZN (u)) under the Miura transform µ : W (π) → U(gl ) ⊗ · · · ⊗ U(gl ) from (3.30) is equal to Z (u) ⊗ · · · ⊗ q1 ql q1 p1 p2 Zql (u). So, since µ is injective, we have to check that µ(u (u − 1) ··· (u − n + pn 1) Cn(u)) also equals Zq1 (u) ⊗ · · · ⊗ Zql (u). By (2.75), we have that µ(Cn(u)) = Cn(u) ⊗ · · · ⊗ Cn(u)(l times). Therefore it just remains to observe in the special case that π consists of a single column of height t ≤ n, i.e. when W (π) = U(glt), that

(u − n + t) ··· (u − n + 2)(u − n + 1)Cn(u) = Zt(u). To see this, one reduces using Lemma 3.2 to the case t = n, when it follows as in [MNO, Remark 2.11]. We can also consider the Harish-Chandra homomorphism ∼ Sq1 ×···×Sql Ψq1 ⊗ · · · ⊗ Ψql : Z(U(h)) → S(dN ) (3.52) for the Levi subalgebra h = gl ⊕· · ·⊕gl , identifying U(h) with U(gl )⊗· · ·⊗U(gl ). q1 ql q1 ql It follows immediately from (3.49) and the explicit computation of µ(ψ(ZN (u))) made in the proof of Lemma 3.7 that the following diagram commutes:

∼ SN Z(U(glN )) −−−−→ S(dN ) ΨN     µ◦ψy y (3.53) ∼ Sq ×···×Sq Z(U(h)) −−−−−−−→ S(dN ) 1 l Ψq1 ⊗···⊗Ψql where the right hand map is the obvious inclusion. Hence the Harish-Chandra homomorphism ΨN factors through the map ψ, as has been observed in much greater generality than this by Lynch [Ly, Proposition 2.6] and Premet [P1, 6.2]. In partic- (1) (N) ular this shows that ψ is injective, so the elements ψ(ZN ), . . . , ψ(ZN ) of Z(W (π)) are actually algebraically independent.

4. Dual canonical bases The appropriate setting for the combinatorics underlying the representation theory of the algebras W (π) is provided by certain dual canonical bases for representations of the Lie algebra gl∞. In this chapter we review these matters following [B] closely. Throughout, π denotes a fixed pyramid (q1, . . . , ql) with row lengths (p1, . . . , pn), and N = p1 + ··· + pn = q1 + ··· + ql. 4.1. Tableaux. By a π-tableau we mean a filling of the boxes of the diagram of π with arbitrary elements of the ground field F. Let Tab(π) denote the set of all such π-tableaux. If π = π0 ⊗ π00 for pyramids π0 and π00 and we are given a π0-tableau A0 and a π00-tableau A00, we write A0 ⊗ A00 for the π-tableau obtained by concatenating A0 and A00. For example, 1 1 1 A = 0 3 2 = 0 3 ⊗ 2 = 0 ⊗ 3 ⊗ 2 . 4 3 1 4 3 1 4 3 1 We always number the rows of A ∈ Tab(π) by 1, . . . , n from top to bottom and the columns by 1, . . . , l from left to right, like for the diagram of π. 36 JONATHAN BRUNDAN AND ALEXANDER KLESHCHEV

We will use two different ways of reading the entries of a π-tableau A to get an N ∗ N-tuple α = (a1, . . . , aN ) ∈ F , i.e. a weight in dN . The first is the row reading denoted ρ(A) defined by reading the entries of A along rows starting with the top row (confusion with (3.11) seems unlikely), the second is the column reading denoted γ(A) defined by reading the entries of A down columns starting with the leftmost column. For example, if A is as in the previous paragraph, ρ(A) = (1, 0, 3, 2, 4, 3, 1) and γ(A) = (1, 0, 4, 3, 3, 2, 1). Define the content θ(A) of A to be the content of either of the weights ρ(A) or γ(A) in the sense of (3.50), an element of the free abelian L group P = εa. a∈F Z We say that two π-tableaux A and B are row equivalent, written A ∼ro B, if one can be obtained from the other by permuting entries within rows. The notion ∼co of column equivalence is defined similarly. Let Row(π) denote the set of all row equivalence classes of π-tableaux. We refer to elements of Row(π) as row symmetrized π-tableaux. Also define the set Col(π) of column strict π-tableaux, namely, the π- tableaux whose entries are strictly increasing up columns from bottom to top in the partial order ≥ on F defined at the end of the introduction. We stress the deliberate asymmetry of these definitions: Col(π) is a subset of Tab(π) but Row(π) is a quotient. We need a couple of variations on the usual definition of the Bruhat ordering on tableaux. First we have the ordering ≥ on the set Row(π), defined as follows. Given π-tableaux A and B, write A ↓ B if B is obtained from A by swapping an entry x in the ith row of A with an entry y in the jth row of A, and moreover we have that i < j and x > y. For example, 2 5 2 3 2 3 7 7 ↓ 7 7 ↓ 7 3 . 3 3 5 3 5 5 7 5 5

Now A ≥ B means that there exists r ≥ 1 and π-tableaux A1,...,Ar such that

A ∼ro A1 ↓ · · · ↓ Ar ∼ro B. (4.1) It is obvious that if A ≥ B then θ(A) = θ(B); conversely, if θ(A) = θ(B) then there exists C ∈ Row(π) such that A ≥ C ≤ B. Similarly, we deﬁne an ordering ≥0 on the set Col(π). Given π-tableaux A and B, write A → B if B is obtained from A by swapping an entry x in the ith column with an entry y in the jth column, such that 0 i < j and x > y. Then A ≥ B means that there exist A1,...,Ar ∈ Tab(π) such that

A ∼co A1 → · · · → Ar ∼co B. (4.2) Also define an equivalence relation k on Col(π) by declaring that A k B if B can be obtained from A by shuffling the columns in such a way that the relative position of all columns belonging to the same coset of F modulo Z remains the same. The partial order ≥0 on Col(π) induces a partial order also denoted ≥0 on k-equivalence classes. It just remains to introduce notions of dominant and of standard π-tableaux. The first of these is easy: call an element A ∈ Row(π) dominant if it has a representative belonging to Col(π) and let Dom(π) denote the set of all such dominant row symmetrized π-tableaux. The notion of a standard π-tableau is more subtle. Suppose first that π is left-justified, when its diagram is a Young diagram in the usual sense.

In that case, a π-tableau A ∈ Col(π) with entries ai,1, . . . , ai,pi in its ith row read REPRESENTATIONS OF SHIFTED YANGIANS 37 from left to right is called standard if ai,j 6> ai,k for all 1 ≤ i ≤ n and 1 ≤ j < k ≤ pi. If A has integer entries (rather than arbitrary elements of F) this is just saying that the entries of A are strictly increasing up columns from bottom to top and weakly increasing along rows from left to right, i.e. it is the usual notion of standard tableau. Lemma 4.1. Assume that π is left-justiﬁed. Then any element A ∈ Dom(π) has a representative that is standard. Proof. By deﬁnition, we can choose a representative for A that is column strict.

Let ai,1, . . . , ai,pi be the entries on the ith row of this representative read from left to right, for each i = 1, . . . , n. We need to show that we can permute entries within rows so that it becomes standard. Proceed by induction on #{(i, j, k) | 1 ≤ i ≤ n, 1 ≤ j < k ≤ pi such that ai,j > ai,k}. If this number is zero then our tableau is already standard. Otherwise we can pick 1 ≤ i ≤ n and 1 ≤ j < k ≤ pi such that ai,j > ai,k, none of ai,j+1, . . . , ai,k−1 lie in the same coset of F modulo Z as ai,j, and either i = n or ai+1,j 6> ai+1,k. Then deﬁne 1 ≤ h ≤ i to be minimal so that k ≤ ph and ar,j > ar,k for all h ≤ r ≤ i. Thus our tableau contains the following entries:

ah−1,j ≤ ah−1,k ah,j > ah,k ah+1,j > ah+1,k ...... ai,j > ai,k ai+1,j ≤ ai+1,k, where entries on the (h − 1)th and/or (i + 1)th rows should be omitted if they do not exist. Now swap the entries ah,j ↔ ah,k, ah+1,j ↔ ah+1,k, . . . ai,j ↔ ai,k and observe that the resulting tableau is still column strict. Finally by the induction hypothesis we get that the new tableau is row equivalent to a standard tableau. To define what it means for A ∈ Col(π) to be standard for more general pyramids π we need to recall the notion of row insertion; see e.g. [F, §1.1]. Suppose we are N given an N-tuple (a1, . . . , aN ) ∈ F . We decide if it is admissible, and if so construct an element of Row(π), according to the following algorithm. Start from the diagram of π with all boxes empty. Insert a1 into some box in the bottom (nth) row. Then if a2 6< a1 insert a2 into the bottom row too; else replace the entry a1 by a2 and insert a1 into the next row up instead. Continue in this way: at the ith step the pyramid π has (i − 1) boxes filled in and we need to insert the entry ai into the bottom row. If ai is 6< all of the entries in this row, simply add it to the row; else find the smallest entry b in the row that is strictly larger than ai, replace this entry b with ai, then insert b into the next row up in similar fashion. If at any stage of this process one gets more than pi entries in the ith row for some i, the algorithm terminates and the tuple (a1, . . . , aN ) is inadmissible; else, the tuple (a1, . . . , aN ) is admissible and we have successfully computed a tableau A ∈ Row(π). Now, for any pyramid π, we say that A ∈ Col(π) is standard if the tuple γ(A) obtained from the column reading of A is admissible. In that case, we define the rectification R(A) ∈ Row(π) to be the row symmetrized π-tableau computed from the tuple γ(A) by the algorithm described in the previous paragraph. Actually it is 38 JONATHAN BRUNDAN AND ALEXANDER KLESHCHEV clear from the algorithm that R(A) belongs to Dom(π), i.e. it has a representative that is column strict. Moreover, if A k B for some B ∈ Col(π), then A is standard if and only if B is standard, and R(A) = R(B). Let Std(π) denote the set of all k-equivalence classes of standard π-tableaux. The rectification map R induces a well- defined map R : Std(π) → Dom(π). (4.3) In the special case that π is left justified, it is straightforward to check that the new definition of standard tableau agrees with the one given before Lemma 4.1, and moreover in this case the map R is simply the map sending a tableau to its row equivalence class. In general, the map R always defines a bijection between the sets Std(π) and Dom(π). This follows in the left-justified case using Lemma 4.1, and then in general by a result of Lascoux and Schützenberger [LS]; see [F, §A.5] and [B, §2]. The partial orders ≤0 on Col(π) and ≤ on Row(π) give partial orders ≤0 on Std(π) and ≤ on Dom(π) too. In the case that π is left-justified, the map R : Std(π) → Dom(π) is actually an isomorphism between the partially ordered sets (Std(π), ≤0) and (Dom(π), ≤); see [B, Lemma 1]. This is definitely not the case for more general pyramids π. We have now introduced four sets Row(π), Col(π), Dom(π) and Std(π) of tableaux which will be needed later on to parametrize the various bases/modules that we will meet. One more thing: for any a ∈ F we write Rowa(π), Cola(π), Doma(π) and Stda(π) for the subsets of Row(π), Col(π), Dom(π) and Std(π) consisting just of the tableaux all of whose entries belong to the same coset of F modulo Z as a. We will often restrict our attention just to the sets Row0(π), Col0(π), Dom0(π) and Std0(π), since this covers the most important situation when all the tableaux have integer entries. In fact, most of the problems that we will meet are reduced in a straightforward fashion to this special situation. Often, we will identify an element A ∈ Row0(π) with its unique representative in Tab(π) whose entries are weakly increasing along rows from left to right.

4.2. Dual canonical bases. Now let gl∞ denote the Lie algebra of matrices with rows and columns labelled by Z, all but finitely many entries of which are zero. It is generated by the usual Chevalley generators ei, fi, i.e. the matrix units ei,i+1 and ei+1,i, together with the diagonal matrix units di = ei,i, for each i ∈ Z. The associated integral weight lattice P∞ is the free abelian group on basis {εi | i ∈ Z}, with simple roots εi−εi+1 for i ∈ Z; we will view P∞ as a subgroup of the group P from (3.44). We let UZ be the Kostant Z-form for the universal enveloping algebra U(gl∞), generated r r di di(di−1)···(di−r+1) by the divided powers ei /r!, fi /r! and the elements r = r! for all i ∈ Z, r ≥ 0. Let V be the natural gl∞-module with usual basis vi (i ∈ Z). Let VZ be the Z-submodule generated by these basis vectors, which is naturally a module for the Z-form UZ. N Consider to start with the UZ-module arising as the Nth tensor power T (VZ) N of VZ. It is a free Z-module with the monomial basis {Mα | α ∈ Z } defined from N Mα = va1 ⊗ · · · ⊗ vaN for α = (a1, . . . , aN ) ∈ Z . We also need the dual canonical N basis {Lα | α ∈ Z }. The best way to define this is to first quantize, then define Lα using a natural bar involution on the q-tensor space, then specialize to q = 1 at the REPRESENTATIONS OF SHIFTED YANGIANS 39 end. We refer to [B, §4] for the details of this construction (which is due to Lusztig [L, ch.27]); the only significant difference is that in [B] the Lie algebra gln is used in place of the Lie algebra gl∞ here. We just content ourselves with writing down an explicit formula for the expansion of Mα as a linear combination of Lβ’s in terms of the usual Kazhdan-Lusztig polynomials Px,y(q) associated to the symmetric group N SN from [KL] evaluated at q = 1. To do this, let SN act on the right on the set Z N in the natural way, and given any α ∈ Z define d(α) ∈ SN to be the unique element of minimal length with the property that α · d(α)−1 is a weakly increasing sequence. Then, by [B, §4], we have that X Mα = Pd(α)w0,d(β)w0 (1)Lβ, (4.4) N β∈Z writing w0 for the longest element of SN . We also need to consider certain tensor products of symmetric and exterior powers N of VZ. Let S (VZ) denote the Nth symmetric power of VZ, defined as a quotient N VN of T (VZ) in the usual way. Also let (VZ) denote the Nth exterior power of VZ, N viewed as the subspace of T (VZ) consisting of all skew-symmetric tensors. Recalling the fixed pyramid π, let

π p1 pn S (VZ) := S (VZ) ⊗ · · · ⊗ S (VZ), (4.5) Vπ Vq1 Vql (VZ) := (VZ) ⊗ · · · ⊗ (VZ). (4.6) π N Vπ Thus, S (VZ) is a quotient of the space T (VZ), while (VZ) is a subspace. Following [B, §5], both of these free Z-modules have two natural bases, a monomial basis and a dual canonical basis, parametrized by the sets Row0(π) and Col0(π), respectively. π First we define these two bases for the space S (VZ). Take A ∈ Row0(π), and identify A with its unique representative in Tab(π) having weakly increasing rows as indicated in the last sentence of the previous section. Define MA to be the image of N Mρ(A) and LA to be the image of Lρ(A) under the canonical quotient map T (VZ) π S (VZ), where ρ(A) is the row reading as defined in the previous section. Then, the π monomial basis for S (VZ) is {MA | A ∈ Row0(π)}, and the dual canonical basis is {LA | A ∈ Row0(π)}. Vπ Now we define the two bases for the space (VZ). For A ∈ Col0(π), let X `(A,B) NA := (−1) Mγ(B), (4.7) B∼coA where `(A, B) denotes the minimal number of transpositions of adjacent elements in the same column needed to get from A to B and γ(B) is the column reading of B as N defined in the previous section. Also let KA denote the vector Lγ(A) ∈ T (VZ). Then, Vπ N both NA and KA belong to the subspace (V ) of T (V ); see [B, §5]. Moreover, Z VπZ {NA |A ∈ Col0(π)} and {KA |A ∈ Col0(π)} are bases for (VZ), giving the monomial basis and the dual canonical basis, respectively. The following formulae, derived in [B, §5] as consequences of (4.4), express the monomial bases in terms of the dual canonical bases and certain Kazhdan-Lusztig 40 JONATHAN BRUNDAN AND ALEXANDER KLESHCHEV polynomials: X MA = Pd(ρ(A))w0,d(ρ(B))w0 (1)LB, (4.8) B∈Row0(π) ! X X `(A,C) NA = (−1) Pd(γ(C))w0,d(γ(B))w0 (1) KB, (4.9) B∈Col0(π) C∼coA for A ∈ Row0(π) and A ∈ Col0(π), respectively. π Notice that S (VZ) is a summand of the commutative algebra S(VZ) ⊗ · · · ⊗ S(VZ), that is, the tensor product of n copies of the symmetric algebra S(VZ). In particular, 0 00 if π = π ⊗ π , the multiplication in this algebra defines a UZ-module homomorphism π0 π00 π µ : S (VZ) ⊗ S (VZ) → S (VZ). (4.10) q1 Decompose π into its individual columns as π = π1 ⊗· · ·⊗πl and note that T (VZ)⊗ ql π1 πl · · · ⊗ T (VZ) = S (VZ) ⊗ · · · ⊗ S (VZ). Now we can define the canonical UZ-module homomorphism Vπ π F : (VZ) → S (VZ) (4.11) Vπ q1 ql to be the composite first of the natural inclusion (VZ) ,→ T (VZ) ⊗ · · · ⊗ T (VZ) π1 πl π then the multiplication map S (VZ) ⊗ · · · ⊗ S (VZ) → S (VZ) obtained by iterating π the map (4.10) a total of (l − 1) times. We define P (VZ) to be the image of the map F just constructed. It is a well known Z-form for the irreducible polynomial representation of gl∞ parametrized by the partition λ = (p1, . . . , pn). For any A ∈ Col0(π), define VA := F (NA). (4.12) λ By [B, Theorem 26], P (VZ) is a free Z-module with standard monomial basis given by the vectors {VA | A ∈ Std0(π)}. Moreover, for A ∈ Col0(π), we have that L if A ∈ Std (π), F (K ) = R(A) 0 (4.13) A 0 otherwise, recalling the rectification map R from (4.3). The vectors {LA | A ∈ Dom0(π)} give π another basis for the submodule P (VZ), which is the dual canonical basis of Lusztig, or Kashiwara’s upper global crystal basis. Finally, by (4.9) and (4.13), we have for any A ∈ Col0(π) that ! X X `(A,C) VA = (−1) Pd(γ(C))w0,d(γ(B))w0 (1) LR(B). (4.14) B∈Std0(π) C∼coA 4.3. Crystals. In this section, we introduce the crystals underlying the modules N Vπ π π N ˜ T (VZ), (VZ),S (VZ) and P (VZ). First, we define a crystal (Z , e˜i, fi, εi, ϕi, θ) N in the sense of Kashiwara [K2], as follows. Take α = (a1, . . . , aN ) ∈ Z and i ∈ Z. The i-signature of α is the tuple (σ1, . . . , σN ) defined from   + if aj = i, σj = − if aj = i + 1, (4.15)  0 otherwise. REPRESENTATIONS OF SHIFTED YANGIANS 41

From this the reduced i-signature is computed by successively replacing subsequences of the form −+ (possibly separated by 0’s) in the signature with 00 until no − appears to the left of a +. Let δj denote the N-tuple (0,..., 0, 1, 0,..., 0) where 1 appears in the jth place. Now deﬁne ∅ if there are no −’s in the reduced i-signature, e˜i(α) := (4.16) α − δj if the leftmost − is in position j; ∅ if there are no +’s in the reduced i-signature, f˜i(α) := (4.17) α + δj if the rightmost + is in position j;

εi(α) = the total number of −’s in the reduced i-signature; (4.18)

ϕi(α) = the total number of +’s in the reduced i-signature. (4.19) N Finally define the weight function θ : Z → P∞ to be the restriction of the map (3.50). N ˜ This completes the definition of the crystal (Z , e˜i, fi, εi, ϕi, θ). It is the N-fold tensor product of the usual crystal associated to the gl∞-module V , but for the opposite tensor product to the one used in [K2]. This crystal carries information about the N action of the Chevalley generators of UZ on the dual canonical basis {Lα | α ∈ Z } N of T (VZ), thanks to the following result of Kashiwara [K1, Proposition 5.3.1]: for N α ∈ Z , we have that X i eiLα = εi(α)Le˜i(α) + xα,βLβ (4.20) N β∈Z εi(β)<εi(α)−1 X f L = ϕ (α)L + yi L (4.21) i α i f˜i(α) α,β β N β∈Z ϕi(β)<ϕi(α)−1 i i for xα,β, yα,β ∈ Z. The right hand side of (4.20) resp. (4.21) should be interpreted as zero if εi(α) = 0 resp. ϕi(α) = 0. π Vπ There are also crystals attached to the modules S (VZ) and (VZ). To define N N them, identify Row0(π) with a subset of Z via the row reading ρ : Row0(π) ,→ Z (again we are viewing elements of Row0(π) as elements of Tab(π) with weakly in- N creasing rows), and identify Col0(π) with a subset of Z via the column reading N γ : Col0(π) ,→ Z . In this way, both Row0(π) and Col0(π) become identified with N ˜ subcrystals of the crystal (Z , e˜i, fi, εi, ϕi, θ) via these maps. This defines crystals (Row0(π), e˜i, f˜i, εi, ϕi, θ) and (Col0(π), e˜i, f˜i, εi, ϕi, θ). These crystals control the action of the Chevalley generators of UZ on the dual canonical bases {LA |A ∈ Row0(π)} and {KA | A ∈ Col0(π)}, just like in (4.20)–(4.21). First, for A ∈ Row0(π), we have that X i eiLA = εi(A)Le˜i(A) + xρ(A),ρ(B)LB, (4.22) B∈Row0(π) εi(B)<εi(A)−1 X f L = ϕ (A)L + yi L . (4.23) i A i f˜i(A) ρ(A),ρ(B) B B∈Row0(π) ϕi(B)<ϕi(A)−1 42 JONATHAN BRUNDAN AND ALEXANDER KLESHCHEV

Second, for A ∈ Col0(π), we have that X i eiKA = εi(A)Ke˜i(A) + xγ(A),γ(B)KB, (4.24) B∈Col0(π) εi(B)<εi(A)−1 X f K = ϕ (A)K + yi K . (4.25) i A i f˜i(A) γ(A),γ(B) B B∈Col0(π) ϕi(B)<ϕi(A)−1 Finally, there is a well known crystal attached to the irreducible polynomial rep- π resentation P (VZ). This has various different realizations, in terms of either the set Dom0(π) or the set Std0(π); the realization as Std0(π) when π is left-justified is the usual description from [KN]. In the first case, we note that Dom0(π) is a subcrystal of the crystal (Row0(π), e˜i, f˜i, εi, ϕi, θ), indeed it is the connected component of this crystal generated by the row equivalence class of ground-state tableau Aπ, that is, the tableau having all entries on row i equal to (1 − i). In the second case, as explained in [B, §2], Std0(π) is a subcrystal of the crystal (Col0(π), e˜i, f˜i, εi, ϕi, θ), indeed again it is the connected component of this crystal generated by the ground- state tableau Aπ. In this way, we obtain two new crystals (Dom0(π), e˜i, f˜i, εi, ϕi, θ) and (Std0(π), e˜i, f˜i, εi, ϕi, θ). The rectification map R : Std0(π) → Dom0(π) is the unique isomorphism between these crystals, and it sends the ground-state tableau Aπ to its row equivalence class. 4.4. Consequences of the Kazhdan-Lusztig conjecture. In this section, we record a representation theoretic interpretation of the dual canonical basis of the N Vπ spaces T (VZ) and (VZ), which is a well known reformulation of the Kazhdan- Lusztig conjecture for glN [BB, BrK]. Later on in the article we will formulate analo- π gous interpretations for the dual canonical bases of the spaces S (VZ) (conjecturally) π and P (VZ). Going back to the notation from §3.8, let O denote the [BGG3] category of all finitely generated glN -modules which are locally finite over bN and semisimple over dN . The basic objects in O are the Verma modules M(α) and their unique N irreducible quotients L(α) for α = (a1, . . . , aN ) ∈ F , using the ρ-shifted notation explained by (3.47). Also recall that we have parametrized the central characters of L U(gl ) by the set of elements θ of P = εa whose coefficients are non-negative N a∈F Z integers summing to N. For θ ∈ P , let O(θ) denote the full subcategory of O consisting of the objects all of whose composition factors are of central character θ, setting O(θ) = 0 by convention if the coefficients of θ are not non-negative integers summing to N. The category O has the following block decomposition: M O = O(θ). (4.26) θ∈P

We will write prθ : O → O(θ) for the natural projection functor. To be absolutely explicit, if the coeﬃcients of θ ∈ P are non-negative integers summing to N, so θ N (1) N−1 (N) corresponds to the polynomial f(u) = u + f u + ··· + f ∈ F[u] according REPRESENTATIONS OF SHIFTED YANGIANS 43 to (3.45)–(3.46), we have that for each r = 1,...,N there exists p 0 prθ(M) = v ∈ M (r) (r) p . (4.27) such that (ZN − f ) v = 0 We have already observed in §3.8 that the Verma module M(α) is of central character N θ(α). Hence, for any θ ∈ P , the modules {L(α) | α ∈ F with θ(α) = θ} form a complete set of pairwise non-isomorphic irreducibles in the category O(θ). L Recall that the integral weight lattice P∞ of gl is the subgroup εi of P . ∞ i∈Z Z Let us restrict our attention from now on to the full subcategory M O0 = O(θ) (4.28)

θ∈P∞⊂P of O corresponding just to integral central characters. The Grothendieck group [O0] N N of this category has the two natural bases {[M(α)] | α ∈ Z } and {[L(α)] | α ∈ Z }. Deﬁne a Z-module isomorphism N j : T (VZ) → [O0],Mα 7→ [M(α)]. (4.29) N Note this isomorphism sends the θ-weight space of T (VZ) isomorphically onto the block component [O(θ)] of [O0], for each θ ∈ P∞. The Kazhdan-Lusztig conjecture [KL], proved in [BB, BrK], can be formulated as follows for the special case of the Lie algebra glN . N Theorem 4.2. The map j sends the dual canonical basis element Lα of T (VZ) to the class [L(α)] of the irreducible module L(α). N Proof. In view of (4.4), it suﬃces to show for α, β ∈ Z that the composition multiplicity of L(β) in the Verma module M(α) is given by the formula

[M(α): L(β)] = Pd(α)w0,d(β)w0 (1). This is well known consequence of the Kazhdan-Lusztig conjecture combined with the translation principle for singular weights, or see [BGS, Theorem 3.11.4]. N Using (4.29) we can view the action of UZ on T (VZ) instead as an action on the Grothendieck group [O0]. The resulting actions of the Chevalley generators ei, fi of UZ on [O0] are in fact induced by some exact functors ei, fi : O0 → O0 on the category O0 itself. Like in [BK1], these functors are certain translation functors arising from ∗ tensoring with the natural glN module VN or its dual VN then projecting onto certain blocks. To be precise, for i ∈ Z, we have that M ∗ ei = prθ+(εi−εi+1) ◦ (? ⊗ VN ) ◦ prθ, (4.30) θ∈P∞ M fi = prθ−(εi−εi+1) ◦ (? ⊗ VN ) ◦ prθ. (4.31) θ∈P∞ Recall that these exact functors are both left and right adjoint to each other in a canonical way. The next lemma is a standard consequence of the tensor identity. N Lemma 4.3. For α ∈ F , the module M(α) ⊗ VN has a ﬁltration with factors M(β) N for all tuples β ∈ F obtained from the tuple α by adding 1 to one of its entries. 44 JONATHAN BRUNDAN AND ALEXANDER KLESHCHEV

∗ Similarly, the module M(α) ⊗ VN has a filtration with factors M(β) for all tuples N β ∈ F obtained from the tuple α by subtracting 1 from one of its entries. N Taking blocks and passing to the Grothendieck group, we deduce for α ∈ Z and i ∈ that Z X [eiM(α)] = [M(β)] (4.32) β N summing over all tuples β ∈ Z obtained from the tuple α by replacing an entry equal to (i + 1) by an i, and X [fiM(α)] = [M(β)] (4.33) β N summing over all tuples β ∈ Z obtained from the tuple α by replacing an entry equal to i by an (i + 1). This verifies that the maps on the Grothendieck group [O0] induced by the exact functors ei, fi really do coincide with the action of the Chevalley generators of UZ from (4.29). Here is an alternative definition of the functors ei and fi, explained in detail in PN [CR, §7.4]. Let Ω = i,j=1 ei,j ⊗ ej,i ∈ U(glN ) ⊗ U(glN ). This element centralizes the image of U(glN ) under the comultiplication ∆ : U(glN ) → U(glN ) ⊗ U(glN ). For any M ∈ O0, fiM is precisely the generalized i-eigenspace of the operator Ω acting on M ⊗ VN , for any M ∈ O0. Similarly, eiM is precisely the generalized ∗ −(N + i)-eigenspace of Ω acting on M ⊗ VN . We need to recall a little more of the setup from [CR]. Define an endomorphism x of the functor ? ⊗ VN by letting xM : M ⊗ VN → M ⊗ VN be left multiplication by Ω, for all g-modules M. Also define an endomorphism s of the functor ? ⊗ VN ⊗ VN by letting sM : M ⊗VN ⊗VN → M ⊗VN ⊗VN be the permutation m⊗vi⊗vj 7→ m⊗vj ⊗vi. By [CR, Lemma 7.21], we have that

sM ◦ (xM ⊗ idVN ) = xM⊗VN ◦ sM − idM⊗VN ⊗VN (4.34) for any g-module M, equality of maps from M ⊗ VN ⊗ VN to itself. It follows that 2 x and s restrict to well-deﬁned endomorphisms of the functors fi and fi ; we denote these restrictions by x and s too. Moreover, we have that

(s1fi ) ◦ (1fi s) ◦ (s1fi ) = (1fi s) ◦ (s1fi ) ◦ (1fi s), (4.35) 2 s = 1 2 , (4.36) fi s ◦ (1f x) = (x1f ) ◦ s − 1 2 , (4.37) i i fi 3 2 2 equality of endomorphisms of fi , fi and fi , respectively. In the language of [CR, §5.2.1], this shows that the category O0 equipped with the adjoint pair of functors 2 (fi, ei) and the endomorphisms x ∈ End(fi) and s ∈ End(fi ) is an sl2-categoriﬁcation for each i ∈ Z. This has a number of important consequences, explored in detail in [CR]. We just record one more thing here, our proof of which also depends on The- orem 4.2; see [Ku] for an independent proof. Recall for the statement the deﬁnition N ˜ of the crystal (Z , e˜i, fi, εi, ϕi, θ) from (4.16)–(4.19). N Theorem 4.4. Let α ∈ Z and i ∈ Z. REPRESENTATIONS OF SHIFTED YANGIANS 45

(i) If εi(α) = 0 then eiL(α) = 0. Otherwise, eiL(α) is an indecomposable module with irreducible socle and cosocle isomorphic to L(˜ei(α)). (ii) If ϕi(α) = 0 then fiL(α) = 0. Otherwise, fiL(α) is an indecomposable module with irreducible socle and cosocle isomorphic to L(f˜i(α)). N 0 k Proof. (i) For α ∈ Z , let εi(α) be the maximal integer k ≥ 0 such that (ei) L(α) 6= 0 0. If εi(α) > 0, then [CR, Proposition 5.23] shows that eiL(α) is an indecomposable 0 0 module with irreducible socle and cosocle isomorphic to L(˜ei(α)) for somee ˜i(α) ∈ N 0 0 0 Z . Moreover, using [CR, Lemma 4.3] too, εi(˜ei(α)) = εi(α) − 1 and all remaining 0 composition factors of eiL(α) not isomorphic to L(˜ei(α)) are of the form L(β) for N 0 0 β ∈ Z with εi(β) < εi(α) − 1. k Observe from (4.20) that εi(α) is the maximal integer k ≥ 0 such that (ei) Lα 6= 0, and assuming εi(α) > 0 we know that eiLα = εi(α)Le˜i(α) plus a linear combination of Lβ’s with εi(β) < εi(α) − 1. Applying Theorem 4.2 and comparing the preceeding 0 0 paragraph, it follows immediately that εi(α) = εi(α), in which casee ˜i(α) =e ˜i(α). This completes the proof. (ii) Similar, or follows from (i) using adjointness. It just remains to extend all of this to the parabolic case. Continuing with the fixed pyramid π = (q1, . . . , ql), recall from (3.2) that h denotes the standard Levi subalgebra gl ⊕ · · · ⊕ gl of g = gl and p is the corresponding standard parabolic q1 ql N subalgebra of g. Let O(π) denote the parabolic category O consisting of all finitely generated g-modules that are locally finite dimensional over p and semisimple over h. Note O(π) is a full subcategory of the category O. To define the basic modules N in O(π), let A ∈ Col(π) be a column strict π-tableau and let α = (a1, . . . , aN ) ∈ F denote the tuple γ(A) obtained from column reading A as in §4.1. Let Y (A) denote the usual finite dimensional irreducible h-module of highest weight α − ρ = a1ε1 + (a2 + 1)ε2 + ··· + (aN + N − 1)εN . View Y (A) as a p-module through the natural projection p h, then form the parabolic Verma module

N(A) := U(g) ⊗U(p) Y (A). (4.38) The unique irreducible quotient of N(A) is denoted K(A); by comparing highest weights we have that K(A) =∼ L(γ(A)). In this way, we obtain two natural bases {[N(A)] | A ∈ Col(π)} and {[K(A)] | A ∈ Col(π)} for the Grothendieck group [O(π)] of the category O(π). The vectors {[N(A)] | A ∈ Col0(π)} and {[K(A)] | A ∈ Col0(π)} form bases for the Grothendieck group [O0(π)] of the full subcategory O0(π) := O(π)∩O0. Moreover, the translation functors ei, fi from (4.30)–(4.31) send modules in O0(π) to modules in O0(π), hence the Grothendieck group [O0(π)] is a UZ-submodule of [O0(π)]. Also recall the deﬁnition of the crystal structure on Col0(π) from §4.3. Vπ Theorem 4.5. There is a unique UZ-module isomorphism i : (VZ) → [O0(π)] such that i(NA) = [N(A)] and i(KA) = [K(A)] for each A ∈ Col0(π). Moreover, for A ∈ Col0(π) and i ∈ Z, the following properties hold: (i) If εi(A) = 0 then eiK(A) = 0. Otherwise, eiK(A) is an indecomposable module with irreducible socle and cosocle isomorphic to K(˜ei(A)). (ii) If ϕi(A) = 0 then fiK(A) = 0. Otherwise, fiK(A) is an indecomposable module with irreducible socle and cosocle isomorphic to K(f˜i(A)). 46 JONATHAN BRUNDAN AND ALEXANDER KLESHCHEV

Vπ Proof. Deﬁne a Z-module isomorphism i : (VZ) → [O0(π)] by setting i(NA) := [N(A)] for each A ∈ Col0(π). We observe that the following diagram commutes: Vπ N (VZ) −−−−→ T (VZ)   i j (4.39) y y

[O0(π)] −−−−→ [O0] where the horizontal maps are the natural inclusions. This is checked by computing the image either way round the diagram of NA: one way round one uses the deﬁnitions (4.7) and (4.29); the other way round uses the Weyl character formula to express [Y (A)] as a linear combination of Verma modules over h, then exactness of the functor U(g)⊗U(p)? to express [N(A)] as a linear combination of [M(α)]’s. Since we already know that all of the maps apart from i are UZ-module homomorphisms, it then follows that i is too. To complete the proof of the ﬁrst statement of the theorem, it just remains to show that i(KA) = [K(A)]. This follows by Theorem 4.2 because ∼ K(A) = L(γ(A)) and KA = Lγ(A). The remaining statements (i) and (ii) follow from Theorem 4.4.

5. Highest weight theory In this chapter, we set up the usual machinery of highest weight theory for the shifted Yangian Yn(σ), exploiting its triangular decomposition. Fix throughout a shift matrix σ = (si,j)1≤i,j≤n.

5.1. Admissible modules. Recall the definition of the subalgebra dn of Yn(σ) from ∗ §2.1, and the root decomposition (2.20). Given a dn-module M and a weight α ∈ dn, the (generalized) α-weight space of M is the subspace for each i = 1, . . . , n there exists p 0 Mα := v ∈ M (1) (1) p . (5.1) such that (Di − α(Di )) v = 0 We say that M is admissible if L (a) M is the direct sum of its weight spaces, i.e. M = ∗ Mα; α∈dn (b) each Mα is finite dimensional; ∗ (c) the set of all α ∈ dn such that Mα is non-zero is contained in a finite union ∗ ∗ of sets of the form D(β) := {α ∈ dn | α ≤ β} for β ∈ dn. For example, all finite dimensional Yn(σ)-modules are automatically admissible. τ t Given an admissible Yn(σ)-module M, let M be the Yn(σ )-module equal as a vector space to τ M ∗ ∗ M := (Mα) ⊆ M (5.2) ∗ α∈dn τ t with action defined by (xf)(v) = f(τ(x)v) for each f ∈ M , v ∈ M and x ∈ Yn(σ ), t where τ : Yn(σ) → Yn(σ ) is the anti-isomorphism from (2.35). It is obvious that M τ is also admissible. Indeed, making the obvious definition on morphisms, τ can be viewed as a contravariant equivalence between the categories of admissible Yn(σ)- t and Yn(σ )-modules. REPRESENTATIONS OF SHIFTED YANGIANS 47

5.2. Gelfand-Tsetlin characters. Next, let Pn denote the set of all power series A(u) = A1(u1)A2(u2) ··· An(un) in indeterminates u1, . . . , un, with the property that −1 −1 each Ai(u) belongs to 1 + u F[[u ]]. Note that Pn is an abelian group under multiplication. For A(u) ∈ Pn, we always write Ai(u) for the ith power series defined (r) −r from the equation A(u) = A1(u1) ··· An(un) and Ai for the u -coefficient of Ai(u). The associated weight of A(u) ∈ Pn is defined by (1) (1) (1) ∗ wt A(u) := A1 ε1 + A2 ε2 + ··· + An εn ∈ dn. (5.3)

Now we form the completed group algebra Zb[Pn]. The elements of Zb[Pn] consist of formal sums S = P m [A(u)] for integers m with the property that A(u)∈Pn A(u) A(u) (a) the set {wt A(u) | A(u) ∈ supp S} is contained in a finite union of sets of the ∗ form D(β) for β ∈ dn; ∗ (b) for each α ∈ dn the set {A(u) ∈ supp S | wt A(u) = α} is finite, where supp S denotes {A(u) ∈ Pn | mA(u) 6= 0}. There is an obvious multiplication on Zb[Pn] extending the rule [A(u)][B(u)] = [A(u)B(u)]. Given an admissible Yn(σ)-module M and A(u) ∈ Pn, the corresponding Gelfand- Tsetlin subspace of M is defined by for each i = 1, . . . , n and r > 0 there exists MA(u) := v ∈ M (r) (r) p . (5.4) p 0 such that (Di − Ai ) v = 0 (r) Since the weight spaces of M are finite dimensional and the operators Di commute ∗ with each other, we have for each α ∈ dn that M Mα = MA(u). (5.5) A(u)∈Pn wt A(u)=α Hence, since M is the direct sum of its weight spaces, it is also the direct sum of its Gelfand-Tsetlin subspaces: M = L M . Now we are ready to introduce A(u)∈Pn A(u) a notion of Gelfand-Tsetlin character of an admissible Yn(σ)-module M, which is analogous to the characters of Knight [Kn] for Yangians in general and of Frenkel and Reshetikhin [FR] in the setting of quantum affine algebras: set X ch M := (dim MA(u))[A(u)]. (5.6) A(u)∈Pn

By the deﬁnition of admissibility, ch M belongs to the completed group algebra Zb[Pn]. For example, the Gelfand-Tsetlin character of the trivial Yn(σ)-module is [1]. 0 00 For the ﬁrst lemma, recall the comultiplication ∆ : Yn(σ) → Yn(σ ) ⊗ Yn(σ ) from (2.68), where σ0 resp. σ00 is the strictly lower resp. upper triangular matrix 0 00 0 such that σ = σ + σ . This allows us to view the tensor product of a Yn(σ )- 0 00 00 module M and a Yn(σ )-module M as a Yn(σ)-module. We will always denote this 0 00 “external” tensor product by M M , to avoid confusion with the usual “internal” tensor product on glN -modules which we will also exploit later on. We point out (1) (1) (1) that ∆(Di ) = Di ⊗ 1 + 1 ⊗ Di , so the α-weight space of M N is equal to P ∗ Mβ ⊗ Mα−β. β∈dn 48 JONATHAN BRUNDAN AND ALEXANDER KLESHCHEV

0 0 00 Lemma 5.1. Suppose that M is an admissible Yn(σ )-module and M is an admis- 00 0 00 sible Yn(σ )-module. Then, M M is an admissible Yn(σ)-module, and 0 00 0 00 ch(M M ) = (ch M )(ch M ). 0 00 Proof. The fact that M M is admissible is obvious. To compute its character, 0 0 order the set of weights of M as α1, α2,... so that αj > αk ⇒ j < k. Let Mj denote P M 0 . Then Theorem 2.5(i) implies that the subspace M 0 ⊗ M 00 of M 0 ⊗ M 00 1≤k≤j αk j (r) is invariant under the action of all Di . Moreover in order to compute the character 0 00 of M M , we can replace it by M 0 00 0 00 M 0 0 00 (Mj ⊗ M )/(Mj−1 ⊗ M ) = (Mj/Mj−1) ⊗ M j≥1 j≥1 with Di(u) acting as Di(u) ⊗ Di(u). The next lemma is concerned with the duality τ on admissible modules from (5.2). τ Lemma 5.2. For an admissible Yn(σ)-module M, we have that ch(M ) = ch M. (r) (r) Proof. τ(Di ) = Di .

5.3. Highest weight modules. A vector v in a Yn(σ)-module M is called a highest weight vector of type A(u) ∈ Pn if (r) (a) Ei v = 0 for all i = 1, . . . , n − 1 and r > si,i+1; (r) (r) (b) Di v = Ai v for all i = 1, . . . , n and r > 0. We call M a highest weight module of type A(u) if it is generated by such a highest weight vector. The following lemma gives an equivalent way to state these deﬁnitions (r) in terms of the elements Ti,j from (2.33).

Lemma 5.3. A vector v in a Yn(σ)-module is a highest weight vector of type A(u) (r) (r) (r) if and only if Ti,j v = 0 for all 1 ≤ i < j ≤ n and r > si,j, and Ti,i v = Ai v for all i = 1, . . . , n and r > 0. (r) Proof. By the deﬁnition (2.33), the left ideal of Yn(σ) generated by {Ei | i = (r) 1, . . . , n − 1, r > si,i+1} coincides with the left ideal generated by {Ti,j | 1 ≤ i < j ≤ (r) (r) n, r > si,j}. Moreover, Ti,i ≡ Di modulo this left ideal. In the next lemma, we write σ = σ0 + σ00 where σ0 resp. σ00 is strictly lower resp. upper triangular. 0 Lemma 5.4. Suppose v is a highest weight vector in a Yn(σ )-module M of type A(u) 00 and w is a highest weight vector in a Yn(σ )-module N of type B(u). Then v ⊗ w is a highest weight vector in the Yn(σ)-module M N of type A(u)B(u). Proof. Apply Theorem 2.5.

To construct the universal highest weight module of type A(u), let FA(u) denote (r) (r) the one dimensional Y(1n)-module on which Di acts as the scalar Ai . Inﬂating REPRESENTATIONS OF SHIFTED YANGIANS 49

] through the epimorphism Y(1n)(σ) Y(1n) from (2.30), we can view FA(u) instead as ] a Y(1n)(σ)-module. Now form the induced module

Mn(σ, A(u)) := Yn(σ) ⊗ ] FA(u). (5.7) Y(1n)(σ) This is a highest weight module of type A(u), generated by the highest weight vector m+ := 1 ⊗ 1. Clearly it is the universal such module, i.e. all other highest weight modules of this type are quotients of Mn(σ, A(u)). In the next theorem we record two natural bases for Mn(σ, A(u)).

Theorem 5.5. For any A(u) ∈ Pn, the following sets of vectors give bases for the module Mn(σ, A(u)): (i) {xm+ | x ∈ X}, where X denotes the collection of all monomials in the (r) elements {Fi,j | 1 ≤ i < j ≤ n, sj,i < r} taken in some ﬁxed order; (ii) {ym+ | y ∈ Y }, where Y denotes the collection of all monomials in the (r) elements {Tj,i | 1 ≤ i < j ≤ n, sj,i < r} taken in some ﬁxed order.

Proof. Let M := Mn(σ, A(u)). ] (i) The isomorphism (2.29) implies that Yn(σ) is a free right Y(1n)(σ)-module with basis X. Hence M has basis {xm+ | x ∈ X}. (ii) Recall the definition of the canonical filtration F0Yn(σ) ⊆ F1Yn(σ) ⊆ · · · of Yn(σ) from §2.2. In view of Lemma 2.1, it may also be defined by declaring that all (r) Ti,j are of degree r. Also introduce a filtration F0M ⊆ F1M ⊆ · · · of M by setting (d) (d) FdM := FdYn(σ)m+. Let X resp. Y denote the set of all monomials in the elements X resp. Y of total degree at most d in the canonical filtration. Applying (d) (i), one deduces at once that the set of all vectors of the form xm+ | x ∈ X form a basis for FdM. On the other hand using Lemmas 2.1 and 5.3, the vectors (d) ym+ | y ∈ Y span FdM. By dimension they must be linearly independent too. S Since M = d≥0 FdM, this implies that the vectors {ym+ | y ∈ Y } give a basis for M itself.

This implies in particular that the wt A(u)-weight space of Mn(σ, A(u)) is one dimensional, spanned by the vector m+, while all other weights are strictly lower in the dominance ordering. Given this, the usual argument shows that Mn(σ, A(u)) has a unique maximal submodule denoted rad Mn(σ, A(u)). Set

Ln(σ, A(u)) := Mn(σ, A(u))/ rad Mn(σ, A(u)). (5.8) This is the unique (up to isomorphism) irreducible highest weight module of type A(u) for the algebra Yn(σ). We also note that

dim EndYn(σ)(Ln(σ, A(u))) = 1 (5.9) for any A(u) ∈ Pn. 5.4. Classiﬁcation of admissible irreducible representations. A natural question arises at this point: the module Mn(σ, A(u)) is certainly not admissible, since all of its weight spaces other than the highest one are inﬁnite dimensional, but the irreducible quotient Ln(σ, A(u)) may well be. 50 JONATHAN BRUNDAN AND ALEXANDER KLESHCHEV

Theorem 5.6. For A(u) ∈ Pn, the irreducible Yn(σ)-module Ln(σ, A(u)) is admissible if and only if Ai(u)/Ai+1(u) is a rational function for each i = 1, . . . , n − 1.

Proof. (⇐). Suppose that each Ai(u)/Ai+1(u) is a rational function. For f(u) ∈ −1 −1 1 + u F[[u ]], the twist of Ln(σ, A(u)) by the automorphism µf from (2.37) is isomorphic to Ln(σ, f(u1 ··· un)A(u)). This allows us to reduce to the case that −1 each Ai(u) is actually a polynomial in u . Assuming this, there certainly exists p l ≥ sn,1 +s1,n such that, on setting pi := l−sn,i −si,n, u i Ai(u) is a monic polynomial in u of degree pi for each i = 1, . . . , n. Let π = (q1, . . . , ql) be the pyramid associated p to the shift matrix σ and the level l. For each i = 1, . . . , n, factorize u i Ai(u) as

(u + ai,1) ··· (u + ai,pi ) for ai,j ∈ F, and write the numbers ai,1, . . . , ai,pi into the boxes on the ith row of π from left to right. For each j = 1, . . . , l, let bj,1, . . . , bj,qj denote the entries in the jth column of the resulting π-tableau read from top to bottom. Let Mj denote the Verma module for the Lie algebra glqj of highest weight bj,1ε1+···+bj,qj εqj , as in (3.47). The tensor product M1 ··· Ml is naturally a W (π)-module, hence a Yn(σ)-module via the quotient map (3.20). Applying Lemmas 5.1 and 5.4, it is an admissible Yn(σ)-module and it contains an obvious highest weight vector of type A(u). (⇒). Assume to start with that the shift matrix σ is the zero matrix, i.e. Yn(σ) is just the usual Yangian Yn. Writing simply Ln(A(u)) instead of Ln(σ, A(u)) in this case, assume that Ln(A(u)) is admissible for some A(u) ∈ Pn. In particular, for each i = 1, . . . , n − 1, the (wt A(u) − εi + εi+1)-weight space of Ln(A(u)) is ﬁnite dimensional. Given this an argument due originally to Tarasov [T1, Theorem 1], see e.g. the proof of [M2, Proposition 3.5], shows that Ai(u)/Ai+1(u) is a rational function for each i = 1, . . . , n − 1. Assume next that σ is lower triangular, and consider the canonical embedding Yn(σ) ,→ Yn. Given A(u) ∈ Pn such that Ln(σ, A(u)) is admissible, the PBW theorem implies that the induced module Yn ⊗Yn(σ) Ln(σ, A(u)) is also admissible and contains a non-zero highest weight vector of type A(u). Hence by the preceeding paragraph Ai(u)/Ai+1(u) is a rational function for each i = 1, . . . , n − 1. Finally suppose that σ is arbitrary. Recalling the isomorphism ι from (2.34), the twist of a highest weight module by ι is again a highest weight module of the same type, and the twist of an admissible module is again admissible. So the conclusion in general follows from the lower triangular case. In view of this result, let us deﬁne Ai(u)/Ai+1(u) is a rational function Qn := A(u) ∈ Pn . (5.10) for each i = 1, . . . , n − 1

Then, Theorem 5.6 implies that the modules {Ln(σ, A(u))|A(u) ∈ Qn} give a full set of pairwise non-isomorphic admissible irreducible Yn(σ)-modules. Moreover, the construction explained in the proof shows that every admissible irreducible Yn(σ)-module can be obtained from an admissible irreducible W (π)-module via the homomorphism

κ ◦ µf : Yn(σ) W (π), (5.11) −1 −1 for some pyramid π associated to the shift matrix σ and some f(u) ∈ 1 +u F[[u ]]. REPRESENTATIONS OF SHIFTED YANGIANS 51

5.5. Composition multiplicities. The final job in this chapter is to make precise the sense in which Gelfand-Tsetlin characters characterize admissible modules. We need to be a little careful here since admissible modules need not possess a composition series. Nevertheless, given admissible Yn(σ)-modules M and L with L irreducible, we define the composition multiplicity of L in M by ∼ [M : L] := sup #{i = 1, . . . , r | Mi/Mi−1 = L} (5.12) where the supremum is taken over all finite filtrations 0 = M0 ⊂ M1 ⊂ · · · ⊂ Mr = M. By general principles, this multiplicity is additive on short exact sequences. Now we repeat some standard arguments from [K, ch. 9]. ∗ Lemma 5.7. Let M be an admissible Yn(σ)-module and α ∈ dn be a fixed weight. Then, there is a filtration 0 = M0 ⊂ M1 ⊂ · · · ⊂ Mr = M and a subset I ⊆ {1, . . . , r} such that ∼ (i) (i) (i) (i) for each i ∈ I, Mi/Mi−1 = Ln(σ, A (u)) for A (u) ∈ Qn with wt A (u) ≥ α; (ii) for each i∈ / I, (Mi/Mi−1)β = 0 for all β ≥ α. In particular, given A(u) ∈ Qn with wt A(u) ≥ α, we have that (i) [M : Ln(σ, A(u))] = #{i ∈ I | A (u) = A(u)}. Proof. Adapt the proof of [K, Lemma 9.6].

Corollary 5.8. For an admissible Yn(σ)-module M, we have that X ch M = [M : Ln(σ, A(u))] ch Ln(σ, A(u)).

A(u)∈Qn Proof. Argue using the lemma exactly as in [K, Proposition 9.7].

Theorem 5.9. Let M and N be admissible Yn(σ)-modules such that ch M = ch N. Then M and N have all the same composition multiplicities.

Proof. This follows from Corollary 5.8 once we check that the ch Ln(σ, A(u))’s are linearly independent in an appropriate sense. To be precise we need to show, given X S = mA(u) ch Ln(σ, A(u)) ∈ Zb[Pn] A(u)∈Qn for coefficients mA(u) ∈ Z satisfying the conditions from §5.2(a),(b), that S = 0 implies each mA(u) = 0. Suppose for a contradiction that mA(u) 6= 0 for some A(u). Amongst all such A(u)’s, pick one with wt A(u) maximal in the dominance ordering. But then, since ch Ln(σ, A(u)) equals [A(u)] plus a (possibly infinite) linear combination of [B(u)]’s for wt(B(u)) < wt(A(u)), the coefficient of [A(u)] in P m ch L (σ, A(u)) is non-zero, which is the desired contradiction. A(u)∈Qn A(u) n

τ ∼ t Corollary 5.10. For A(u) ∈ Qn, we have that Ln(σ, A(u)) = Ln(σ ,A(u)). t Proof. Using (2.34), it is clear that Ln(σ, A(u)) and Ln(σ ,A(u)) have the same τ t formal characters. Hence by Lemma 5.2 so do Ln(σ, A(u)) and Ln(σ ,A(u)). 52 JONATHAN BRUNDAN AND ALEXANDER KLESHCHEV

6. Verma modules Now we turn our attention to studying highest weight modules over the algebras W (π) themselves. Fix throughout the chapter a pyramid π = (q1, . . . , ql) of height ≤ n, let (p1, . . . , pn) be the tuple of row lengths, and choose a corresponding shift matrix σ = (si,j)1≤i,j≤n as usual. Notions of weights, highest weight vectors etc... are exactly as in the previous chapter, viewing W (π)-modules as Yn(σ)-modules via the quotient map κ : Yn(σ) W (π) from (3.20). 6.1. Parametrization of highest weights. Our ﬁrst task is to understand the universal highest weight module of type A(u) ∈ Pn for the algebra W (π). This module is obviously the unique largest quotient of the Yn(σ)-module Mn(σ, A(u)) from (5.7) on which the kernel of the homomorphism κ : Yn(σ) W (π) from (3.20) acts as zero. In other words, it is the W (π)-module W (π) ⊗Yn(σ) Mn(σ, A(u)). We will abuse notation and write simply m+ instead of 1 ⊗ m+ for the highest weight vector in W (π) ⊗Yn(σ) Mn(σ, A(u)).

Theorem 6.1. For A(u) ∈ Pn, W (π) ⊗Yn(σ) Mn(σ, A(u)) is non-zero if and only if p u i Ai(u) ∈ F[u] for each i = 1, . . . , n. In that case, the following sets of vectors give bases for W (π) ⊗Yn(σ) Mn(σ, A(u)): (i) {xm+ | x ∈ X}, where X denotes the collection of all monomials in the (r) elements {Fi,j | 1 ≤ i < j ≤ n, sj,i < r ≤ Sj,i} taken in some fixed order; (ii) {ym+ | y ∈ Y }, where Y denotes the collection of all monomials in the (r) elements {Tj,i | 1 ≤ i < j ≤ n, sj,i < r ≤ Sj,i} taken in some fixed order. Proof. (ii) Let us work with the following reformulation of the definition (5.7): the module Mn(σ, A(u)) is the quotient of Yn(σ) by the left ideal J generated by (r) (r) (r) the elements {Ei | i = 1, . . . , n − 1, r > si,i+1} ∪ {Di − Ai | i = 1, . . . , n, r > 0}. Equivalently, by Lemma 5.3, J is the left ideal of Yn(σ) generated by the elements (r) (r) (r) P := {Ti,j | 1 ≤ i < j ≤ n, si,j < r} ∪ {Ti,i − Ai | 1 ≤ i ≤ n, si,i < r}. (r) Also let Q := {Tj,i | 1 ≤ i < j ≤ n, sj,i < r}. Pick an ordering on P ∪ Q so that the elements of Q preceed the elements of P . Obviously all ordered monomials in the elements P ∪Q containing at least one element of P belong to J. Hence by Lemma 2.1 and Theorem 5.5(ii), the ordered monomials in the elements P ∪Q containing at least one element of P in fact form a basis for J.

Now it is clear that W (π) ⊗Yn(σ) Mn(σ, A(u)) is the quotient of W (π) by the image ¯ (r) J of J under the map κ : Yn(σ) W (π). If Ai 6= 0 for some 1 ≤ i ≤ n and r > pi, (r) (r) pi ¯ i.e. u Ai(u) ∈/ F[u], then the image of Ti,i −Ai gives us a unit in J by Theorem 3.5, pi hence W (π) ⊗Yn(σ) Mn(σ, A(u)) = 0 in this case. On the other hand, if all u Ai(u) belong to F[u], we let ¯ (r) (r) (r) P := {Ti,j | 1 ≤ i < j ≤ n, si,j < r ≤ Si,j} ∪ {Ti,i − Ai | 1 ≤ i ≤ n, si,i < r ≤ Si,i}, ¯ (r) Q := {Tj,i | 1 ≤ i < j ≤ n, sj,i < r ≤ Sj,i}. REPRESENTATIONS OF SHIFTED YANGIANS 53

Then Theorem 3.5 implies that J¯ is spanned by all ordered monomials in the elements P¯ ∪ Q¯ containing at least one element of P¯. By Lemma 3.6, these monomials are also linearly independent, hence form a basis for J¯. It follows that the image of Y gives a basis for W (π)/J¯, proving (ii). (i) This follows from (ii) by reversing the argument used to deduce (ii) from (i) in the proof of Theorem 5.5.

Now suppose that m+ is a non-zero highest weight vector in some W (π)-module

M. By Theorem 6.1, there exist elements (ai,j)1≤i≤n,1≤j≤pi of F such that p1 u D1(u)m+ = (u + a1,1)(u + a1,2) ··· (u + a1,p1 )m+, (6.1) p2 (u − 1) D2(u − 1)m+ = (u + a2,1)(u + a2,2) ··· (u + a2,p2 )m+, (6.2) . .

pn (u − n + 1) Dn(u − n + 1)m+ = (u + an,1)(u + an,2) ··· (u + an,pn )m+. (6.3)

In this way, the highest weight vector m+ deﬁnes a row symmetrized π-tableau A in the sense of §4.1, namely, the unique element of Row(π) with entries ai,1, . . . , ai,pi on its ith row. From now on, we will say simply that the highest weight vector m+ is of type A if these equations hold.

Conversely, suppose that we are given A ∈ Row(π) with entries ai,1, . . . , ai,pi on its ith row. Deﬁne the corresponding Verma module M(A) to be the universal highest weight module of type A, i.e.

M(A) := W (π) ⊗Yn(σ) Mn(σ, A(u)) (6.4) p where A(u) = A1(u1) ··· An(un) is defined from (u − i + 1) i Ai(u − i + 1) = (u + ai,1)(u + ai,2) ··· (u + ai,pi ) for each i = 1, . . . , n. Theorem 6.1 then shows that the vector m+ ∈ M(A) is a non-zero highest weight vector of type A. Moreover, M(A) is admissible and, as in §5.3, it has a unique maximal submodule denoted rad M(A). The quotient ∼ L(A) := M(A)/ rad M(A) = W (π) ⊗Yn(σ) Ln(σ, A(u)) (6.5) is the unique (up to isomorphism) irreducible highest weight module of type A. The modules {L(A) | A ∈ Row(π)} give a complete set of pairwise non-isomorphic irreducible admissible representations of the algebra W (π). An important point to make here is that although all these definitions required us at the outset to make fixed choices of n and k in order to realize W (π) as a quotient of Yn(σ), the notion of highest weight vector of type A is actually an intrinsic notion independent of these choices, as follows from Lemma 3.2. Hence the Verma module M(A) and the irreducible module L(A) also only depend (up to isomorphism) on the π-tableau A, not on the particular choices of n or k. Let us describe in detail the situation when the pyramid π consists of a single column of height n. In this case we have simply that W (π) = U(gln) according to the definition (3.8). Let A be a π-tableau with entries a1, . . . , an ∈ F read from top n to bottom and let α = (a1, . . . , an) ∈ F . A highest weight vector for W (π) of type A means a vector m+ with the properties (a) ei,jm+ = 0 for all 1 ≤ i < j ≤ n; 54 JONATHAN BRUNDAN AND ALEXANDER KLESHCHEV

(b) (ei,i − i + 1)m+ = aim+ for all i = 1, . . . , n. This is the usual deﬁnition of a highest weight vector of weight (α − ρ) = a1ε1 + (a2 + 1)ε2 + ··· + (an + n − 1)εn for the Lie algebra gln. Hence the module M(A) here coincides with the Verma module M(α) from (3.47). For another example, the trivial W (π)-module, which we deﬁned earlier as the restriction of the U(p)-module Fρ from (3.11), is isomorphic to the module L(Aπ) where Aπ is the ground state tableau from §4.3, i.e. the tableau having all entries on its ith row equal to (1 − i).

6.2. Characters of Verma modules. By the character chn M of an admissible W (π)-module M, we mean its Gelfand-Tsetlin character when viewed as a Yn(σ)- module via the homomorphism κ : Yn(σ) W (π). Thus chn M is an element of the completed group algebra Zb[Pn] from §5.2. We include the extra subscript n here to emphasize that this notion deﬁnitely does depend on the particular choice of n (though it is independent of the other choice k made when determining the shift matrix σ). Given a decomposition π = π0 ⊗ π00 with π0 of level l0 and π00 of level l00, the 0 comultiplication ∆l0,l00 from (3.31) allows us to view the tensor product of a W (π )- 0 00 00 0 00 module M and a W (π )-module M as a W (π)-module, denoted M M . Assuming 0 00 0 00 M and M are both admissible, Lemma 5.1 and (3.32) imply that M M is also admissible and 0 00 0 00 chn(M M ) = (chn M )(chn M ). (6.6) Lemma 5.4 also carries over in an obvious way to this setting. Introduce the following shorthand for some special elements of the completed group algebra Zb[Pn]: −1 xi,a := [1 + (ui + a + i − 1) ], (6.7) −1 yi,a := [1 + (a + i − 1)ui ], (6.8) for 1 ≤ i ≤ n and a ∈ F. We note that

yi,a/yi,a−k = xi,a−1xi,a−2 ··· xi,a−k (6.9) for any k ∈ N. The following theorem implies in particular that the character of any admissible W (π)-module actually belongs to the completion of the subalgebra of ±1 Zb[Pn] generated just by the elements {yi,a | i = 1, . . . , n, a ∈ F}.

Theorem 6.2. For A ∈ Row(π) with entries ai,1, . . . , ai,pi on its ith row for each i = 1, . . . , n, we have that

n pi ( n ) X Y Y Y yk,ai,j −(ci,j,k+1+···+ci,j,n) ch M(A) = y n i,ai,j −(ci,j,i+1+···+ci,j,n) y c i=1 j=1 k=i+1 k,ai,j −(ci,j,k+···+ci,j,n) where the sum is over all tuples c = (ci,j,k)1≤i

Corollary 6.3. Let A1,...,Al be the columns of any representative of A ∈ Row(π), so that A ∼ro A1 ⊗ · · · ⊗ Al. Then, chn M(A) = (chn M(A1)) × · · · × (chn M(Al)) = chn(M(A1) ··· M(Al)). Proof. This follows from the theorem on interchanging the ﬁrst two products on the right hand side. In order to derive the next corollary we need to explain an alternative way of managing the combinatorics in Theorem 6.2. Continue with A ∈ Row(π) with entries ai,1, . . . , ai,pi on its ith row as in the statement of the theorem. By a tabloid we mean an array t = (ti,j,a)1≤i≤n,1≤j≤pi,a

(a) · · · ≤ ti,j,ai,j −3 ≤ ti,j,ai,j −2 ≤ ti,j,ai,j −1; (b) ti,j,a = i for a ai,j; for each 1 ≤ i ≤ n, 1 ≤ j ≤ pi. Draw a diagram with rows parametrized by pairs (i, j) for 1 ≤ i ≤ n, 1 ≤ j ≤ pi such that the (i, j)th row consists of a strip of infinitely many boxes, one in each of the columns parametrized by the numbers . . . , ai,j − 3, ai,j − 2, ai,j − 1. Then the tabloid t can be recorded on the diagram by writing the number ti,j,a into the box in the ath column of the (i, j)th row. In this way tabloids can be thought of as fillings of the boxes of the diagram by integers from the set {1, . . . , n} so that the entries on each row are weakly increasing and all but finitely many entries on row (i, j) are equal to i.

Given a tabloid t = (ti,j,a)1≤i≤n,1≤j≤pi,a

n pi ( n ) n pi Y Y Y yk,ai,j −(ci,j,k+1+···+ci,j,n) Y Y Y yi,ai,j −(ci,j,i+1+···+ci,j,n) = xti,j,a,a, yk,a −(c +···+c ) i=1 j=1 k=i+1 i,j i,j,k i,j,n i=1 j=1 a

This may be proved by reducing ﬁrst to the case that all ai belong to the same coset of F modulo Z, then to the case that all ai are equal. After these reductions it follows from the obvious fact that there are at most N! diﬀerent N-part compositions with prescribed transpose partition.

Remark 6.5. On analyzing the proof of the corollary more carefully, one sees that this upper bound p1!(p1 + p2)! ··· (p1 + ··· + pn−1)! for the dimensions of the Gelfand- Tsetlin subspaces of M(A) is attained if and only if all entries in the first (n−1) rows of the tableau A belong to the same coset of F modulo Z. At the other extreme, all Gelfand-Tsetlin subspaces of M(A) are one dimensional if and only if all entries in the first (n − 1) rows of the tableau A belong to different cosets of F modulo Z. 6.3. The linkage principle. Our next application of Theorem 6.2 is to prove a “linkage principle” showing that the row ordering from (4.1) controls the types of composition factors that can occur in a Verma module. In the special case that π consists of a single column of height n, i.e. W (π) = U(gln), this result is [BGG2, Theorem A1]; even in this case the proof given here is quite different.

Lemma 6.6. Suppose A ↓ B. Then chn M(A) = chn M(B) + (∗) where (∗) is the character of some admissible W (π)-module. Proof. In view of Corollary 6.3, it suﬃces prove this in the special case that π consists of a single column, i.e. W (π) = U(glt) for some t > 0. But in that case it is well known that A ↓ B implies that there is an embedding M(B) ,→ M(A); see [BGG1] or [Di, Lemma 7.6.13].

Theorem 6.7. Let A, B ∈ Row(π) with entries ai,1, . . . , ai,pi and bi,1, . . . , bi,pi on their ith rows, respectively. The following are equivalent: (i) A ≥ B; (ii) [M(A): L(B)] 6= 0; REPRESENTATIONS OF SHIFTED YANGIANS 57

(iii) there exists a tuple c = (ci,j,k)1≤i

Proof. (i)⇒(ii). If A ≥ B then Lemma 6.6 implies that chn M(A) = chn M(B)+(∗), where (∗) is the character of some admissible W (π)-module. Hence [M(A): L(B)] ≥ [M(B): L(B)] = 1. (ii)⇒(iii). Suppose that [M(A): L(B)] 6= 0. The highest weight vector m+ Qn Qpi of L(B) contributes i=1 j=1 yi,bi,j to the formal character chn L(B). Hence, by Qn Qpi Corollary 5.8, we see that chn M(A) also involves i=1 j=1 yi,bi,j with non-zero coeﬃcient. In view of Theorem 6.2 this implies (iii). (iii)⇒(i). Suppose that

n pi n pi ( n ) Y Y Y Y Y yk,ai,j −(ci,j,k+1+···+ci,j,n) y = y i,bi,j i,ai,j −(ci,j,i+1+···+ci,j,n) y i=1 j=1 i=1 j=1 k=i+1 k,ai,j −(ci,j,k+···+ci,j,n) for some tuple c = (ci,j,k)1≤i

pi2 pi2 i2−1 pi Y Y Y Y yi2,ai,j yi2,bi ,j = yi2,ai ,j × . 2 2 yi ,a −c j=1 j=1 i=1 j=1 2 i,j i,j,i2

Hence there exist 1 ≤ i1 < i2, 1 ≤ j1 ≤ pi1 and 1 ≤ j2 ≤ pi2 such that ai2,j2 = ¯ ai1,j1 − ci1,j1,i2 6= ai1,j1 . Let A = (¯ai,j)1≤i≤n,1≤j≤pi be the π-tableau obtained from

A by swapping the entries ai1,j1 and ai2,j2 . Deﬁne a new tuple (¯ci,j,k)1≤i

n pi ( n ) Y Y Y yk,a¯i,j −(¯ci,j,k+1+···+¯ci,j,n) y = i,a¯i,j −(¯ci,j,i+1+···+¯ci,j,n) y i=1 j=1 k=i+1 k,a¯i,j −(¯ci,j,k+···+¯ci,j,n)

n pi ( n ) n pi Y Y Y yk,ai,j −(ci,j,k+1+···+ci,j,n) Y Y y = y . i,ai,j −(ci,j,i+1+···+ci,j,n) y i,bi,j i=1 j=1 k=i+1 k,ai,j −(ci,j,k+···+ci,j,n) i=1 j=1 P P Since c¯i,j,k < ci,j,k we deduce by induction that A¯ ≥ B. Since A ↓ A¯ this completes the proof.

Corollary 6.8. For A ∈ Row(π) with entries ai,1, . . . , ai,pi on its ith row, the following are equivalent: (i) M(A) is irreducible; (ii) A is minimal with respect to the ordering ≥;

(iii) ai1,j1 6> ai2,j2 for every 1 ≤ i1 < i2 ≤ n, 1 ≤ j1 ≤ pi1 and 1 ≤ j2 ≤ pi2 . 58 JONATHAN BRUNDAN AND ALEXANDER KLESHCHEV

Moreover, assuming (i)–(iii) hold, let A1,...,Al be the columns of any representative of A read from left to right, so that A ∼ro A1 ⊗ · · · ⊗ Al. Then, we have that ∼ M(A) = M(A1) ··· M(Al). Proof. The equivalence of (i) and (ii) follows from Theorem 6.7. The equivalence of (ii) and (iii) is clear from the deﬁnition of the Bruhat ordering. The ﬁnal statement follows from Corollary 6.3 and Theorem 5.9.

6.4. The center of W (π). Our final application of Theorem 6.2 is to prove that the (1) (N) center Z(W (π)) is a polynomial algebra on generators ψ(ZN ), . . . , ψ(ZN ), notation as in §3.8. In the case that π is an n × l rectangle, when W (π) is the Yangian of level l, this result is due to Cherednik [C1, C2]; see also [M3, Corollary 4.1]. For the first lemma, recall the definition of the Miura transform µ : W (π) ,→ U(h) from (3.30). Lemma 6.9. µ(Z(W (π))) ⊆ Z(U(h)). Proof. Take z ∈ Z(W (π)) and u ∈ U(h). We need to show that [µ(z), u] = 0. This follows by [Di, Theorem 8.4.4] if we can show that [µ(z), u] annihilates a generic Verma module for h. Corollary 6.8 shows that a generic Verma module for h is irreducible when viewed as a W (π)-module via µ. Hence µ(z) acts on it as a scalar by (5.9). So certainly [µ(z), u] acts as zero.

Theorem 6.10. The map ψ : Z(U(glN )) → Z(W (π)) from (3.51) is an isomorphism. (1) (N) Hence, the elements ψ(ZN ), . . . , ψ(ZN ) are algebraically independent and generate the center Z(W (π)). Proof. In view of Lemma 6.9 and the commutativity of the diagram (3.53), we just need to show that the image of an element z ∈ Z(W (π)) under the map (Ψq1 ⊗ · · · ⊗

Ψql ) ◦ µ is invariant under the action of all of SN . This follows from Theorem 5.9,

Corollary 6.3 and the deﬁnition of the map Ψq1 ⊗ · · · ⊗ Ψql . We remark that there is now a quite diﬀerent proof of this, valid in arbitrary type, due to Ginzburg. For a sketch of the argument, see the footnote to [P2, Question 5.1]. (1) (2) Corollary 6.11. The elements Cn ,Cn ,... of Yn(σ) are algebraically independent and generate the center Z(Yn(σ)). Moreover, the epimorphism κ : Yn(σ) W (π) from (3.20) maps Z(Yn(σ)) surjectively onto Z(W (π)).

Proof. This is immediate from the theorem on recalling that Yn(σ) is a ﬁltered inverse limit of W (π)’s as explained in [BK5, Remark 6.4]. We are grateful to one of the referees of [BK5] for pointing out that we are already in a position to apply [FO] to obtain the following generalization of a theorem of Kostant from [Ko1]. In the case W (π) is the Yangian of level l this result is [FO, Theorem 2]. Theorem 6.12. The algebra W (π) is free as a module over its center. REPRESENTATIONS OF SHIFTED YANGIANS 59

Proof. Recall by the PBW theorem that the associated graded algebra gr W (π) is free commutative on generators (3.35)–(3.37), in particular W (π) is a special ﬁltered algebra in the sense of [FO]. Let A be the quotient of gr W (π) by the ideal generated (r) (r) (r) by the elements (3.36)–(3.37). Let di resp. cn denote the image of grr Di resp. (r) (r) grr Cn in A. Thus, A is the free polynomial algebra F[di |i = 1, . . . , n, r = 1, . . . , pi]. (r) Moreover by Theorem 3.5 and (2.33) we have that di = 0 for r > pi. It follows from this and (2.70) that if we set

pi X (r) pi−r di(u) = di u , r=0 N X (r) N−r cn(u) = cn u r=0 then cn(u) = d1(u)d2(u) ··· dn(u). Now applying [FO, Theorem 1] as in the proof of (1) (2) (N) [FO, Theorem 2], it suffices to show that cn , cn , . . . , cn is a regular sequence in A. Hence by [FO, Proposition 1(5)] we need to check that the variety Z = (1) (N) N V (cn , . . . , cn ) is equidimensional of dimension 0. Consider the morphism ϕ : F → N (r) F mapping a point (xi )1≤i≤n,1≤r≤pi to the coefficients of the following monic polynomial: n Y pi (1) pi−1 (pi) (u + di u + ··· + di ). i=1 −1 N p p Obviously Z = ϕ (0). Since F[u] is a unique factorization domain, u = u 1 ··· u n is the unique decomposition of uN as a product of monic polynomials of degrees p1, . . . , pn. Hence Z = {0}. In view of Theorem 6.10, the center of W (π) is canonically isomorphic to the center of U(glN ). So we can parametrize the central characters of W (π) in exactly L the same way as we did for U(gl ) in §3.8, by the set of θ ∈ P = εa whose N a∈F Z coefficients are non-negative integers summing to N. Given such an element θ, define N (1) N−1 (N) f(u) = u + f u + ··· + f ∈ F[u] according to (3.45)–(3.46). Then, for an admissible W (π)-module M, define for each r = 1,...,N there exists p 0 prθ(M) := v ∈ M (r) (r) p . (6.11) such that ψ(ZN − f ) v = 0 Equivalently, by (2.70) and Lemma 3.7, we have that M prθ(M) = MA(u) (6.12) A(u) where the direct sum is over all A(u) ∈ Pn such that

p1 p2 pn u (u − 1) ··· (u − n + 1) A1(u)A2(u − 1) ··· An(u − n + 1) = f(u). 60 JONATHAN BRUNDAN AND ALEXANDER KLESHCHEV

Since the admissible W (π)-module M is the direct sum of its Gelfand-Tsetlin subspaces, it follows that M M = prθ(M), (6.13) θ∈P with the convention that prθ(M) = 0 if the coeﬃcients of θ are not non-negative integers summing to N. This is clearly a decomposition of M as a W (π)-module. Lemma 6.13. All highest weight W (π)-modules of type A ∈ Row(π) are of central character θ(A).

Proof. Suppose that the entries on the ith row of A are ai,1, . . . , ai,pi . By (2.70), Lemma 3.7 and the deﬁnition (6.1)–(6.3), ψ(ZN (u)) acts on any highest weight mod- Qn Qpi ule of type A as the scalar i=1 j=1(u + ai,j).

6.5. Proof of Theorem 6.2. Letπ ¯ denote the pyramid obtained from π by re- moving the bottom row. The tuple of row lengths corresponding to the pyramid π¯ is (p1, . . . , pn−1) and the submatrixσ ¯ = (si,j)1≤i,j≤n−1 of the shift matrix σ = (si,j)1≤i,j≤n chosen for π gives a shift matrix forπ ¯. It is elementary to see from the relations that there is a homomorphism W (¯π) → W (π) mapping the genera- (r) (r) (r) tors Di (i = 1, . . . , n − 1, r > 0), Ei (i = 1, . . . , n − 2, r > si,i+1) and Fi (i = 1, . . . , n−2, r > si+1,i) of W (¯π) to the elements with the same names in W (π). By the PBW theorem this map is in fact injective, allowing us to view W (¯π) as a subalgebra of W (π). We will in fact prove the following branching theorem for Verma modules.

Theorem 6.14. Let A ∈ Row(π) with entries ai,1, . . . , ai,pi on its ith row for each i = 1, . . . , n. There is a ﬁltration 0 = M0 ⊂ M1 ⊂ · · · of M(A) as a W (¯π)-module S with i≥0 Mi = M(A) and subquotients isomorphic to the Verma modules M(B) for

B ∈ Row(¯π) such that B has the entries (ai,1 − ci,1),..., (ai,pi − ci,pi ) on its ith row for each i = 1, . . . , n − 1, one for each tuple (ci,j)1≤i≤n−1,1≤j≤pi of natural numbers. Let us ﬁrst explain how to deduce Theorem 6.2 from this. Proceed by induction on n, the case n = 1 being trivial. For the induction step, we have by Theorem 6.14 W (π) and the induction hypothesis that the character of resW (¯π) M(A) equals

n−1 pi ( n−1 ) X Y Y Y yk,ai,j −(ci,j,k+1+···+ci,j,n) y , i,ai,j −(ci,j,i+1+···+ci,j,n) y c i=1 j=1 k=i+1 k,ai,j −(ci,j,k+···+ci,j,n) where the ﬁrst sum is over all tuples c = (ci,j,k)1≤i

Lemma 6.15. The following relations hold in W (π).

(i) For all i < j, [Fi,j(u)Di(u),Fi,j(v)Di(v)] = 0. (ii) For all i < j < k, (u − v)[Fj,k(u),Fi,j(v)] equals X r X (−1) Fir,ir+1 (u) ··· Fi1,i2 (u)(Fi,i1 (v) − Fi,i1 (u)). r≥0 i

(iii) For all i < j and k j, [Dk(u),Fi,j(v)] = 0. (iv) For all i < j, (u − v)[Di(u),Fi,j(v)] = (Fi,j(u) − Fi,j(v))Di(u). (v) For all i < j, (u − v)[Dj(u),Fi,j(v)] equals X r X (−1) Fir,ir+1 (u) ··· Fi1,i2 (u)(Fi,i1 (v) − Fi,i1 (u))Dj(u). r≥0 i

(vi) For all i < j < k, (u − v)[Dj(u),Fi,k(v)] equals X r X (−1) Fir,ir+1 (u) ··· Fi1,i2 (u)(Fi,i1 (v) − Fi,i1 (u))Fj,k(u)Dj(u). r≥0 i

(vii) For all i < j < k, (u − v)[Fj,k(u)Dj(u),Fi,k(v)] equals X r X (−1) Fir,ir+1 (u) ··· Fi1,i2 (u)(Fi,i1 (v) − Fi,i1 (u))Fj,k(u)Dj(u). r≥0 i

pi X (r) pi−r pi Li(u) = Li u := u Tn,i(u) ∈ W (π)[u] (6.14) r=0 for each 1 ≤ i < n. Also for h ≥ 0 set 1 dh L (u) := L (u). (6.15) i,h h! duh i We will apply the following simple observation repeatedly from now on: given a vector m of weight α in a W (π)-module M with the property that α +εj −εi is not a weight p of M for any 1 ≤ j < i, we have by (2.32) that Li(u)m = u i Fi,n(u)Di(u)m. Lemma 6.16. Suppose we are given 1 ≤ i < n and a vector m of weight α in a W (π)-module M such that

(i) α − d(εi − εn) + εj − εi is not a weight of M for any 1 ≤ j < i and d ≥ 0; pi 0 (ii) u Di(u)m ≡ (u+a1) ··· (u+api )m (mod M [u]) for some scalars a1, . . . , api ∈ 0 F and some subspace M of M. For j = 1, . . . , pi, deﬁne mj := Li,h(j)(−aj)m where h(j) = #{k = 1, . . . , j − 1 | ak = aj}. Then we have that

pi u Di(u)mj ≡ (u + a1) ··· (u + aj−1)(u + aj − 1)(u + aj+1) ··· (u + api )mj pi X (u + a1) ··· (u + ap ) X (r) − i m (mod L M 0[u]). (u + a )h(j)−h(k)+1 k i k=1,...,j−1 j r=1 ak=aj 62 JONATHAN BRUNDAN AND ALEXANDER KLESHCHEV

Moreover, the subspace of M spanned by the vectors m1, . . . , mpi coincides with the (1) (pi) subspace spanned by the vectors Li m, . . . , Li m. Proof. By Lemma 6.15(iv) and the assumptions (i)–(ii), we have that

pi (u − v)[u Di(u),Li(v)]m ≡ (v + a1) ··· (v + api )Li(u)m pi X (r) 0 − (u + a1) ··· (u + api )Li(v)m (mod Li M [u, v]). r=1 Hence,

(u + a1) ··· (u + ap )Li(v) upi D (u)L (v)m ≡ (u + a ) ··· (u + a )L (v)m − i m i i 1 pi i u − v (v + a ) ··· (v + a )L (u) + 1 pi i m. u − v h(j) Apply the operator 1 d to both sides using the Leibnitz rule then set v := −a h(j)! dvh(j) j to deduce that

pi u Di(u)Li,h(j)(−aj)m ≡ (u + a1) ··· (u + api )Li,h(j)(−aj)m h(j) X (u + a1) ··· (u + ap ) − i L (−a )m. (u + a )h(j)−k+1 i,k j k=0 j p The left hand side equals u i Di(u)mj by definition. The right hand side simplifies to give h(j)−1 X (u + a1) ··· (u + ap ) (u + a ) ··· (u + a − 1) ··· (u + a )m − i L (−a )m 1 j pi j (u + a )h(j)−k+1 i,k j k=0 j which is exactly what we need to prove the first part of the lemma. For the second part, we observe that the transition matrix between the vectors (1) (pi) Li(u1)m, ··· ,Li(upi )m and Li m, ··· ,Li m is a Vandermonde matrix with deter- h(j) minant Q (u − u ). Apply 1 d for j = 1, . . . , p to deduce that the de- 1≤j

Evaluate this expression at uj = −aj for each j = 1, . . . , pi to get

h(1)+···+h(pi) Y (−1) (ak − aj) 6= 0.

1≤j

(1) (pi) Hence the transition matrix between the vectors m1, . . . , mpi and Li m, ··· ,Li m is invertible, so they span the same space. REPRESENTATIONS OF SHIFTED YANGIANS 63

Lemma 6.17. Under the same assumptions as Lemma 6.16, let Cd denote the set of all pi-tuples c = (c1, . . . , cpi ) of natural numbers summing to d. Put a total order on 0 0 Cd so that c < c if c is lexicographically greater than c. For c ∈ Cd let

pi cj Y Y mc := Li,hc(j,k)(−aj + k − 1)m j=1 k=1 where hc(j, k) = #{l = 1, . . . , j − 1 | al − cl = aj − k + 1}. Then,

pi 0 u Di(u)mc ≡ (u + a1 − c1) ··· (u + api − cpi )mc (mod Mc[u])

0 0 (r1) (rd) 0 where Mc is the subspace of M spanned by all mc0 for c < c and by all Li ··· Li M for 1 ≤ r1, . . . , rd ≤ pi. Moreover, the vectors {mc | c ∈ Cd} span the same subspace (r1) (rd) of M as the vectors Li ··· Li m for all 1 ≤ r1, . . . , rd ≤ pi.

Proof. Note first that the definition of the vectors mc does not depend on the order taken in the products, thanks to Lemma 6.15(i). Now proceed by induction on d, the case d = 1 being precisely the result of the previous lemma. For d > 1, define 0 vectors m1, . . . , mpi according to the preceeding lemma. For r = 1, . . . , pi, let Mr be (s) 0 the subspace spanned by m1, . . . , mr−1 and Li M for all s = 1, . . . , pi. Then the preceeding lemma shows that

pi 0 u Di(u)mr ≡ (u + a1) ··· (u + ar − 1) ··· (u + api )mr (mod Mr[u])

(1) (pi) and that m1, . . . , mpi span the same space as the vectors Li m, . . . , Li m. For c ∈ Cd−1 and r = 1, . . . , pi, let

pi cj Y Y mr,c := Li,hr,c(j,k)(−aj + δj,r + k − 1)mr j=1 k=1 0 where hr,c(j, k) := #{l = 1, . . . , j − 1 | al − δr,l − cl = aj − δr,j − k + 1}. Let Mr,c be 0 (r1) (rd−1) 0 the subspace of M spanned by all mr,c0 for c < c together with Li ··· Li Mr for all 1 ≤ r1, . . . , rd−1 ≤ pi. Then by the induction hypothesis,

pi 0 u Di(u)mr,c ≡ (u+a1 −c1) ··· (u+ar −1−cr) ··· (u+api −cpi )mr,c (mod Mr,c[u]).

Moreover the vectors {mr,c | c ∈ Cd−1} span the same subspace of M as the vectors (r1) (rd−1) Li ··· Li mr for all 1 ≤ r1, . . . , rd−1 ≤ pi. Now observe that if c ∈ Cd−1 satisﬁes c1 = ··· = cr−1 = 0, then mr,c = mc+δr where c+δr ∈ Cd is the tuple (c1, . . . , cr−1, cr+ 0 0 1, cr+1, . . . , cpi ); otherwise, mr,c lies in the subspace spanned by the ms,c for s < r, c ∈ Cd−1. The lemma follows. At last we can complete the proof of Theorem 6.14. Let C denote the set of all Ppi tuples c = (ci,j)1≤i≤n−1,1≤j≤pi of natural numbers. Writing |c|i for j=1 ci,j and |c| 0 for |c|1 + |c|2 + ··· + |c|n−1, we put a total order on C so that c ≤ c if any of the following hold: (a) |c0| < |c|; 0 0 0 0 0 (b) |c | = |c| but |c |n−1 = |c|n−1,|c |n−2 = |c|n−2,. . . , |c |i+1 = |c|i+1 and |c |i > |c|i for some i ∈ {1, . . . , n − 1}; 64 JONATHAN BRUNDAN AND ALEXANDER KLESHCHEV

equal to the tuple (ci,1, . . . , ci,pi ) for every i = 1, . . . , n − 1. Now let M := M(A) for short. For each c ∈ C, deﬁne a vector mc ∈ M by   n−1 pi ci,j Y Y Y  mc := Li,hc(i,j,k)(−ai,j + k − 1) m+ i=1 j=1 k=1  where hc(i, j, k) = #{l = 1, . . . , j − 1 | ai,l − ci,l = ai,j − k + 1} and the ﬁrst product is taken in order of increasing i from left to right. The second part of Lemma 6.17 and Theorem 6.1(ii) imply that the vectors {ymc | y ∈ Y, c ∈ C} form a basis for M, (r) where Y here denotes the set of all monomials in the elements {Tj,i | 1 ≤ i < j ≤ 0 n − 1, r = 1, . . . , pi}. For each c ∈ C, let Mc resp. Mc denote the subspace of M 0 0 spanned by the vectors {ymc0 | y ∈ Y, c ≤ c} resp. {ymc0 | y ∈ Y, c < c}. Clearly S M = c∈C Mc. Now we complete the proof of Theorem 6.14 by showing that each 0 ∼ Mc is actually a W (¯π)-submodule of M with Mc/Mc = M(B) for B ∈ Row(¯π) such that B has entries (ai,1 −ci,1),..., (ai,pi −ci,pi ) on its ith row for each i = 1, . . . , n−1. Proceeding by induction on the total ordering on C, the induction hypothesis allows 0 us to assume that Mc is a W (¯π)-submodule of M. Then the vectors 0 0 {ymc0 + Mc | y ∈ Y, c ≥ c} 0 0 form a basis for the W (¯π)-module M/Mc. Hence the vector mc := mc +Mc is a vector 0 (r) of maximal weight in M/Mc, so it is annihilated by all Ei (i = 1, . . . , n−2, r > si,i+1). Moreover, using Lemma 6.15(iii),(vi) and (vii), Lemma 6.17 and the PBW theorem [ for Y(1n)(σ), one checks that

pi u Di(u)mc = (u + ai,1 − ci,1) ··· (u + ai,pi − ci,pi )mc. 0 Hence, mc ∈ M/Mc is a highest weight vector of type B as claimed. Now it follows easily using Theorem 6.1(ii) and the universal property of Verma modules that Mc is 0 ∼ a W (¯π)-submodule of M and Mc/Mc = M(B). 7. Standard modules In this chapter, we begin by classifying the irreducible finite dimensional representations of W (π) and of Yn(σ), following the argument in the case of the Yangian Yn itself due to Tarasov [T2] and Drinfeld [D]. Then we define and study another family of finite dimensional W (π)-modules which we call standard modules. 7.1. Two rows. In this section we assume that n = 2 and let π be any pyramid with just two rows of lengths p1 ≤ p2. We will represent the π-tableau with entries a1···ap1 a1, . . . , ap on its first row and b1, . . . , bp on its second row by . The first lemma 1 2 b1···bp2 is well known; see e.g. [CP1]. We reproduce here the detailed argument following [M2, Proposition 3.6] since we need to slightly weaken the hypotheses later on.

Lemma 7.1. Assume p1 = p2 = l and a1, . . . , al, b1, . . . , bl, a, b ∈ F. (i) If ai > b implies that ai ≥ a > b for each i = 1, . . . , l, then all highest weight a1···al a vectors in the module L(b1···bl ) L(b) are scalar multiples of m+ ⊗ m+. REPRESENTATIONS OF SHIFTED YANGIANS 65

(ii) If a > bi implies that a > b ≥ bi for each i = 1, . . . , l, then all highest weight a a1···al vectors in the module L(b) L(b1···bl ) are scalar multiples of m+ ⊗ m+.

Proof. (i) Abbreviate e := e1,2, d2 := e2,2 and f := e2,1 in the Lie algebra gl2. (r) r a Let f denote f /r!. Recall that the irreducible gl2-module L(b) of highest weight (2) (a−b−1) (aε1 +(b+1)ε2) has basis m+, fm+, f m+,... if a 6> b or m+, fm+, . . . , f m+ (r+1) (r) if a > b. Also ef m+ = (a − b − r − 1)f m+. a1···al a Suppose that L(b1···bl )L(b) contains a highest weight vector v that is not a scalar multiple of m+ ⊗ m+. We can write k X (k−i) v = mi ⊗ f m+ i=0 for vectors m0 6= 0, m1, . . . , mk and k ≥ 0 with k < a − b in case a > b. The element (r+1) (r+1) (r) (r) T1,2 acts on the tensor product as T1,2 ⊗1+T1,2 ⊗d2 +T1,1 ⊗e ∈ W (π)⊗U(gl2). (r+1) (k) Apply T1,2 to the vector v and compute the ?⊗y m+-coefficient to deduce that (r+1) (r) T1,2 m0 + (b + k + 1)T1,2 m0 = 0 (r) for all r ≥ 0. It follows that T1,2 m0 = 0 for all r > 0, hence m0 is a scalar multiple of a1···al the canonical highest weight vector m+ of L(b1···bl ). Moreover we must in fact have that k ≥ 1 since v is not a multiple of m+ ⊗ m+. (k−1) (r+1) Next compute the ? ⊗ f m+-coefficient of T1,2 v to get that (r+1) (r) (r) T1,2 m1 + (b + k)T1,2 m1 + (a − b − k)T1,1 m0 = 0. Multiply by (−(b + k))l−r and sum over r = 0, 1, . . . , l to deduce that l (l+1) X l−r (r) T1,2 m1 + (a − b − k) (−(b + k)) T1,1 m0 = 0. r=0 (l+1) But T1,2 = 0 in W (π) by a trivial special case of Theorem 3.5. Moreover, by the Pl l−r (r) definition (6.1), we have that r=0 u T1,1 m0 = (u + a1) ··· (u + al)m0. So we have shown that

(a − b − k)(a1 − b − k)(a2 − b − k) ··· (al − b − k) = 0. Since k ≥ 1 and k < a − b in case a > b, we have that (a − b − k) 6= 0. Hence we must have that ai = b + k for some i = 1, . . . , l, i.e. ai > b and either a 6> b or ai < a. This is a contradiction. (ii) Similar.

Corollary 7.2. Assume p1 = p2 = l and a1, . . . , al, b1, . . . , bl, a, b ∈ F. a a1···al (i) If b < ai implies that b < a ≤ ai for each i = 1, . . . , l, then L(b) L(b1···bl ) is a highest weight module generated by the highest weight vector m+ ⊗ m+. a1···al a (ii) If bi < a implies that bi ≤ b < a for each i = 1, . . . , l, then L(b1···bl ) L(b) is a highest weight module generated by the highest weight vector m+ ⊗ m+. 66 JONATHAN BRUNDAN AND ALEXANDER KLESHCHEV

a1···al a Proof. (i) By Lemma 7.1(i), L(b1···bl )L(b) has simple socle generated by the highest weight vector m+ ⊗ m+. Now apply the duality τ from (5.2) using Corollary 5.10 and (3.33) to deduce that a1···al a τ ∼ a a1···al (L(b1···bl ) L(b)) = L(b) L(b1···bl ) has a unique maximal submodule and that the highest weight vector m+ ⊗ m+ does not belong to this submodule. Hence it is a highest weight module generated by the vector m+ ⊗ m+. (ii) Similar.

a1···al Remark 7.3. The module L(b1···bl ) in the statement of Corollary 7.2 can in fact a1···al be replaced by any non-zero quotient of the Verma module M(b1···bl ). This follows a1···al because the only property of L(b1···bl ) needed for the proof of Lemma 7.1 is that all its highest weight vectors are scalar multiples of m+; any non-zero submodule of the a1···al τ dual Verma module M(b1···bl ) also has this property.

Lemma 7.4. Assume p1 ≤ p2 and a1, . . . , ap1 , b1, . . . , bp2 , b ∈ F. (i) If ai > b implies that ai > bi ≥ b for each i = 1, . . . , p1 then all highest weight a1···ap1 vectors in the module L( ) L( ) are scalar multiples of m+ ⊗ m+. b1···bp2 b a ···a (ii) All highest weight vectors in the module L( ) L( 1 p1 ) are scalar multiples b b1···bp2 of m+ ⊗ m+.

Proof. Let σ = (si,j)1≤i,j≤2 be a shift matrix corresponding to the pyramid π. Also note that L(b) is just the one dimensional gl1-module with basis m+ such that e1,1m+ = bm+. a1···ap1 (i) Suppose that m ⊗ m+ is a non-zero highest weight vector in L( ) L( ). b1···bp2 b (r+1) So we have that E1 (m ⊗ m+) = 0 for all r > s1,2 and p1 u D1(u)(m ⊗ m+) = (u + c1)(u + c2) ··· (u + cp1 )(m ⊗ m+) for some scalars c1, . . . , cp1 ∈ F. (r+1) By [BK5, Lemma 11.3] and [BK5, Theorem 4.1(i)], we have that ∆p2,1(E1 ) = (r+1) (r) (r+1) (r) E1 ⊗1+E1 ⊗(e1,1 +1) for all r > s1,2. Hence E1 m+(b+1)E1 m = 0 for all 0 (s1,2+1) (s1,2+r+1) r 0 r > s1,2. On setting m := E1 m, we deduce that E1 m = (−(b + 1)) m for all r ≥ 0, i.e. u−s1,2−1 E (u)m = (1 − (b + 1)u−1 + (b + 1)2u−2 − · · · )u−s1,2−1m0 = m0. 1 1 + (b + 1)u−1 0 (r) If m = 0 then we have that E1 m = 0 for all r > s1,2, hence m is a scalar multiple 0 of m+ as required. So assume from now on that m 6= 0 and aim for a contradiction. (r) (r) Since ∆p2,1(D1 ) = D1 ⊗ 1 for all r > 0 we have that −1 −1 −1 D1(u)m = (1 + c1u )(1 + c2u ) ··· (1 + cp1 u )m. REPRESENTATIONS OF SHIFTED YANGIANS 67

(s1,2+1) s1,2 The last two equations combined with the identity [D1(u),E1 ] = u D1(u)E1(u) in W (π)[[u−1]] show that (1 + c u−1) ··· (1 + c u−1)(1 + (b + 1)u−1) D (u)m0 = 1 p1 m0. 1 1 + bu−1 (r) Since D1 = 0 for r > p1 it follows from this that b = ci for some 1 ≤ i ≤ p1. Without loss of generality we may as well assume that b = c1. Then we have shown that 0 −1 −1 −1 0 D1(u)m = (1 + (c1 + 1)u )(1 + c2u ) ··· (1 + cp1 u )m .

a1···ap1 Now we claim that if we have any non-zero vector in L( ) on which D1(u) acts b1···bp2 −1 −1 as the scalar (1+d1u ) ··· (1+dp1 u ) then there exists a permutation w ∈ Sp1 such that ai ≥ dwi and moreover if ai > bi then dwi > bi, for each i = 1, . . . , p1. To prove a ···a this, we may as well replace the module L( 1 p1 ) with the tensor product L(a1) b1···bp2 b1 ap1 a1···ap1 ··· L( ) L(bp +1) ··· L(bp ), since that contains L( ) (possibly twisted bp1 1 2 b1···bp2 by the isomorphism ι) as a subquotient. Now the claim follows from Lemma 5.1 and a the familiar fact that if we have a non-zero vector in the irreducible gl2-module L(b) −1 on which D1(u) acts as the scalar (1 + du ) then a ≥ d and moreover if a > b then d > b. a ···a Applying the claim to the non-zero vectors m and m0 of L( 1 p1 ), we deduce (after b1···bp2 reordering if necessary) that there exists a permutation w ∈ Sp1 such that (a) a1 ≥ c1 + 1 and moreover if a1 > b1 then c1 + 1 > b1; a2 ≥ c2 and moreover if

a2 > b2 then c2 > b2;...; ap1 ≥ cp1 and moreover if ap1 > bp1 then cp1 > bp1 ; (b) a1 ≥ cw1 and moreover if a1 > b1 then cw1 > b1; a2 ≥ cw2 and moreover

if a2 > b2 then cw2 > b2;...; ap1 ≥ cwp1 and moreover if ap1 > bp1 then

cwp1 > bp1 . From this we can derive the required contradiction, as follows. Suppose that we know that ci > b for some i. Then ai ≥ ci > b, hence by the hypothesis from the statement of the lemma ai ≥ cwi > bi ≥ b. Hence cwi > b. Now we do know that c1 = b. Hence a1 ≥ c1 + 1 > b, so a1 ≥ cw1 > b1 ≥ b. Hence cw1 > b. Combining this with the preceeding observation we deduce that cwk1 > b for all k ≥ 1, hence in particular c1 > b. (r) (r) (ii) We have that ∆1,p2 (E1 ) = 1 ⊗ E1 for all r > s1,2. So if m+ ⊗ m is a highest a1···ap1 (r) weight vector in L( ) L( ) then E m = 0 for all r > s1,2. Hence m is a scalar b b1···bp2 1 multiple of m+ as required.

Corollary 7.5. Assume p1 ≤ p2 and a1, . . . , ap1 , b1, . . . , bp2 , b ∈ F. (i) If b < ai implies that b ≤ bi < ai for each i = 1, . . . , p1 then the module a ···a L( ) L( 1 p1 ) is a highest weight module generated by the highest weight b b1···bp2 vector m+ ⊗ m+. a ···a (ii) The module L( 1 p1 ) L( ) is a highest weight module generated by the b1···bp2 b highest weight vector m+ ⊗ m+. Proof. Argue using the duality τ exactly as in the proof of Corollary 7.2. 68 JONATHAN BRUNDAN AND ALEXANDER KLESHCHEV

a ···a Remark 7.6. As in Remark 7.3, the module L( 1 p1 ) in the statement of Corol- b1···bp2 a ···a lary 7.5(ii) can be replaced by any non-zero quotient of the Verma module M( 1 p1 ). b1···bp2 We can not quite say the same thing for Corollary 7.5(i), but by the proof we a ···a can at least replace L( 1 p1 ) by any non-zero quotient M of the Verma mod- b1···bp2 a ···a ule M( 1 p1 ) with the property that all of its Gelfand-Tsetlin weights, i.e. the b1···bp2 A(u) ∈ P2 such that MA(u) 6= 0, are also Gelfand-Tsetlin weights of the module a1 ap1 L( ) ··· L( ) L(b ) ··· L(b ). b1 bp1 p1+1 p2

Now we can prove the main theorem of the section. This is new only if p1 6= p2.

Theorem 7.7. Assume p1 ≤ p2 and a1, . . . , ap1 , b1, . . . , bp2 ∈ F are scalars such that the following property holds for each i = 1, . . . , p1: If the set {aj − bk | i ≤ j ≤ p1, i ≤ k ≤ p2 such that aj > bk} is non-empty then (ai − bi) is its smallest element. a ···a Then the irreducible W (π)-module L( 1 p1 ) is isomorphic to the tensor product of b1···bp2 the modules a1 ap1 L( ),...,L( ),L(bp +1),...,L(bp ) b1 bp1 1 2 taken in any order that makes sense. Proof. Assume to start with that the pyramid π is left-justiﬁed. First we show for p1 > 0 that a ···a a ···a L( 1 p1 ) =∼ L(a1) L( 2 p1 ). b1···bp1 b1 b2···bp1 Since a1 > bi implies that a1 > b1 ≥ bi for all i = 2, . . . , p1, Lemma 7.1(ii) implies that m+ ⊗ m+ is the unique (up to scalars) highest weight vector in the module on the right hand side. Since b1 < ai implies b1 < a1 ≤ ai, Corollary 7.2(i) shows that this vector generates the whole module. Hence it is irreducible, so isomorphic to a1···ap1 L( ) by Lemma 5.4. Next we show for p2 > p1 that b1···bp1 a ···ap a ···a 1 1 ∼ 1 p1 L(b ···b ) = L( ) L(bp ). 1 p2 b1···bp2−1 2

Since ai > bp2 implies ai > bi ≥ bp2 Lemma 7.4(i) implies that m+ ⊗m+ is the unique (up to scalars) highest weight vector in the module on the right hand side. But by Corollary 7.5(ii) this vector generates the whole module, hence it is irreducible. Using these two facts, it follows by induction on p2 that a ···a a 1 p1 ∼ a1 p1 L( ) = L( ) ··· L( ) L(bp +1) ··· L(bp ). b1···bp2 b1 bp1 1 2 This proves the theorem for one particular ordering of the tensor product and for one particular choice of the pyramid π with row lengths (p1, p2). The theorem for all other orderings and pyramids follows from this by character considerations. Suppose finally that we are given an arbitrary two row tableau A with entries a1, . . . , ap1 on row one and b1, . . . , bp2 on row two. We can always reindex the entries in the rows so that the hypothesis of Theorem 7.7 is satisfied: first reindex to ensure if possible that a1 − b1 is the minimal positive integer difference amongst all the df- ferences ai − bj, then inductively reindex the remaining entries a2, . . . , ap1 , b2, . . . , bp2 . Hence Theorem 7.7 shows that every irreducible admissible W (π)-module can be REPRESENTATIONS OF SHIFTED YANGIANS 69 realized as a tensor product of irreducible gl2- and gl1-modules. This remarkable observation was first made by Tarasov [T2] in the case p1 = p2.

a1···ap1 Corollary 7.8. If L( ) is finite dimensional for scalars a1, . . . , ap , b1, . . . , bp ∈ b1···bp2 1 2 F then there exists a permutation w ∈ Sp2 such that a1 > bw1, a2 > bw2, . . . , ap1 > bwp1 . Proof. Reindexing if necessary, we may assume that the hypothesis of Theorem 7.7 ai is satisfied. Then by the theorem we must have that L(bi ) is finite dimensional for each i = 1, . . . , p1, i.e ai > bi for each such i.

7.2. Classification of finite dimensional irreducible representations. Now assume that π = (q1, . . . , ql) is an arbitrary pyramid with row lengths (p1, . . . , pn). Recall the definitions of the sets Row(π) of row symmetrized π-tableaux, Col(π) of column strict π-tableaux and Dom(π) of dominant row symmetrized π-tableaux from §4.1. Theorem 7.9. For A ∈ Row(π), the irreducible W (π)-module L(A) is finite dimensional if and only if A is dominant, i.e. it has a representative belonging to Col(π).

Proof. Suppose ﬁrst that L(A) is ﬁnite dimensional. Let σ = (si,j)1≤i,j≤n be a shift matrix corresponding to π, so that W (π) is canonically a quotient of the shifted Yangian Yn(σ). For each i = 1, . . . , n − 1, let σi denote the 2 × 2 submatrix si,i si,i+1 si+1,i si+1,i+1 of the matrix σ. Also let ai,1, . . . , ai,pi be the entries in the ith row of A for each i = 1, . . . , n. The map ψi−1 from (2.66) obviously induces an embedding of the shifted Yangian Y2(σi) into Yn(σ). The highest weight vector m+ ∈ L(A) is also a highest weight vector in the restriction of L(A) to Y2(σi) using this embedding. Hence by Corollary 7.8 there exists w ∈ Spi+1 such that

ai,1 > ai+1,w1, ai,2 > ai+1,w2, . . . , ai,pi > ai+1,wpi , for each i = 1, . . . , n − 1. Hence A has a representative belonging to Col(π). Conversely, suppose that A has a representative belonging to Col(π). Let A1,...,Al be the columns of this representative, so that A ∼ro A1 ⊗ · · · ⊗ Al. Since each Ai is column strict, the irreducible glqi -module L(Ai) is finite dimensional. By Lemma 5.4 the tensor product L(A1) ··· L(Al) is then a finite dimensional W (π)-module containing a highest weight vector of type A. Hence L(A) is finite dimensional. Hence, the modules {L(A)|A ∈ Dom(π)} give a full set of pairwise non-isomorphic finite dimensional irreducible W (π)-modules. As a corollary, we have the following result classifying the finite dimensional irreducible representations of the shifted Yan- gians Yn(σ) themselves. Since every finite dimensional Yn(σ)-module is admissible, it is enough for this to determine which of the irreducible modules Ln(σ, A(u)) from (5.8) is finite dimensional.

Corollary 7.10. Let σ = (si,j)1≤i,j≤n be a shift matrix and set di := si,i+1 + si+1,i. For A(u) ∈ Pn, the irreducible Yn(σ)-module Ln(σ, A(u)) is finite dimensional if 70 JONATHAN BRUNDAN AND ALEXANDER KLESHCHEV and only if there exist (necessarily unique) monic polynomials P1(u),...,Pn−1(u), Q1(u),...,Qn−1(u) ∈ F[u] such that Qi(u) is of degree di, (Pi(u),Qi(u)) = 1 and A (u) P (u) udi i = i × Ai+1(u) Pi(u − 1) Qi(u) for each i = 1, . . . , n − 1. Proof. Recall from the proof of Theorem 5.6 that every admissible irreducible Yn(σ)-module may be obtained by inflating an admissible irreducible W (π)-module through the map (5.11), for some pyramid π with shift matrix σ and some f(u) ∈ 1+ −1 −1 u F[[u ]]. Given this and Theorem 7.9, we see that Ln(σ, A(u)) is finite dimensional −1 −1 if and only if there exist l ≥ sn,1 + s1,n, f(u) ∈ 1 + u F[[u ]] and scalars ai,j ∈ F for 1 ≤ i ≤ n, 1 ≤ j ≤ pi := l − sn,i − si,n such that −1 −1 (a) Ai(u) = f(u)(1 + ai,1u ) ··· (1 + ai,pi u ) for each i = 1, . . . , n; (b) ai,j ≥ ai+1,j for each i = 1, . . . , n − 1 and j = 1, . . . , pi. Following the proof of [M2, Theorem 2.8], these conditions are equivalent to the existence of monic polynomials P1(u),...,Pn−1(u),Q1(u),...,Qn−1(u) ∈ F[u] such that Qi(u) is of degree di and A (u) P (u) udi i = i × Ai+1(u) Pi(u − 1) Qi(u) for each i = 1, . . . , n − 1. Finally to get uniqueness of the Pi(u)’s and Qi(u)’s we have to insist in addition that (Pi(u),Qi(u)) = 1. From Corollary 7.10 and (2.77), it also follows that the isomorphism classes of irreducible SYn(σ)-modules are parametrized in the same fashion by monic polynomials P1(u),...,Pn−1(u), Q1(u),...,Qn−1(u) ∈ F[u] such that Qi(u) is of degree di and (Pi(u),Qi(u)) = 1 for each i = 1, . . . , n − 1. In the case σ is the zero matrix, each Qi(u) is of course just equal to 1, so we recover the classification from [D] of finite dimensional irreducible representations of the Yangian of sln by their Drinfeld polynomials P1(u),...,Pn−1(u); see also [M2, §2] once more.

7.3. Tensor products. Assume throughout the section that π = (q1, . . . , ql) is a fixed pyramid and make a corresponding choice σ = (si,j)1≤i,j≤n of shift matrix. Also set t := ql for short. For A ∈ Col(π) with columns A1,...,Al read from left to right, let V (A) := L(A1) ··· L(Al). (7.1) We will refer to the modules {V (A)|A ∈ Col(π)} as standard modules. As we observed already in the proof of Theorem 7.9, each V (A) is a finite dimensional W (π)-module, and the vector m+ ⊗ · · · ⊗ m+ ∈ V (A) is a highest weight vector of type equal to the row equivalence class of A. We wish to give a sufficient condition for V (A) to be a highest weight module generated by this highest weight vector, following an argument due to Chari [C] in the context of quantum affine algebras. The key step is provided by the following lemma; in its statement we work with the usual action of the symmetric group St on finite dimensional irreducible glt-modules, and s1, . . . , st−1 ∈ St denote the basic transpositions. REPRESENTATIONS OF SHIFTED YANGIANS 71

Lemma 7.11. Suppose that we are given a π-tableau A with columns A1,...,Al from −1 −1 left to right, together with 1 ≤ k < t = ql and w ∈ St such that k ≥ w k < w (k+1). Letting a1, . . . , ap resp. c1, . . . , cq, b1, . . . , bp denote the entries in the (n−t+k)th resp. the (n − t + k + 1)th row of A read from left to right, assume that

(i) ai > bi for each i = 1, . . . , p; (ii) ai 6> aj for each 1 ≤ i < j ≤ p; (iii) either ci 6< aj or ci ≤ bj for each i = 1, . . . , q and j = 1, . . . , p; (iv) none of the elements c1, . . . , cq lie in the same coset of F modulo Z as ap; (v) Al is column strict. Then, the vector m+ ⊗ · · · ⊗ m+ ⊗ skwm+ lies in the W (π)-submodule of L(A1) ··· L(Al−1) L(Al) generated by the vector m+ ⊗ · · · ⊗ m+ ⊗ wm+. Since this is technical, let us postpone the proof until the end of the section and explain the applications. For the first one, recall from §4.1 the definition of the set Std(π) of standard π-tableaux in the case that π is left-justified. Theorem 7.12. Assume that the pyramid π is left-justified and let A ∈ Std(π). Then the W (π)-module V (A) is a highest weight module generated by the highest weight vector m+ ⊗ · · · ⊗ m+.

Proof. Let A1,...,Al denote the columns of A from left to right, and set M := L(A1) ··· L(Al−1),L := L(Al) for short. By induction on l, M is a highest weight module generated by the vector m+ ⊗ · · · ⊗ m+. Fix the following choice of reduced expression for the longest element w0 of the symmetric group St:

w0 = sih ··· si1 where (i1, . . . , ih) = (t − 1; t − 2, t − 1; ... ; 2, . . . , t − 1; 1, . . . , t − 1).

For r = 0, . . . , h let mr := sir ··· si1 m+ ∈ L. Note by the choice of reduced expression that ir+1 ≥ si1 ··· sir (ir+1) < si1 ··· sir (ir+1 + 1). So, taking w = sir ··· si1 and k = ir+1 for some r = 0, . . . , h−1, the hypotheses of Lemma 7.11 are satisﬁed. Hence the lemma implies that the vector m+ ⊗ · · · ⊗ m+ ⊗ mr+1 lies in the W (π)-submodule of M L generated by the vector m+ ⊗ · · · ⊗ m+ ⊗ mr. Since this is true for all r = 0, . . . , h − 1 and mh = w0m+, this shows that the vector m+ ⊗ · · · ⊗ m+ ⊗ w0m+ lies in the W (π)-submodule of M L generated by the highest weight vector m+ ⊗ · · · ⊗ m+ ⊗ m+. Now to complete the proof we show that M L is generated as a W (π)-module by the vector m+ ⊗ · · · ⊗ m+ ⊗ w0m+. Let Md denote the span of all weight spaces of M of weight λ − (εj1 − εj1+1) − · · · − (εjd − εjd+1) for 1 ≤ j1, . . . , jd < n, where λ is the weight of the highest weight vector m+ ⊗ · · · ⊗ m+ of M. We will prove by induction on d ≥ 0 that Md ⊗ L is contained in the W (π)-submodule of M L generated by the vector (m+ ⊗ · · · ⊗ m+) ⊗ w0m+. Note to start with for any vector y ∈ L and 1 ≤ i < t that (1) En−t+i((m+ ⊗ · · · ⊗ m+) ⊗ y) = (m+ ⊗ · · · ⊗ m+) ⊗ (ei,i+1y).

Since L is generated as a glt-module by the lowest weight vector w0m+ this is enough to verify the base case. Now for the induction step we know already that M is a highest weight module, hence it suﬃces to show that every vector of the form 72 JONATHAN BRUNDAN AND ALEXANDER KLESHCHEV

(r) (Fi x) ⊗ y for 1 ≤ i < n, r > 0, x ∈ Md−1 and y ∈ L lies in the W (π)-submodule of M L generated by Md−1 ⊗ L. But for this we have that (r) (r) Fi (x ⊗ y) ≡ (Fi x) ⊗ y (mod Md−1 ⊗ L) by Theorem 2.5(iii).

For the second application, we return to an arbitrary pyramid π = (q1, . . . , ql). The following theorem reduces the problem of computing the characters of all finite dimensional irreducible W (π)-modules to that of computing the characters just of the modules L(A) where all entries of A lie in the same coset of F modulo Z. Twisting moreover with the automorphism ηa from (3.28) using Lemma 3.1 one can reduce further to the case that all entries of A actually lie in Z itself, i.e. A ∈ Dom0(π). Theorem 7.13. Suppose π = π0 ⊗ π00 for pyramids π0 and π00. Pick A0 ∈ Dom(π0) 00 00 0 and A ∈ Dom(π ) such that no entry of A lies in the same coset of F modulo Z 00 0 00 as an entry of A . Then the W (π)-module L(A ) L(A ) is irreducible with highest weight vector m+ ⊗ m+. Proof. By character considerations, we may assume for the proof that the pyramid π0 is right-justified of level l0 and the pyramid π00 is left-justified of level l00. Pick 00 00 a standard π -tableau representing A and let Al0+1,Al0+2,...,Al be its columns 0 read from left to right. We claim that L(A ) L(Al0+1) ··· L(Al) is a highest weight module generated by the highest weight vector m+ ⊗ m+ ⊗ · · · ⊗ m+. The theorem follows from this claim as follows. By Theorem 7.12, L(A00) is a quotient 0 00 of L(Al0+1) ··· L(Al). Hence we get from the claim that L(A ) L(A ) is a highest weight module generated by the highest weight vector m+ ⊗ m+. Similarly 00 τ 0 τ so is L(A ) L(A ) , hence on twisting with τ we see that m+ ⊗ m+ is actually the 0 00 0 00 unique (up to scalars) highest weight vector in L(A ) L(A ). Thus L(A ) L(A ) is irreducible.

To prove the claim, ﬁx the same reduced expression w0 = sih ··· si1 for the longest element of St as in the proof of Theorem 7.12. Let mr := sir ··· si1 m+ ∈ L(Al). We 0 will actually show that mr+1 lies in the W (π)-submodule of L(A ) L(Al0+1) ··· L(Al) generated by the vector m+ ⊗ m+ ⊗ · · · ⊗ m+ ⊗ mr for each r = 0, . . . , h − 1. Given this, it follows that m+ ⊗ m+ ⊗ · · · ⊗ m+ ⊗ w0m+ lies in the W (π)-submodule generated by the highest weight vector. Since we already know by induction that 0 L(A )L(Al0+1)···L(Al−1) is highest weight, the argument can then be completed in the same way as in last paragraph of the proof of Theorem 7.12.

So ﬁnally ﬁx a choice of r = 0, . . . , h − 1. Let w := sir ··· si1 and k := ir+1. Pick a 0 representative for A so that, letting a1, . . . , ap resp. c1, . . . , cq, b1, . . . , bp denote the entries in its (n − t + k)th resp. (n − t + k + 1)th row read from left to right, we have that

(i) ai > bi for each i = 1, . . . , p; (ii) ai 6> aj for each 1 ≤ i < j ≤ p; (iii) either ci 6< aj or ci ≤ bj for each i = 1, . . . , q and j = 1, . . . , p. To see that this is possible, it is easy to arrange things so that (i) and (ii) are satisfied. Then if p > 0 we can rearrange the (n−t+i+1)th row so that the difference ap −bp is the smallest of all the positive integers in the set {ap −b1, . . . , ap −bp, ap −c1, . . . , ap − REPRESENTATIONS OF SHIFTED YANGIANS 73 cq}. The condition (iii) is then automatic for j = p, and the remaining entries c1, . . . , cq, b1, . . . , bp−1 can then be rearranged inductively to get (iii) in general. Let A1,...,Al0 denote the columns of this representative from left to right. It then follows by Lemma 7.11 that the vector m+ ⊗ · · · ⊗ m+ ⊗ mr+1 lies in the W (π)-submodule of L(A1) ··· L(Al−1) ⊗ L(Al) generated by the vector m+ ⊗ · · · ⊗ m+ ⊗ mr. Since 0 L(A ) is a quotient of the submodule of L(A1) ··· L(Al0 ) generated by the highest weight vector m+ ⊗ · · · ⊗ m+, this completes the proof. We still need to explain the proof of Lemma 7.11. Let the notation be as in the statement of the lemma and abbreviate (n − t + k) by i. Let π0 be the pyramid consisting just of the ith and (i + 1)th rows of π. The 2 × 2 submatrix σ0 consisting just of the ith and (i + 1)th rows and columns of σ gives a choice of shift matrix for 0 π . As in the proof of Theorem 7.9, the map ψi−1 from (2.66) induces an embedding 0 ϕ : Y2(σ ) ,→ Yn(σ). For j = 1, . . . , l, let  2 if n − qj < i, 0  qj := 1 if n − qj = i,  0 if n − qj > i. So, numbering the columns of the pyramid π0 by 1, . . . , l in the same way as in the 0 0 0 pyramid π, its columns are of heights q1, q2, . . . , ql from left to right (including possibly some empty columns at the left hand edge). Recall the quotient map κ : Yn(σ) W (π) from (3.20) and also the Miura transform µ : W (π) ,→ U(gl ) ⊗ · · · ⊗ U(gl ) q1 ql 0 0 0 from (3.30). Similarly we have the quotient map κ : Y2(σ ) W (π ) and the Miura 0 0 transform µ : W (π ) ,→ U(gl 0 ) ⊗ · · · ⊗ U(gl 0 ). For each j = 1, . . . , l, define an q1 ql 0 algebra embedding ϕj : U(gl 0 ) ,→ U(gl ) so that if q = 2 then qj qj j

e1,1 7→ eqj −n+i,qj −n+i + n − qj, e1,2 7→ eqj −n+i,qj −n+i+1,

e2,1 7→ eqj −n+i+1,qj −n+i, e2,2 7→ eqj −n+i+1,qj −n+i+1 + n − qj, 0 and if qj = 1 then e1,1 7→ e1,1 + n − qj − 1. We have now defined all the maps in the following diagram: 0 0 0 κ 0 µ Y2(σ ) −−−−→ W (π ) −−−−→ U(gl 0 ) ⊗ · · · ⊗ U(gl 0 ) q1 ql   ϕ ϕ ⊗···⊗ϕ (7.2) y y 1 l µ Y (σ) −−−−→κ W (π) −−−−→ U(gl ) ⊗ · · · ⊗ U(gl ) n q1 ql 0 This diagram definitely does not commute. So the actions of Y2(σ ) on the U(glq1 ) ⊗ · · · ⊗ U(gl )-module L(A ) ··· L(A ) defined either using the homomorphism ql 1 l µ ◦ κ ◦ ϕ or using the homomorphism ϕ1 ⊗ · · · ⊗ ϕl ◦ µ ◦ κ are in general different. In the proof of the following lemma we will see that in fact the two actions coincide on special vectors. 0 Lemma 7.14. The subspaces (µ ◦ κ ◦ ϕ)(Y2(σ ))(m+ ⊗ · · · ⊗ m+ ⊗ wm+) and (ϕ1 ⊗ 0 0 0 · · · ⊗ ϕl ◦ µ ◦ κ )(Y2(σ ))(m+ ⊗ · · · ⊗ m+ ⊗ wm+) of L(A1) ··· L(Al−1) L(Al) are equal.

Proof. For j = 1, . . . , l − 1, let mj be an element of L(Aj) whose weight is equal to the weight of the highest weight vector m+ of L(Aj) minus some multiple of the ith 74 JONATHAN BRUNDAN AND ALEXANDER KLESHCHEV

∗ simple root εi − εi+1 ∈ dn. Also let ml be any element of L(Al). We claim for any 0 element x of Y2(σ ) that 0 0 (µ ◦ κ ◦ ϕ)(x)(m1 ⊗ · · · ⊗ ml) = (ϕ1 ⊗ · · · ⊗ ϕl ◦ µ ◦ κ )(x)(m1 ⊗ · · · ⊗ ml). Clearly the lemma follows from this claim. The advantage of the claim is that it 0 suﬃces to prove it for x running over a set of generators for the algebra Y2(σ ), since the vector on the right hand side of the equation can obviously be expressed as a 0 0 0 linear combination of vectors of the form m1 ⊗ · · · ⊗ ml where again the weight of mj is equal to the weight of m+ minus some multiple of εi −εi+1 for each j = 1, . . . , l −1, (r) (r) (r) (r) So now we proceed to prove the claim just for x = D1 ,D2 ,E1 and F1 and all meaningful r. For each of these choices for x, explicit formulae for each of κ0(x) ∈ W (π0) and of κ ◦ ϕ(x) ∈ W (π) are recorded in (3.13) and (3.20). On applying the Miura transforms one obtains explicit formulae for (µ ◦ κ ◦ ϕ)(x) and for (ϕ1 ⊗ · · · ⊗ ϕ ◦ µ0 ◦ κ0)(x) as elements of U(gl ) ⊗ · · · ⊗ U(gl ). By considering these formulae l q1 ql 0 0 directly, one observes ﬁnally that (µ ◦ κ ◦ ϕ)(x) − (ϕ1 ⊗ · · · ⊗ ϕl ◦ µ ◦ κ )(x) is a linear combination of terms of the form x1 ⊗ · · · ⊗ xl such that some xj (j = 1, . . . , l − 1) annihilates mj by weight considerations, which proves the claim. Let us explain this (r) last step in detail just in the case x = D2 , all the other cases being entirely similar. In this case, we have that

0 0 X #{t=1,...,r−1 | row(jt)≤i} (µ ◦ κ ◦ ϕ − ϕ1 ⊗ · · · ⊗ ϕl ◦ µ ◦ κ )(x) = (−1) e¯i1,j1 ··· e¯ir,jr i1,...,ir j1,...,jr ⊗(c−1) where for 1 ≤ s, t ≤ qc the notatione ¯q1+···+qc−1+s,q1+···+qc−1+t here denotes 1 ⊗ e ⊗ 1⊗(l−c−1) + δ (n − q ) ∈ U(gl ) ⊗ · · · ⊗ U(gl ), and the sum is over all 1 ≤ s,t s,t c q1 ql ir, . . . , ir, j1, . . . , jr ≤ n with (a) row(i1) = row(jr) = i + 1; (b) col(it) = col(jt) for all t = 1, . . . , r; (c) row(jt) = row(it+1) for all t = 1, . . . , r − 1; (d) if row(jt) ≥ i + 1 then col(jt) < col(it+1) for all t = 1, . . . , r − 1; (e) if row(jt) ≤ i then col(jt) ≥ col(it+1) for all t = 1, . . . , r − 1; (f) row(jt) ∈/ {i, i + 1} for at least one t = 1, . . . , r − 1.

Take such a monomiale ¯i1,j1 ··· e¯ir,jr . Let c be minimal such that there exists jt with col(jt) = c and row(jt) ∈/ {i, i + 1}, then take the maximal such t. Consider the component ofe ¯i1,j1 ··· e¯ir,jr in the cth tensor position. If row(jt) > i + 1, then by the choices of c and t, this component is of the form ue¯it,jt v where row(it) ≤ i+1 < row(jt) and the weight of v is some multiple of εi − εi+1. Similarly if row(jt) row(it+1) and the weight of v is some multiple of εi −εi+1. In either case, this component annihilates the vector mc ∈ L(Ac) by weight considerations.

Now let Lj be the irreducible U(gl 0 )-submodule of L(Aj) generated by the highest qj weight vector m+ for each j = 1, . . . , l − 1, embedding U(gl 0 ) into U(gl ) via ϕj. qj qj Similarly, let Ll be the U(gl 0 )-submodule of L(Al) generated by the vector wm+. ql Recall by the hypotheses in Lemma 7.11 that the tableau Al is column strict and REPRESENTATIONS OF SHIFTED YANGIANS 75

−1 −1 k ≥ w (k) < w (k + 1). It follows that the vector wm+ ∈ Ll is a highest weight vector for the action of U(gl 0 ) with e1,1 acting as (a + i − 1) and e2,2 − 1 acting as ql (b + i − 1), for some b < a ≥ ap. In particular Ll is also irreducible. So in our usual 0 notation the W (π )-module L1 ··· Ll is isomorphic to the tensor product L( ) ··· L( ) L(a1+i−1) ··· L(ap−1+i−1) L(a+i−1) c1+i−1 cq+i−1 b1+i−1 bp−1+i−1 b+i−1 for some b > a ≤ ap. Using the remaining hypotheses (i)–(iv) from Lemma 7.11, we apply Corollaries 7.2(i) and 7.5(i), or rather the slightly stronger versions of these corollaries described in Remarks 7.3 and 7.6, repeatedly to this tensor product working 0 from right to left to deduce that L1 ··· Ll is actually a highest weight W (π )- module generated by the highest weight vector m+ ⊗ · · · ⊗ m+ ⊗ wm+. Hence in particular, since skwm+ ∈ Ll, we get that 0 m+ ⊗ · · · ⊗ m+ ⊗ skwm+ ∈ W (π )(m+ ⊗ · · · ⊗ wm+). In view of Lemma 7.14, this completes the proof of Lemma 7.11. 7.4. Characters of standard modules. We wish to explain how to compute the Gelfand-Tsetlin characters of the standard modules {V (A) | A ∈ Col(π)} from (7.1). In view of (6.6) it suffices just to explain how to compute the character of V (A) when A consists just of a single column with entries a1 > ··· > an read from top to bottom, i.e. the character of the finite dimensional irreducible gln-module of highest weight a1ε1 + (a2 + 1)ε2 + ··· (an + n − 1)εn. Choose an arbitrary scalar c ∈ F so that an + n − 1 ≥ c. Then, (a1 − c, a2 + 1 − c, . . . , an + n − 1 − c) is a partition. Draw its Young diagram in the usual English way and define the residue of the box in the ith row and jth column to be (c + j − i). For example, if n = 3, c = 0 and (a1, a2 + 1, a3 + 2) = (5, 3, 2) then the Young diagram with boxes labelled by their residues is as follows 0 1 2 3 4 -1 0 1 -2 -1 Given a filling t of the boxes of this diagram with the integers {1, . . . , n} we associate the monomial n a +i−1−c Y i Y x(t) := xti,j ,c+j−i ∈ Zb[Pn] (7.3) i=1 j=1 where ti,j denotes the entry of t in the ith row and jth column and xi,a and yi,a are as in (6.7)–(6.8). Then we have that X chn V (A) = y1,cy2,c−1 ··· yn,c−n+1 × x(t) (7.4) t summing over all fillings t of the boxes of the diagram with integers {1, . . . , n} such that the entries are weakly increasing along rows from left to right and strictly increasing down columns from top to bottom. (Slightly more generally, for any m ≥ n, chm V (A) can be deduced from (7.4) by simply replacing each xi,a by xm−n+i,a and each yi,a by ym−n+i,a.) For example, suppose that the entries of A 76 JONATHAN BRUNDAN AND ALEXANDER KLESHCHEV are 1, −1, −2,..., 1 − n from top to bottom. Then V (A) is the n-dimensional natural representation Vn of gln of highest weight ε1. The possible fillings of the Young diagram are 1 , 2 ,. . . , n. Hence

chn Vn = x1,0 + x2,0 + ··· + xn,0. (7.5) The proof of the formula (7.4) is standard, based like the proof of Theorem 6.1 on branching V (A) from gln to gln−1. This time however the restriction is completely understood by the classical branching theorem for finite dimensional representations of gln, so everything is easy. The closest reference that we could find in the literature is [NT, Lemma 2.1]; see also [GT, C1] and [FM, Lemma 4.7] (the last of these references greatly influenced our choice of notation here). Let us make a few further comments. By (6.9), we have that  x x ··· x if a > 1 − i,  i,−i+1 i,−i+2 i,a−1 yi,a = 1 if a = 1 − i, (7.6) −1 −1 −1  xi,a xi,a+1 ··· xi,−i if a < 1 − i for each i = 1, . . . , n. Hence if the scalar c in (7.4) is an integer, i.e. if the representation V (A) is a rational representation of gln, then chn V (A) belongs to the subalgebra ±1 Z[xi,a | i = 1, . . . , n, a ∈ Z] of Zb[Pn]. Moreover, the character of a rational representation of gln in the usual sense can be deduced from its Gelfand-Tsetlin character by applying the algebra homomorphism ±1 ±1 Z[xi,a | i = 1, . . . , n, a ∈ Z] → Z[xi | i = 1, . . . , n], xi,a 7→ xi. (7.7) Finally, if one can choose the scalar c in (7.4) to be 0, i.e. if the representation V (A) is actually a polynomial representation of gln, then the formula (7.4) is especially simple since the leading monomial y1,cy2,c−1 ··· yn,c−n+1 is equal to 1. So the Gelfand- Tsetlin character of any polynomial representation of gln belongs to the subalgebra Z[xi,a | i = 1, . . . , n, a ∈ Z] of Zb[Pn]. 7.5. Grothendieck groups. Let us at long last introduce some categories of W (π)- modules. First, let M(π) denote the category of all finitely generated, admissible W (π)-modules. Obviously M(π) is an abelian category closed under taking finite direct sums. Note that the duality τ from (5.2) defines a contravariant equivalence M(π) → M(πt). Also, for any other pyramidπ ˙ with the same row lengths as π, the isomorphism ι from (3.23) induces an isomorphism M(π) → M(π ˙ ). Lemma 7.15. Every module in the category M(π) has a composition series. Proof. Copying the standard proof that modules in the usual category O have composition series, it suffices to prove the lemma for the Verma module M(A), A ∈ Row(π). In that case it follows because all the weight spaces of M(A) are finite dimensional, and moreover there are only finitely many irreducibles L(B) with the same central character as M(A) by Lemma 6.13. Hence, the Grothendieck group [M(π)] of the category M(π) is the free abelian group on basis {[L(A)]|A ∈ Row(π)}. By Theorem 6.7, we have that [M(A)] = [L(A)] plus an N-linear combination of [L(B)]’s for B < A. It follows that the Verma modules REPRESENTATIONS OF SHIFTED YANGIANS 77

{[M(A)] | A ∈ Row(π)} also form a basis for [M(π)]. By Theorem 5.9, the character map chn deﬁnes an injective map

chn :[M(π)] ,→ Zb[Pn]. (7.8) 0 00 0 00 Now suppose π = π ⊗π for pyramids π and π . Then the tensor product induces a multiplication µ :[M(π0)] ⊗ [M(π00)] → [M(π)]. (7.9) To see that this makes sense, the only difficulty is to see that the tensor product 0 00 0 0 00 00 M M of M ∈ M(π ) and M ∈ M(π ) is finitely generated. It suffices to check this for Verma modules. So take A0 ∈ Row(π0) and A00 ∈ Row(π00). Then, by 0 00 0 00 Corollary 6.3, we have that chn(M(A ) M(A )) = chn M(A) where A ∼ro A ⊗ A . 0 00 In view of Theorem 5.9 and Lemma 7.15, this shows that M(A ) M(A ) has a composition series, hence it is finitely generated as required. Moreover, µ([M(A0)] ⊗ [M(A00)]) = [M(A)]. (7.10) Recalling the decomposition (6.13), the category M(π) has the following block decomposition M M(π) = M(π, θ) (7.11) θ∈P where M(π, θ) is the full subcategory of M(π) consisting of objects all of whose composition factors are of central character θ; by convention, we set M(π, θ) = 0 if the coefficients of θ are not non-negative integers summing to N. Like in (4.28), we now restrict our attention just to modules with integral central characters: let M M0(π) := M(π, θ). (7.12)

θ∈P∞⊂P

The Grothendieck group [M0(π)] has the two natural basis {[M(A)] | A ∈ Row0(π)} and {[L(A)] | A ∈ Row0(π)}. π Next recall the deﬁnition of the UZ-module S (VZ) from (4.5). This is also a free abelian group, with two natural bases {MA | A ∈ Row0(π)} and {LA | A ∈ Row0(π)}. Deﬁne an isomorphism of abelian groups π k : S (VZ) → [M0(π)],MA 7→ [M(A)] (7.13) π for each A ∈ Row0(π). Under this isomorphism, the θ-weight space of S (VZ) corresponds to the block component [M(π, θ)] of [M0(π)], for each θ ∈ P∞. Moreover, the isomorphism is compatible with the multiplications µ arising from (4.10) and (7.9) in the sense that for every decomposition π = π0 ⊗ π00 the following diagram commutes: π0 π00 µ π S (VZ) ⊗ S (VZ) −−−−→ S (VZ)     k⊗ky yk (7.14) 0 00 [M0(π )] ⊗ [M0(π )] −−−−→ [M0(π)] µ Now we can formulate the following conjecture, which may be viewed as a more precise formulation in type A of [VD]. Note this conjecture is true if π consists of a single column; see Theorem 4.2. 78 JONATHAN BRUNDAN AND ALEXANDER KLESHCHEV

π Conjecture 7.16. For each A ∈ Row0(π), the isomorphism k : S (VZ) → [M0(π)] maps the dual canonical basis element LA to the class [L(A)] of the irreducible module L(A). In other words, for every A, B ∈ Row0(π), we have that

[M(A): L(B)] = Pd(ρ(A))w0,d(ρ(B))w0 (1), notation as in (4.8). Let us turn our attention to ﬁnite dimensional W (π)-modules. Let F(π) denote the category of all ﬁnite dimensional W (π)-modules, a full subcategory of the category M(π). Let F0(π) = F(π) ∩ M0(π). Like in (7.11)–(7.12), we have the block decompositions M F(π) = F(π, θ), (7.15) θ∈P M F0(π) = F(π, θ). (7.16)

θ∈P∞⊂P By Theorem 7.9, the Grothendieck group [F(π)] has basis {[L(A)] | A ∈ Dom(π)} coming from the simple modules. Hence [F0(π)] has basis {[L(A)] | A ∈ Dom0(π)}; we will refer to these L(A) ∈ F0(π) as the rational irreducible representations of W (π). π π Recall the subspace P (VZ) of S (VZ) from §4.3. Comparing (4.12) and (7.1) and π using (7.14), it follows that the map k : S (VZ) → [M0(π)] maps VA to [V (A)]. Hence π there is a well-deﬁned map j : P (VZ) → [F0(π)] such that VA 7→ [V (A)] for each A ∈ Col0(π). Moreover, the following diagram commutes: π π P (VZ) −−−−→ S (VZ)     jy yk (7.17)

[F0(π)] −−−−→ [M0(π)] where the horizontal maps are the natural inclusions. π Lemma 7.17. The map j : P (VZ) → [F0(π)],VA 7→ [V (A)] is an isomorphism of abelian groups. Proof. Arguing with the isomorphism ι, it suﬃces to prove this in the special case that π is left-justiﬁed. In this case, recall from (4.3) that R(A) denotes the row equivalence class of A ∈ Std0(π). By Theorem 7.12, for each A ∈ Std0(π) the standard module V (A) is a quotient of M(R(A)), hence we have that V (A) = L(R(A)) plus an N-linear combination of L(B)’s for B < A. It follows that {[V (A)] | A ∈ π Std0(π)} is a basis for [F0(π)]. Since the map j : P (VZ) → [F0(π)] maps the basis π {VA | A ∈ Std0(π)} of P (VZ) onto this basis of [F0(π)], it follows that j is indeed an isomorphism.

This lemma implies that {[V (A)] | A ∈ Std0(π)} gives a basis for the Grothendieck group [F0(π)]. In particular, this means that the Gelfand-Tsetlin character of any ±1 module in F0(π) belongs to the subalgebra Z[xi,a |i = 1, . . . , n, a ∈ Z] of Zb[Pn], since we know already that this is true for the standard modules. In the next lemma we extend this “standard basis” from [F0(π)] to all of the Grothendieck group [F(π)]. REPRESENTATIONS OF SHIFTED YANGIANS 79

Recall for the statement the definition of the set Std(π) from §4.1; its elements are k-equivalence classes rather than elements of Col(π). For any A ∈ Std(π), we define [V (A)] ∈ [F(π)] to be the class of the standard module parametrized by any representative of A. This definition is independent of the choice of representative, since if A, B ∈ Col(π) satisfy A k B then we obviously have that [V (A)] = [V (B)] in the Grothendieck group. (Actually we believe here that A k B implies V (A) =∼ V (B) but have been unable to prove this stronger statement except in special cases, e.g. it is true if π is left-justified by Theorem 7.13.) Lemma 7.18. The elements {[V (A)] | A ∈ Std(π)} form a basis for [F(π)]. In particular, the elements {[V (A)] | A ∈ Std0(π)} form a basis for [F0(π)].

Proof. We have already proved the second statement about F0(π). The ﬁrst statement follows from this and Theorem 7.13.

8. Finite dimensional representations

Throughout the chapter, we fix a pyramid π = (q1, . . . , ql) of height ≤ n. Conjec- π ture 7.16 immediately implies that the isomorphism j : P (VZ) → [F0(π)] from (7.17) π maps the dual canonical basis of the polynomial representation P (VZ) to the basis of the Grothendieck group [F0(π)] arising from irreducible modules. In this chapter, we will give an independent proof of this statement. Hence we can in principle compute the Gelfand-Tsetlin characters of all finite dimensional irreducible W (π)-modules. 8.1. Skryabin’s theorem. We begin by recalling the relationship between the algebra W (π) and the representation theory of g. Let Qχ denote the generalized Gelfand- Graev representation Qχ := U(g) ⊗U(m) Fχ (8.1) induced from the one dimensional m-module Fχ associated to χ. To avoid confusion, we will denote the element 1⊗1 ∈ Qχ by 1χ. Obviously the map U(p) → Qχ, u 7→ u1χ is a vector space isomorphism. As explained in the introduction of [BK5], W (π) is op naturally identified with the endomorphism algebra EndU(g)(Qχ) , so that w ∈ W (π) corresponds to the endomorphism of Qχ mapping u1χ 7→ uw1χ for any u ∈ U(g). So Qχ is a (U(g),W (π))-bimodule. Let W(π) denote the category of generalized Whittaker modules of type π, that is, the category of all g-modules on which (x − χ(x)) acts locally nilpotently for all x ∈ m. If V ∈ W(π) then the subspace m ∼ V := {v ∈ V | (x − χ(x))v = 0 for all x ∈ m} = HomU(g)(Qχ,V ) (8.2) is invariant under the action of W (π), hence ?m is a functor from W(π) to the category W (π) -mod of all left W (π)-modules. Conversely, Qχ⊗W (π)? is a functor from W (π) -mod to W(π). Skryabin’s theorem [Sk] asserts that the functors m ? and Qχ⊗W (π)? are quasi-inverse equivalences between the categories W(π) and W (π) -mod. Skryabin also proved that Qχ is a free right W (π)-module, and explained how to write down an explicit basis. Since we want to keep track of certain additional weight information, we must refine this basis slightly. Let c denote the subalgebra of g = glN 80 JONATHAN BRUNDAN AND ALEXANDER KLESHCHEV consisting of all diagonal matrices that centralize the nilpotent matrix e from (3.6). So c is spanned by the matrices X ej,j (8.3) 1≤j≤N row(j)=i for each i = 1, . . . , n. Clearly, c ⊂ W (π), so we can view any W (π)-module (for instance, the restriction of a U(p)- or U(g)-module) as a c-module too. Since (1) X X Di = e˜j,j = (ej,j + (n − qcol(j) − qcol(j)+1 − · · · − ql)), (8.4) 1≤j≤N 1≤j≤N row(j)=i row(j)=i the weight space decomposition of a W (π)-module with respect to c coincides with its usual weight space decomposition with respect to the subalgebra dn of W (π) spanned (1) (1) by D1 ,...,Dn . Now let b1, . . . , bk be a homogeneous basis for m such that bi of ∗ degree −di and of ad c-weight −γi ∈ c . The elements [b1, e],..., [bk, e] are again linearly independent, and [bi, e] is of degree (1 − di) and of ad c-weight −γi. Hence there exist elements a1, . . . , ak ∈ p such that ai is of degree (di − 1), of ad c-weight γi, and ([ai, bj], e) = (ai, [bj, e]) = δi,j. (8.5) We will refer to an ordered set of elements a1, . . . , ak chosen in this way as a Skryabin basis. Given such a choice, it follows from [Sk] that Qχ is a free right W (π)-module i1 i on basis (a1 − ρ(a1)) ··· (ak − ρ(ak)) k 1χ | i1, . . . , ik ≥ 0 , where ρ is the map from (3.11). Let p : Qχ W (π) (8.6) denote the unique right W (π)-module homomorphism with 1 if i = ··· = i = 0, (a − ρ(a ))i1 ··· (a − ρ(a ))ik 1 7→ 1 k 1 1 k k χ 0 otherwise.

Applying the functor ? ⊗W (π) M to p, we get induced a surjective linear map

pM : Qχ ⊗W (π) M M (8.7) for any W (π)-module M. Note that p, hence each pM , is actually a c-module homomorphism. In the ﬁrst lemma, for any vector space M, Hom(U(m),M) denotes the r space of all linear maps from U(m) to M which annihilate (Iχ) for r 0, recalling that Iχ is the two-sided ideal of U(m) generated by the elements {x − χ(x) | x ∈ m}. We view Hom(U(m),M) as an m-module with action deﬁned by (xf)(u) = f(ux) for x ∈ m, u ∈ U(m). Lemma 8.1. For M ∈ W (π) -mod, there is a natural m-module isomorphism

ϕM : Qχ ⊗W (π) M → Hom(U(m),M) deﬁned by ϕM (x)(u) = pM (ux) for x ∈ Qχ ⊗W (π) M and u ∈ U(m). Moreover ϕM is c-equivariant for the action of c on Hom(U(m),M) deﬁned by (hf)(u) = h(f(u)) − f([h, u]) for h ∈ c, u ∈ U(m) and f : U(m) → M. REPRESENTATIONS OF SHIFTED YANGIANS 81

Proof. The fact that ϕM is an m-module isomorphism follows from Skryabin’s proof [Sk]. Using the additional fact that pM is c-equivariant, one checks easily that ϕM is c-equivariant too.

8.2. Tensor identity. Let V be a finite dimensional g-module. Given any M ∈ W(π), it is clear that M ⊗ V (the usual tensor product of g-modules) also belongs to the category W(π). Thus ? ⊗ V gives an exact functor from W(π) to W(π). Using Skryabin’s equivalence of categories, we can transport this functor directly to the category W (π) -mod: for a W (π)-module M, let m M ~ V := ((Qχ ⊗W (π) M) ⊗ V ) . (8.8) This defines an exact functor ? ~ V : W (π) -mod → W (π) -mod. ∗ ∗ Writing V for the dual g-module, the functor ? ~ V is right adjoint to ? ~ V . To write down a canonical adjunction, let v1, . . . , vr be a basis for V and let f1, . . . , fr be the dual basis for V ∗. Then, the unit of the canonical adjunction is the map ∗ Pr η : Id → (? ~ V ) ~ V defined on a W (π)-module M by m 7→ i=1(1χ ⊗ m) ⊗ vi ⊗ fi, ∗ and the counit ε : (?~V )~V → Id is the restriction of the map (u1χ ⊗m)⊗f ⊗v 7→ f(v)u1χ ⊗ m. The following lemma is a more precise formulation of [Ly, Theorem 4.2].

Lemma 8.2. Let V be a finite dimensional U(g)-module on basis v1, . . . , vr. Define ∗ Pr ci,j ∈ U(g) from uvj = i=1 ci,j(u)vi for i, j = 1, . . . , r. Also fix a choice of Skryabin basis, determining projections p and pM as in as in (8.6)–(8.7). For any W (π)-module M, the restriction of the map pM ⊗ idV :(Qχ ⊗W (π) M) ⊗ V → M ⊗ V defines a natural c-module isomorphism ∼ χM,V : M ~ V −→ M ⊗ V. Pr The inverse isomorphism maps m ⊗ vj to i=1(xi,j1χ ⊗ m) ⊗ vi, where (xi,j)1≤i,j≤r is the invertible matrix with entries in U(p) determined uniquely by the properties

(i) p(xi,j1χ) = δi,j1; r X (ii) prχ([x, xi,j]) + ci,k(x)xk,j = 0 for all x ∈ m. k=1

Proof. Let Homm(U(m),V ) denote the space of all m-module homomorphisms from U(m) to V which annihilate some power of Iχ, where the left action of m on U(m) here is by (xf)(u) = f(u(χ(x) − x)) for x ∈ m, u ∈ U(m) and f : U(m) → V . Evaluation at 1U(m) defines an isomorphism Homm(U(m),V ) → V . Combining this with Lemma 8.1, we get natural isomorphisms of c-modules m ∼ m M ~ V = ((Qχ ⊗W (π) M) ⊗ V ) −→ (Hom(U(m),M) ⊗ V ) ∼ m ∼ ∼ −→ M ⊗ (Hom(U(m), F) ⊗ V ) −→ M ⊗ Homm(U(m),V ) −→ M ⊗ V. Let χM,V : M ~ V → M ⊗ V denote the composite isomorphism. Now assume that M = W (π), the regular W (π)-module. In this case, the inverse Pr image of 1⊗vj under the isomorphism χM,V must be of the form i=1(xi,j1χ ⊗1)⊗vi 82 JONATHAN BRUNDAN AND ALEXANDER KLESHCHEV for unique elements xi,j ∈ U(p). Moreover, computing explicitly from the definition of the map χM,V , each xi,j is determined by the property that ∗ p(uxi,j1χ) = ci,j(u )1 (8.9) for all u ∈ U(m), where ∗ : U(m) → U(m) here is the antiautomorphism with x∗ = χ(x) − x for each x ∈ m. Taking u = 1 in (8.9), we see that p(xi,j1χ) = δi,j1, as Pr in property (i). Moreover, i=1(xi,j1χ ⊗ 1) ⊗ vi is m-invariant, which is equivalent to property (ii). Conversely, one checks that properties (i) and (ii) imply (8.9), hence they also determine the xi,j’s uniquely. To see that the matrix (xi,j)1≤i,j≤r is invertible, we may assume without loss of generality that the basis v1, . . . , vr has the property that xvi ∈ Fv1 + ··· + Fvi−1 for each i = 1, . . . , r and x ∈ m, i.e. ci,j(x) = 0 for i ≥ j. But then, if one replaces xi,j by δi,j for all i ≥ j, the new elements still satisfy (8.9). Hence by uniqueness we must already have that xi,j = δi,j for i ≥ j, i.e. the matrix (xi,j)1≤i,j≤r is unitriangular, so it is invertible. Pr Finally return to general M. Property (ii) implies that i=1(xi,j1χ ⊗ m) ⊗ vi belongs to M ~ V for any m ∈ M. By functoriality, the image of this element under the isomorphism χM,V constructed in the first paragraph must equal m ⊗ vj. By property (i) this is also its image under the restriction of the map pM ⊗ idV . This shows that the isomorphism χM,V coincides with the restriction of pM ⊗idV in general, completing the proof.

Corollary 8.3. For any finite dimensional g-module V , the functor ? ~ V maps admissible W (π)-modules to admissible W (π)-modules. Now we can prove the following important “tensor identity”. Theorem 8.4. Let M be any p-module and V be a finite dimensional g-module on basis v1, . . . , vr. The restriction of the map (Qχ ⊗W (π) M) ⊗ V → M ⊗ V sending (u1χ⊗m)⊗v to um⊗v for each u ∈ U(p), m ∈ M, v ∈ V defines a natural isomorphism ∼ µM,V : M ~ V −→ M ⊗ V Pr of W (π)-modules. The inverse map sends m ⊗ vk to i,j=1(xi,j1χ ⊗ yj,km) ⊗ vi, where (xi,j)1≤i,j≤r is the matrix defined in Lemma 8.2 and (yi,j)1≤i,j≤r is the inverse matrix.

Proof. The map (Qχ ⊗W (π) M) ⊗ V → M ⊗ V, (x1χ ⊗ m) ⊗ v 7→ xm ⊗ v is a p- module homomorphism, hence its restriction µM,V is a W (π)-module homomorphism. To prove that µM,V is an isomorphism, just note by Lemma 8.2 that there is a well- Pr defined map M ⊗ V → M ~ V, m ⊗ vk 7→ i,j=1(xi,j1χ ⊗ yj,km) ⊗ vi. This is a two-sided inverse to χM,V . We should also make some comments about associativity of ~. Suppose that we are given another finite dimensional g-module V 0. For any W (π)-module M, the restriction of the linear map 0 0 (Qχ ⊗W (π) ((Qχ ⊗W (π) M) ⊗ V )) ⊗ V → (Qχ ⊗W (π) M) ⊗ V ⊗ V , 0 0 0 0 (u 1χ ⊗ ((u1χ ⊗ m) ⊗ v)) ⊗ v 7→ (u ((u1χ ⊗ m) ⊗ v)) ⊗ v REPRESENTATIONS OF SHIFTED YANGIANS 83 defines a natural isomorphism 0 0 aM :(M ~ V ) ~ V → M ~ (V ⊗ V ) (8.10) of W (π)-modules. If M is actually a p-module, it is straightforward to check that the following diagram commutes: 0 aM 0 (M ~ V ) ~ V −−−−→ M ~ (V ⊗ V )   id ⊗µ ⊗id 0  µ 0 Qχ M,V V y y M,V ⊗V (8.11) 0 0 (M ⊗ V ) ~ V −−−−−−→ M ⊗ V ⊗ V . µM⊗V,V 0 Finally, given a third finite dimensional module V 000, the following diagram commutes: a 0 00 M~V 0 00 (((M ~ V ) ~ V ) ~ V ) −−−−→ (M ~ V ) ~ (V ⊗ V )   id ⊗a ⊗id 00  a Qχ M V y y M (8.12) 0 00 0 00 (M ~ (V ⊗ V )) ~ V −−−−→ M ~ (V ⊗ V ⊗ V ). aM 8.3. Translation functors. In this section we extend the definition of the translation functors ei, fi on the category O0(π) from §4.4 to the category M0(π) from §7.5. We begin by defining an endomorphism

x :? ~ VN →? ~ VN (8.13) of the functor ? ~ VN : W (π) -mod → W (π) -mod. On a W (π)-module M, xM is the m endomorphism of M ~ VN = ((Qχ ⊗W (π) M) ⊗ VN ) defined by left multiplication by PN Ω = i,j=1 ei,j ⊗ ej,i ∈ U(g) ⊗ U(g). Here, we treat the g-module Qχ ⊗W (π) M as the PN first tensor position and VN as the second, so Ω((u1χ ⊗m)⊗v) means i,j=1(ei,ju1χ ⊗ m) ⊗ ej,iv. Next, we define an endomorphism s : (? ~ VN ) ~ VN → (? ~ VN ) ~ VN (8.14) of the functor (? ~ VN ) ~ VN : W (π) -mod → W (π) -mod. For the definition, we use the isomorphism a : (? ~ VN ) ~ VN →? ~ (VN ⊗ VN ) from (8.10). Then, s is the −1 composite a ◦s◦a, where s is the endomorphism of the functor ?~(VN ⊗VN ) defined b b m by letting sbM be the endomorphism of M ~(VN ⊗VN ) = ((Qχ ⊗W (π) M)⊗VN ⊗VN ) defined by left multiplication by Ω[2,3], i.e. Ω acting on the second and third tensor [2,3] 0 PN 0 positions so Ω ((u1χ ⊗ m) ⊗ v ⊗ v ) means i,j=1(u1χ ⊗ m) ⊗ ei,jv ⊗ ej,iv = 0 (u1χ ⊗m)⊗v ⊗v. Actually these definitions are just the natural translations through Skryabin’s equivalence of categories of the endomorphisms x and s from §4.4 of the functors ? ⊗ VN and ? ⊗ VN ⊗ VN on the category W(π). More generally, suppose that we are given d ≥ 1, and introduce the following d endomorphisms of the dth power (? ~ VN ) : for 1 ≤ i ≤ d and 1 ≤ j < d, let x := (1 )d−ix(1 )i−1, s := (1 )d−j−1s(1 )j−1. (8.15) i ?~VN ?~VN j ?~VN ?~VN There is an easier description of these endomorphisms. To formulate this, we exploit (d−1) d ⊗d the natural isomorphism a between the functor (?~VN ) and the functor ?~VN 84 JONATHAN BRUNDAN AND ALEXANDER KLESHCHEV defined by iterating the map a from (8.10). To be precise, set a(0) := 1 and then ?~VN inductively define a(d−1) for d > 1 by picking 1 ≤ k ≤ d − 1 and letting a(d−1) := a ◦ (a(d−k−1)a(k−1)), (8.16) ⊗(d−k) ⊗k ⊗k ⊗(d−k) where a here is the isomorphism a : (?~VN )◦(?~VN ) →?~(VN ⊗VN ) from (8.10). By the commutativity of the diagram (8.12), this definition is independent of the particular choice of k. Now for 1 ≤ i ≤ d and 1 ≤ j < d, let xbi and sbj denote the ⊗d endomorphisms of the functor ? ~ VN defined by left multiplication by the elements Pi [h,i+1] [j+1,j+2] h=1 Ω and Ω , respectively, notation as above. Then, we have that (d−1)−1 (d−1) (d−1)−1 (d−1) xi = a ◦ xbi ◦ a , sj = a ◦ sbj ◦ a . (8.17) Using this alternate description, the following identities are straightforward to check:

xixj = xjxi, (8.18)

sixi = xi+1si − 1, (8.19) 2 si = 1, (8.20) sisj = sjsi if |i − j| > 1, (8.21)

sisi+1si = si+1sisi+1. (8.22) These commutation relations, which are the deﬁning relations of the degenerate aﬃne Hecke algebra Hd as in [CR], can also be deduced directly from (4.35)–(4.37) using Skryabin’s equivalence of categories. ∗ Let us next bring the adjoint functor ? ~ VN into the picture. Actually, we want at the same time to restrict our attention just to the subcategory M0(π) of W (π) -mod, ∗ but we do not yet know that the functors ? ~ VN and ? ~ VN leave this subcategory invariant. Lemma 8.5. For A ∈ Row(π), we have that P (i) chn(M(A) ~ VN ) = B chn M(B) summing over all B obtained from A by adding 1 to one of its entries; ∗ P (ii) chn(M(A) ~ VN ) = B chn M(B) summing over all B obtained from A by subtracting 1 from one of its entries.

Proof. (i) It follows by Corollary 8.3 that M(A) ~ VN is admissible, hence it makes sense to consider its Gelfand-Tsetlin character. In view of Corollary 6.3, Theorem 5.9 and exactness of the functor ? ~ VN , we can replace M(A) by M(A1) ··· M(Al), where A1 ⊗ · · · ⊗ Al is some representative for A. But this is just the restriction from U(p) to W (π) of the inflation, M say, of a Verma module over the Levi subalgebra h. ∼ By Theorem 8.4, we have that M ~ VN = M ⊗ VN as W (π)-modules. Now observe that VN has a filtration as a p-module with factors Vq1 ,...,Vql being the natural modules of the blocks gl ,..., ql of h. Hence M ⊗ V has a filtration with factors q1 ql N M ⊗ Vqi . Now apply Lemma 4.3 to each of these factors in turn, to deduce that the Gelfand-Tsetlin character of M(A) ~ VN is equal to l X X chn(M(A1) ··· M(Ai−1) M(Bi) M(Ai+1) ··· M(Al)), i=1 Bi REPRESENTATIONS OF SHIFTED YANGIANS 85 where the second sum is over all Bi obtained from the column tableau Ai by adding 1 to one of its entries. Now we are done on applying Corollary 6.3 once more. (ii) Similar.

∗ Corollary 8.6. The functors ? ~ VN and ? ~ VN map objects in M0(π) to objects in M0(π). ∗ Proof. It suﬃces to check that M(A) ~ VN and M(A) ~ VN belong to M0(π), for each A ∈ Row0(π). We have already observed that M(A) ~ VN is admissible. By Lemma 8.5(i) and Theorem 5.9, it has a composition series and all of its composition factors are of integral central character. Hence it belongs to M0(π). Similarly so ∗ does M(A) ~ VN .

For θ ∈ P∞, let prθ : M0(π) → M(π, θ) be the projection functor along the decomposition (7.12). Explicitly, for a module M ∈ M0(π), we have that prθ(M) is the summand of M defined by (6.11), or prθ(M) = 0 if the coefficients of θ are not non-negative integers summing to N. In view of Corollary 8.6, it makes sense to define exact functors ei, fi : M0(π) → M0(π) by setting M ∗ ei := prθ+(εi−εi+1) ◦ (? ~ VN ) ◦ prθ, (8.23) θ∈P∞ M fi := prθ−(εi−εi+1) ◦ (? ~ VN ) ◦ prθ. (8.24) θ∈P∞

Note ei is right adjoint to fi, indeed, the canonical adjunction ﬁxed earlier between ∗ ? ~ VN and ? ~ VN induces a canonical adjunction between fi and ei. Similarly, fi is also right adjoint to ei. Moreover, applying Lemma 8.5 and taking blocks, we see for A ∈ Row0(π) and i ∈ Z that X [eiM(A)] = [M(B)] (8.25) B summing over all B obtained from A by replacing an entry equal to (i + 1) by an i, and X [fiM(A)] = [M(B)] (8.26) B summing over all B obtained from A by replacing an entry equal to i by an (i + 1); cf. (4.32)–(4.33). Hence if we identify the Grothendieck group [M0(π)] with the π UZ-module S (VZ) via the isomorphism (7.13), the maps on the Grothendieck group induced by the exact functors ei, fi coincide with the action of ei, fi ∈ UZ.

Lemma 8.7. For M ∈ M0(π), fiM coincides with the generalized i-eigenspace of xM ∈ EndW (π)(M ~ VN ).

Proof. It suﬃces to check this on a Verma module M(A) for A ∈ Row0(π). Say the entries of A in some order are a1, . . . , aN and let B be obtained from A by replacing 86 JONATHAN BRUNDAN AND ALEXANDER KLESHCHEV the entry at by at + 1, for some 1 ≤ t ≤ N. Recall the elements N (1) X ZN = (ei,i − N + i), i=1 (2) X ZN = ((ei,i − N + i)(ej,j − N + j) − ei,jej,i) i

Theorem 8.8. Let A ∈ Row0(π) and i ∈ Z. 0 k (i) Define εi(A) to be the maximal integer k ≥ 0 such that (ei) L(A) 6= 0. 0 Assuming εi(A) > 0, eiL(A) has irreducible socle and cosocle isomorphic 0 0 0 0 0 to L(˜ei(A)) for some e˜i(A) ∈ Row0(π) with εi(˜ei(A)) = εi(A) − 1. The 0 0 multiplicity of L(˜ei(A)) as a composition factor of eiL(A) is equal to εi(A), and all other composition factors are of the form L(B) for B ∈ Row0(π) with 0 0 εi(B) < εi(A) − 1. 0 k (ii) Define ϕi(A) to be the maximal integer k ≥ 0 such that (fi) L(A) 6= 0. 0 Assuming ϕi(A) > 0, fiL(A) has irreducible socle and cosocle isomorphic ˜0 ˜0 0 ˜0 0 to L(fi (A)) for some fi (A) ∈ Row0(π) with ϕi(fi (A)) = ϕi(A) − 1. The ˜0 0 multiplicity of L(fi (A)) as a composition factor of fiL(A) is equal to ϕi(A), and all other composition factors are of the form L(B) for B ∈ Row0(π) with 0 0 ϕi(B) < ϕi(A) − 1. Proof. This follows from [CR, Lemma 4.3] and [CR, Proposition 5.23], as in the first paragraph of the proof of Theorem 4.4. REPRESENTATIONS OF SHIFTED YANGIANS 87

Remark 8.9. This theorem gives a representation theoretic definition of a crystal 0 ˜0 0 0 structure (Row0(π), e˜i, fi , εi, ϕi, θ) on the set Row0(π). In §4.3, we gave a combinatorial definition of another crystal structure (Row0(π), e˜i, f˜i, εi, ϕi, θ) on the same underlying set. If Conjecture 7.16 is true, then it follows by (4.22)–(4.23) (as in the proof of Theorem 4.4) that these two crystal structures are in fact equal, that is, 0 0 0 ˜0 ˜ εi(A) = εi(A), ϕi(A) = ϕi(A), e˜i(A) =e ˜i(A) and fi (A) = fi(A) for all A ∈ Row0(π). Even without Conjecture 7.16, one can show using [CR, Lemma 4.3] and [BK, §4] 0 ˜0 0 0 ˜ that the two crystals (Row0(π), e˜i, fi , εi, ϕi, θ) and (Row0(π), e˜i, fi, εi, ϕi, θ) are at least isomorphic. However, there is a labelling problem here: without invoking Con- jecture 7.16 we do not know how to prove that the identity map on the underlying set Row0(π) is an isomorphism between the two crystals. An analogous labelling problem arises in a number of other situations; compare for example [BK2] and [BK3]. 8.4. Characters of finite dimensional irreducible representations. Following Kostant [Ko2] and Lynch [Ly], we now consider the following important functor from g-modules to W (π)-modules: for a g-module M, let D(M) := (M ∗)m, (8.27) the space of all generalized Whittaker vectors in the full dual space M ∗ (viewed as a g-module in the standard way). Making the obvious definition on morphisms, this defines a contravariant functor D : g -mod → W (π) -mod. The first lemma can be found already in Lynch’s thesis [Ly, Lemma 4.6]; Lynch attributes it to N. Wallach. Lemma 8.10. The functor D sends short exact sequences of g-modules that are finitely generated over m to short exact sequences of finite dimensional W (π)-modules. Proof. Note first for any g-module M that is finitely generated over m that ∼ ∗ ∼ D(M) = Homm(Fχ,M ) = Homm(M, F−χ), which is finite dimensional. Now consider instead the functor Db : g -mod → W(π) mapping an object M to the largest submodule of M ∗ that lies in the category W(π). The functor D is the composite of Db and Skryabin’s equivalence of categories ?m. So to complete the proof of the lemma, it is enough to show that the functor Db is exact on short exact sequences of g-modules that are finitely generated over m. Well, for a g-module M that is finitely generated over m, there are natural isomorphisms of vector spaces D(M) ∼ lim Hom (U(m)/Ir ,M ∗) b = −→ m χ ∼ lim Hom (M, (U(m)/Ir )∗) ∼ Hom (M, lim((U(m)/Ir )∗)), = −→ m χ = m −→ χ where the last isomorphism here depends on the finite generation of M to see that the r ∗ direct limit stabilizes after finitely many terms. Now we observe that lim((U(m)/Iχ) ) ∗ r −→ is the set of all f ∈ U(m) such that f(Iχ) = 0 for r 0, with m-action given by (xf)(u) = −f(xu) for x ∈ m, u ∈ U(m) and f : U(m) → F. This is isomorphic to ∗ r the set of all f ∈ U(m) such that f(I−χ) = 0 for r 0, with m-action given instead by (xf)(u) = f(ux). Hence by [Sk, Assertion 2], we get that lim((U(m)/Ir )∗) is an −→ χ injective m-module. The desired exactness follows. 88 JONATHAN BRUNDAN AND ALEXANDER KLESHCHEV

Recall the definition of the parabolic category O(π) from the end of §4.4. By the PBW theorem, the parabolic Verma modules in O(π) are finitely generated over m, hence so are the simple modules. Since every object in O(π) is of finite length, it follows that all M ∈ O(π) are finitely generated over m. Hence the functor D restricts to a well-defined exact contravariant functor D : O(π) → F(π), (8.28) where F(π) is the category of all finite dimensional W (π)-modules as in §7.5. The next two lemmas give some further properties of this functor D. Lemma 8.11. For any p-module M, there is a natural isomorphism of W (π)-modules ∗ between D(U(g) ⊗U(p) M) and the restriction of the dual p-module M from U(p) to W (π). Proof. We have that ∼ ∗ ∼ D(U(g) ⊗U(p) M) = Homm(Fχ, (U(g) ⊗U(p) M) ) = Homm(U(g) ⊗U(p) M, F−χ) ∼ ∗ = Homm(U(m) ⊗ M, F−χ) = Hom(M, F) = M . It just remains to check that the action of W (π) corresponds under these isomorphisms to the restriction of the usual U(p)-action on the dual space M ∗.

Lemma 8.12. For a g-module M and a finite dimensional g-module V , there is a ∗ natural isomorphism of W (π)-modules between D(M ⊗ V ) and D(M) ~ V Proof. We obviously have that (M ⊗ V ∗)∗ =∼ M ∗ ⊗ V . Hence, D(M ⊗ V ∗) =∼ (M ∗ ⊗ V )m. We can replace M ∗ here by its largest submodule belonging to the category W(π), which by Skryabin’s equivalence of categories is isomorphic to Qχ ⊗W (π) ∗ m ∗ ∼ m ((M ) ). Hence, D(M ⊗ V ) = ((Qχ ⊗W (π) D(M)) ⊗ V ) = D(M) ~ V . Vπ π We are ready to categorify the linear map F : (VZ) → S (VZ) from (4.11). t Letting π denote the transpose pyramid (ql, . . . , q1), define an exact functor γ : O(π) → O(πt) (8.29) mapping a module M to the same vector space but with new action defined by the −1 t twist x · m := −(wπ xwπ) m, for x ∈ g and m ∈ M. Here, wπ is the permutation matrix defined in §3.5. Also recall the duality τ from (5.2), which gives us an exact contravariant functor τ : F(πt) → F(π). Let D : O(πt) → F(πt) be the exact contravariant functor from (8.28) with π replaced by πt, and consider the composite functor F := τ ◦ D ◦ γ : O(π) → F(π). (8.30) This is an exact covariant functor. In the next lemma, we adapt Lemma 8.11 to this functor F . For the statement, recall the definitions of the 1-dimensional p-modules Fρ from (3.11) and Fρ¯ from (3.25). Lemma 8.13. For any finite dimensional p-module M, there is a natural isomorphism of W (π)-modules between F (U(g) ⊗U(p) (Fρ¯ ⊗ M)) and the restriction of the p-module Fρ ⊗ M from U(p) to W (π). REPRESENTATIONS OF SHIFTED YANGIANS 89

−1 t Proof. Let γ : U(g) → U(g) be the automorphism sending x 7→ −(wπ xwπ) and let S : U(g) → U(g) be the antiautomorphism sending x 7→ −x, for each x ∈ g. Recall that τ : W (π) → W (πt) is the restriction of the map τ : U(p) → U(pτ ) from (3.26). The composite S ◦ γ ◦ τ : U(p) → U(p) maps x ∈ p to x + (ρ − ρ¯)(x). By Lemma 8.11 and the deﬁnition of F , F (U(g) ⊗U(p) (Fρ¯ ⊗ M)) is naturally isomorphic to Fρ¯ ⊗ M viewed as a W (π)-module with action x · m = (S ◦ γ ◦ τ)(x)m for each x ∈ W (π), m ∈ Fρ¯ ⊗ M. This is the restriction of the U(p)-module Fρ ⊗ M.

Corollary 8.14. For A ∈ Col(π), we have that F (N(A)) =∼ V (A). Proof. This is immediate from Lemma 8.13 and the deﬁnitions of N(A) and V (A) from (4.38) and (7.1). For a while now, we will restrict our attention to integral central characters. Note Corollary 8.14 implies that the functor F restricts to a well-deﬁned exact functor

F : O0(π) → F0(π). (8.31)

We also write F :[O0(π)] → [F0(π)] for the induced map at the level of Grothendieck Vπ groups. Recall also the isomorphism i : (VZ) → [O0(π)],NA 7→ [N(A)] from the π proof of Theorem 4.5, and the isomorphisms j : P (VZ) → [F0(π)],VA 7→ [V (A)] and π k : S (VZ) → [M0(π)],MA 7→ [M(A)] from (7.17). We observe that the following diagram commutes: Vπ F π π (VZ) −−−−→ P (VZ) −−−−→ S (VZ)       iy yj yk (8.32)

[O0(π)] −−−−→ [F0(π)] −−−−→ [M0(π)], F where the top F is the map from (4.11) and the two unnamed maps are the natural inclusions. To see this, we already checked in (7.17) that the right hand square commutes, and Corollary 8.14 is exactly what is needed to check that the left hand square does too. ∼ τ ∼ τ Lemma 8.15. There are isomorphisms of functors F ◦ei = ei ◦F and F ◦fi = fi ◦F , τ τ where ei := τ ◦ ei ◦ τ and fi := τ ◦ fi ◦ τ. ∼ Proof. For M ∈ O0(π), there are natural isomorphisms γ(M ⊗ VN ) = γ(M) ⊗ ∼ ∗ γ(VN ) = γ(M) ⊗ VN . Hence applying Lemma 8.12, there is an isomorphism of ∼ ∼ τ functors D ◦γ ◦(?⊗VN ) = (?~VN )◦D ◦γ. Equivalently, F ◦(?⊗VN ) = (?~VN ) ◦F. The lemma for fi follows because in view of Corollary 8.14 we already know that both functors map O(π, θ) to F(π, θ) for each θ ∈ P∞. The proof for ei is similar.

Remark 8.16. With substantially more work, one can show that the functors ei and τ ∼ τ ∼ fi commute with τ, i.e. ei = ei and fi = fi in the notation of this lemma. In fact, the functor F is a morphism of sl2-categorifications in the sense of [CR]. We will not need these stronger statements here. Now we are ready to invoke Theorem 4.5, or rather, to invoke the Kazhdan-Lusztig conjecture, since Theorem 4.5 was a direct consequence of it. For the statement of the 90 JONATHAN BRUNDAN AND ALEXANDER KLESHCHEV following theorem, recall the definition of the bijection R : Std0(π) → Dom0(π) from §4.3; in the case that π is left-justified the rectification R(A) of a standard π-tableau A simply means its row equivalence class.

Theorem 8.17. For A ∈ Col0(π), we have that L(R(A)) if A is standard, F (K(A)) =∼ 0 otherwise. Proof. Note that it suffices to prove the theorem in the special case that π is left- Vπ justified. Indeed, in view of Theorem 4.5, the properties of the map F : (VZ) → π P (VZ) and the commutativity of the left hand square in (8.32), the theorem follows if we can show that j(LA) = [L(A)] for every A ∈ Dom0(π). This last statement is independent of the particular choice of π, thanks to the existence of the isomorphism ι. So assume from now on that π is left-justified. Using Theorem 4.5 and the commutativity of the left hand square in (8.32) again, we know already for A ∈ Col0(π) that F (K(A)) 6= 0 if and only if A ∈ Std0(π). Let Aπ ∈ Col0(π) be the ground state, with all entries on row i equal to (1 − i). Since the crystal (Std0(π), e˜i, f˜i, εi, ϕi, θ) is connected, it makes sense to define the height of A ∈ ˜ Std0(π) to be the minimal number of applications of the operatorse ˜i, fi (i ∈ Z) needed ∼ to map A to Aπ. We proceed to prove that F (K(A)) = L(R(A)) for A ∈ Std0(π) by induction on height. For the base case, observe that no other elements of Col0(π) have the same content as Aπ, hence N(Aπ) = K(Aπ). Similarly, V (R(Aπ)) = L(R(Aπ)). ∼ Hence by Corollary 8.14, we have that F (K(Aπ)) = L(R(Aπ)). Now for the induction step, take B ∈ Std0(π) of height > 0. We can write B as eithere ˜i(A) or as f˜i(A), where A ∈ Std0(π) is of strictly smaller height. We will assume that the first case holds, i.e. that B =e ˜i(A), since the argument in the second case is entirely similar. By the induction hypothesis, we know already that F (K(A)) =∼ L(R(A)). We need to show that F (K(B)) =∼ L(R(B)). Note by Corollary 8.14 that F (N(B)) =∼ V (B), and by exactness of the functor F , we know that F (K(B)) is a non-zero quotient of V (B). Since B ∈ Std0(π), Theorem 7.12 shows that V (B) is a highest weight module of type R(B). Hence F (K(B)) is a highest weight module of type R(B) too. Also K(B) is both a quotient and a submodule of eiK(A) by Theorem 4.5. Hence by Lemma 8.15, F (K(B)) is both ∼ τ a quotient and a submodule of F (eiK(A)) = ei L(R(A)). In particular, L(R(B)) is a τ quotient of ei L(R(A)) and F (K(B)) is a non-zero submodule of it. τ Finally, we know by Theorem 8.8 that the socle and cosocle of ei L(R(A)) are irreducible and isomorphic to each other. Since we know already that L(R(B)) is a quo- τ τ tient of ei L(R(A)), it follows that the socle of ei L(R(A)) is isomorphic to L(R(B)). τ Since F (K(B)) embeds into ei L(R(A)), this means that F (K(B)) has irreducible socle isomorphic to L(R(B)) too. But F (K(B)) is a highest weight module of type R(B). These too statements together imply that F (K(B)) is indeed irreducible.

π Corollary 8.18. The isomorphism j : P (VZ) → [F0(π)] maps LA to [L(A)] for each A ∈ Dom0(π). Hence, for A ∈ Col0(π) and B ∈ Std0(π) X `(A,C) [V (A): L(R(B))] = (−1) Pd(γ(C))w0,d(γ(B))w0 (1), C∼coA REPRESENTATIONS OF SHIFTED YANGIANS 91 notation as in (4.14). Proof. The ﬁrst statement follows from the theorem and the commutativity of the diagram (8.32). The second two statement then follows by (4.14).

Corollary 8.19. For A ∈ Dom0(π) and i ∈ Z, the following properties hold: (i) If εi(A) = 0 then eiL(A) = 0. Otherwise, eiL(A) is an indecomposable module with irreducible socle and cosocle isomorphic to L(˜ei(A)). (ii) If ϕi(A) = 0 then fiL(A) = 0. Otherwise, fiL(A) is an indecomposable module with irreducible socle and cosocle isomorphic to L(f˜i(A)). Proof. Argue using (4.22)–(4.23), Theorem 8.8 and Corollary 8.18, like in the proof of Theorem 4.4. Since the Gelfand-Tsetlin characters of standard modules are known, one can now in principle compute the characters of the finite dimensional irreducible W (π)- modules with integral central character, by inverting the unitriangular square submatrix ([V (A): L(R(B))])A,B∈Std0(π) of the decomposition matrix from Corollary 8.18. Using Theorem 7.13 too, one can deduce from this the characters of arbitrary finite dimensional irreducible W (π)-modules. All the other combinatorial results just formulated can also be extended to arbitrary central characters in similar fashion. We just record here the extension of Theorem 8.17 itself to arbitrary central characters. Corollary 8.20. For A ∈ Col(π), we have that L(R(A)) if A is standard, F (K(A)) =∼ 0 otherwise. Thus, the functor F : O(π) → F(π) sends irreducible modules to irreducible modules or to zero. Proof. This is a consequence of Corollary 8.18, Corollary 8.14, Theorem 7.13 and [BG, Proposition 5.12]. The final result gives a criterion for the irreducibility of the standard module V (A), in the spirit of [LNT]. Note as a special case of this corollary, we recover the main result of [M4] concerning Yangians. Following [LZ, Lemma 3.8], we say that two sets A = {a1, . . . , ar} and B = {b1, . . . , bs} of numbers from F are separated if (a) r < s and there do not exist a, c ∈ A − B and b ∈ B − A such that a b > c > d; (c) r > s and there do not exist c ∈ A − B and b, d ∈ B − A such that b > c > d.

Say that a π-tableau A ∈ Col(π) is separated if the sets Ai and Aj of entries in the ith and jth columns of A, respectively, are separated for each 1 ≤ i < j ≤ l. Theorem 8.21. For A ∈ Col(π), the standard module V (A) is irreducible if and only if A is separated, in which case it is isomorphic to L(B) where B ∈ Dom(π) is the row equivalence class of A.

Proof. Using Theorem 7.13, the proof reduces to the special case that A ∈ Col0(π). In that case, we apply [LZ, Theorem 1.1] and the main result of Leclerc, Nazarov and 92 JONATHAN BRUNDAN AND ALEXANDER KLESHCHEV

Thibon [LNT, Theorem 31]; see also [Ca]. These references imply that VA is equal to LB for some B ∈ Dom0(π) if and only if A is separated. Actually, the references cited only prove the q-analog of this statement, but it follows at q = 1 too by the positivity of the structure constants from [B, Remark 24]; see the argument from the proof of [LNT, Proposition 15]. By Theorem 8.17, this shows that V (A) is irreducible if and only if A is separated. Finally, when this happens, we must have that V (A) =∼ L(B) where B is the row equivalence class of A, since V (A) always contains a highest weight vector of that type.

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Department of Mathematics, University of Oregon, Eugene, OR 97403, USA E-mail address: [email protected], [email protected]