Lie Algebras of Differential Operators and D-Modules by Dmitry Donin A

Lie algebras of differential operators and D-modules

Dmitry Donin

A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Graduate Department of Mathematics University of Toronto

Lie algebras of diﬀerential operators and D-modules

Dmitry Donin

Doctor of Philosophy

Graduate Department of Mathematics

University of Toronto

2008

In our thesis we study the algebras of differential operators in algebraic and geometric terms. We consider two problems in which the algebras of differential operators naturally arise. The first one deals with the algebraic structure of differential and pseudodifferential operators. We define the Krichever-Novikov type Lie algebras of differential operators and pseudodifferential symbols on Riemann surfaces, along with their outer derivations and central extensions. We show that the corresponding algebras of meromorphic differential operators and pseudodifferential symbols have many invariant traces and central extensions, given by the logarithms of meromorphic vector fields. We describe which of these extensions survive after passing to the algebras of operators and symbols holomorphic away from several fixed points. We also describe the associated Manin triples, emphasizing the similarities and differences with the case of smooth symbols on the circle.

The second problem is related to the geometry of differential operators and its connection with representations of semi-simple Lie algebras. We show that the semiregular module, naturally associated with a Z-graded semi-simple complex Lie algebra g, can be realized in geometric terms, using the Brion’s construction of degeneration of the diagonal in the square of the flag variety of g. Namely, we consider the Beilinson-Bernstein localization of the semiregular module and show that it is isomorphic to the D-module obtained by applying the Emerton-Nadler-Vilonen geometric Jacquet functor to the D- module of distributions on the square of the flag variety with support on the diagonal.

ii ACKNOWLEDGEMENTS

I would like to express profound gratitude to my advisors, Prof. Boris Khesin and

Prof. Sergey Arkhipov for their invaluable support, encouragement, supervision and useful suggestions throughout this research work. I am highly thankful to Prof. Pierre

Milman for his valuable suggestions and supervision during the whole course of my study.

The moral support and continuous guidance of these people enabled me to complete my work successfully.

I also would like to express my gratitude to Graduate Coordinator of the Department of Mathematics, Ida Bulat, for her continuous attention and kindness during my whole stay in the University of Toronto.

iii Contents

1 Introduction 1

1.1 Krichever-Novikov type algebras ...... 2

1.2 A geometric realization of semiregular modules ...... 6

I Krichever-Novikov algebras of diﬀerential operators 12

2 History and Context 13

3 Meromorphic pseudodiﬀerential symbols on Riemann surfaces 16

3.1 The algebras of meromorphic diﬀerential and pseudodiﬀerential symbols . 16

3.2 Outer derivations of pseudodiﬀerential symbols ...... 18

3.3 The traces ...... 21

3.4 The logarithmic 2-cocycles ...... 23

3.5 Proof of the theorem on outer derivations ...... 28

3.6 The double extension of the meromorphic symbols and Manin triples . . 29

4 Holomorphic pseudodiﬀerential symbols on Riemann surfaces 31

4.1 The Krichever-Novikov algebra ...... 31

4.2 Holomorphic diﬀerential operators and pseudodiﬀerential symbols . . . . 32

4.3 Holomorphic pseudodiﬀerential symbols and the spaces of densities . . . 34

4.4 The density of holomorphic symbols in the smooth ones ...... 35

iv 4.5 The property of generalized grading ...... 37

4.6 Cocycles and extensions ...... 39

4.7 Manin triples for holomorphic pseudodiﬀerential symbols ...... 44

5 Appendix A: Poisson–Lie groups, Lie bialgebras, and Manin triples 46

6 Appendix B: The Krichever-Novikov basis 48

6.1 The Riemann-Roch theorem for line bundles and the Krichever-Novikov

basis ...... 48

6.2 Construction of the Krichever-Novikov basis ...... 50

6.3 Relation to the Lie algebra of vector ﬁelds on the circle ...... 51

II A geometric realization of semiregular modules 53

7 Notations and Preliminaries 54

8 Categories of g-modules 56

8.1 Category O ...... 56

8.2 Verma modules ...... 58

8.3 Action of the Center and Harish-Chandra’s Theorem ...... 60

8.4 Decomposition of the category O with respect to the center and duality . 61

8.5 Projective objects in O, standard ﬁltrations, and BGG reciprocity . . . . 63

8.6 Harish-Chandra modules and the functor H ...... 65

9 Semiregular modules 70

9.1 Semiregular module ...... 70

9.2 Twisting functors ...... 73

9.3 Example ...... 75

10 The Jacquet functor 77

v 11 Algebraic results 81

11.1 Semiregular module as the image of the Jacquet functor ...... 81

11.2 Filtrations ...... 86

12 Geometric results 90

12.1 The De Concini-Procesi compactiﬁcation ...... 90

12.2 Degeneration of the diagonal in the square of the ﬂag variety ...... 91

12.3 Example ...... 92

12.4 The geometric Jacquet functor ...... 93

12.5 A geometric realization of semiregular modules ...... 95

13 Appendix A: Representable functors and Yoneda lemma 97

14 Appendix B: D-modules and the Beilinson-Bernstein localization 99

14.1 Basics on D-modules ...... 99

14.2 Equivariant D-modules and the Beilinson-Bernstein localization . . . . . 102

15 Appendix C: The nearby cycles functor 105

15.1 The Kashiwara-Malgrange ﬁltration ...... 105

15.2 Specializable modules and nearby cycles ...... 106

vi Chapter 1

Introduction

Diﬀerential operators appear naturally in many constructions of representation theory. In this thesis we explore certain algebraic and geometric constructions related to them and focus on the following two scopes of problems emphasizing these two sides respectively.

In the first part of the thesis we study the Krichever-Novikov type Lie algebras of differential operators, as well as algebraic structures, related to them. We describe generalizations of the Krichever-Novikov algebras of holomorphic vector fields on a punctured

Riemann surface to the case of differential operators and pseudodifferential symbols on such a surface. These algebras have a rich source of cocycles, traces, central extensions, and Manin triples. They can be defined in both the meromorphic and holomorphic contexts, have special density and almost grading properties.

The second part of the thesis deals with the geometric side of diﬀerential operators and representations of semi-simple Lie algebras. We study semiregular representations associated with semi-simple Lie algebras in geometric terms. Using the Brion’s construction of a degeneration of the diagonal in the square of the ﬂag variety of the Lie algebra, we realize it as a certain degeneration of D-modules which is obtained by applying the

Emerton-Nadler-Vilonen geometric Jacquet functor to the D-module of distributions on the square of the ﬂag variety with support on the diagonal.

1 Chapter 1. Introduction 2

1.1 Krichever-Novikov type algebras

The Krichever-Novikov algebras are the centrally extended Lie algebras of meromorphic vector ﬁelds on a Riemann surface Σ, which are holomorphic away from several ﬁxed points [35, 36], see also [42, 47]. They are natural generalizations of the Virasoro algebra, which corresponds to the case of Σ = CP 1 with two punctures. A central extension of the

Lie algebras of vector fields on a given Riemann surface is defined by the Gelfand-Fuchs cocycle in a fixed projective structure (that is a class of coordinates related by projective transformations).

In our thesis we deal with two generalizations of the Krichever-Novikov (KN) algebras. The first one is the Lie algebras of all meromorphic differential operators and pseudodifferential symbols on a Riemann surface, while the second one is the Lie algebras of meromorphic differential operators and pseudodifferential symbols which are holomorphic away from several fixed points. The main tool which we employ is fixing a reference meromorphic vector field instead of a projective structure on Σ. It turns out that such a choice allows one to write more explicit formulas for the corresponding cocycles, both for the Krichever-Novikov algebra of vector fields and for its generalizations to operators and symbols.

Several features of these algebras of meromorphic symbols make them different from their smooth analogue, the algebra of pseudodifferential symbols with smooth coefficients on the circle. First of all, this is the existence of many invariant traces on the former algebras: one can associate such a trace to every point on the surface. Furthermore, we show that the logarithm log X of any meromorphic pseudodifferential symbol X defines an outer derivation of the Lie algebra of meromorphic symbols:

Theorem 3.8. All derivations of the algebra of meromorphic pseudodifferential symbols defined by log X for any meromorphic pseudodifferential symbol X are outer and equivalent to a linear combination of derivations given by log Dv for meromorphic vector fields v. Chapter 1. Introduction 3

In turn, the combination of invariant traces and outer derivations produces a variety of independent non-trivial 2-cocycles on the Lie algebras of meromorphic pseudodiﬀerential symbols and diﬀerential operators:

Theorem 3.18. All the logarithmic 2-cocycles cv are nontrivial and non-cohomologous to each other on the algebra of meromorphic pseudodifferential symbols for different choices of meromorphic fields v 6= 0.

Note that the above mentioned scheme of generating 2-cocycles in the meromorphic case, which involve log X for any meromorphic pseudodiﬀerential symbol X, provides a natural unifying framework for the existence of two independent cocycles (generated by log ∂/∂x and log x) in the smooth case, cf. [34, 32].

The second type of algebras under consideration, those of holomorphic diﬀerential operators and pseudodiﬀerential symbols, are more direct generalizations of the Krichever-

Novikov algebras of holomorphic vector ﬁelds on a punctured Riemann surface. For them we prove the density and ﬁltered generalized grading properties, similarly to the corresponding properties of the KN algebras [35, 36]:

Theorem 4.6. The restriction of the algebra of holomorphic pseudodifferential symbols to a sufficiently small circle around a puncture in the surface Σ is dense in the algebra of smooth pseudodifferential symbols on this circle.

Furthermore, one can adapt the notion of a local cocycle suggested in [35] to the

filtered algebras of (pseudo)differential symbols. It turns out that all logarithmic cocycles become linearly dependent when we confine ourselves to local cocycles on holomorphic differential operators (HDO). On the other hand, for holomorphic pseudodifferential symbols (HΨDS) the local cocycles are shown to form a two-dimensional space:

Theorem 4.15. The cohomology space of logarithmic local 2-cocycles on the Lie algebra

HDO is 1-dimensional and on the algebra HΨDS is 2-dimensional.

We put the exposition in a form which emphasizes the similarities with and differences from the algebras of (pseudo)differential symbols with smooth coefficients on the circle, Chapter 1. Introduction 4 developed in [22, 32]. In many respects the algebras of holomorphic symbols extended by local 2-cocycles turn out to be similar to their smooth counterparts. On the other hand, by giving up the condition of locality, one obtains higher-dimensional extensions of the

Lie algebras of holomorphic symbols by means of the 2-cocycles related to diﬀerent paths on the surface. This way one naturally comes to holomorphic analogues of the algebras of “smooth symbols on graphs”, which also have central extensions given by 2-cocycles on diﬀerent loops in the graphs.

Let us outline the structure of Part I of the thesis.

In Section 3.1 of Chapter 3 we introduce the algebras of meromorphic differential operators (MDO) and meromorphic pseudodifferential symbols (MΨDS) on a Riemann surface Σ. We define these algebras as certain subalgebras in the algebra of formal series with meromorphic coefficients on powers of Dv, the operator of the Lie derivative along a meromorphic vector field v on Σ. The associative and Lie structures on these algebras are defined by the natural extension of the Leibnitz rule. For different choices of the meromorphic vector field v the corresponding algebras are naturally isomorphic.

In Section 3.2 we introduce an important class of derivations of the algebra MΨDS by taking the logarithms log X of an arbitrary pseudodifferential symbol X. It turns out that all these derivations are outer and equivalent to a linear combination of derivations of the form log Dv for various choice of meromorphic fields v on Σ (Theorem 3.8). In Section 3.4 we define the logarithmic cocycles of the algebras MDO and MΨDS.

These are the generalizations of the Gelfand-Fuchs cocycle on the algebra of meromorphic vector ﬁelds on the Riemann sphere, holomorphic away from the south and the north poles (0 and ∞). We associate the logarithmic cocycle to any pseudodiﬀerential symbol

X and describe equivalent ones (Theorem 3.8, Corollary 3.17).

In Section 3.6 we show that the algebra MΨDS and its double extension M^ΨDS

(by means of both the central term and the outer derivation) can be naturally split with respect to a certain pairing on symbols. This way we obtain the Manin triples associated Chapter 1. Introduction 5 with both of these algebras (Theorem 3.22). As a corollary we obtain Lie bialgebra structures on the subalgebras of integral symbols in MΨDS and M^ΨDS, which are dual to the algebras of diﬀerential operators.

In Chapter 4 we focus on holomorphic symbols, which are allowed to have poles in two distinguished points on Σ, called punctures and study the corresponding algebras of holomorphic diﬀerential operators (HDO) and holomorphic pseudodiﬀerential symbols

(HΨDS) (see Section 4.2). These algebras contain the Krichever-Novikov algebra of holomorphic vector ﬁelds as a Lie subalgebra.

In Section 4.3 we relate the graded structure on the algebra HΨDS to the spaces of holomorphic densities. The algebra HΨDS is naturally ﬁltered by the degree of symbols and we show that each of the associated graded spaces of HΨDS is naturally isomorphic to the corresponding space of holomorphic densities (Proposition 4.4).

In Section 4.4 we prove an analog of the density theorem for the KN algebra ([35, 36]) for the algebra HΨDS (Theorem 4.6). It asserts that if we restrict the symbols on Σ to a small contour encircling the puncture, we get a dense subalgebra in the algebra of smooth symbols on this contour.

In Section 4.5 we establish the generalized grading property of the algebras HDO and HΨDS (Theorem 4.9).

In Section 4.6 we study 2-cocycles and the corresponding extensions of the algebra of holomorphic symbols. We recall the definition of a local 2-cocycle on a generalized graded Lie algebra. The 1-dimensional central extension corresponding to a local 2- cocycle is also generalized graded. Since the algebras HDO and HΨDS do not have natural generalized grading property, but only filtered generalized graded, we need to adapt the definition of a local 2-cocycle to this setting, see Definition 4.14. We show

(Theorem 4.15) that the cohomology space of local logarithmic 2-cocycles on the algebra

HDO is 1-dimensional, while on the algebra HΨDS it is 2-dimensional. Finally, in

Section 4.7 we discuss obstructions to the existence of a natural Manin triple in the case Chapter 1. Introduction 6 of holomorphic symbols.

The main results of this part of the thesis are published in [19].

1.2 A geometric realization of semiregular modules

Semiregular modules ﬁrst appeared in connection with the semi-inﬁnite cohomology theory, originated by Feigin and then developed by Voronov [51] and Arkhipov [3, 4] in a formal homological setting. In our thesis we study semiregular modules in algebraic and geometric terms.

A semiregular module is a certain module S over a semi-simple graded Lie algebra g, which plays the role of the universal enveloping algebra in the semi-inﬁnite cohomology theory and interpolates between the regular and coregular modules.

Throughout this part of the thesis we work both in algebraic and geometric context, so that our main result can be subdivided into two parts. An algebraic part has its aim to prove the following assertion:

Theorem 11.6. The semiregular module S can be realized as the image of a certain

Harish-Chandra module U∆ under the classical Jacquet functor. The proof of Theorem 11.6 relies on the technique of representation theory of semi- simple complex Lie algebras. In Chapters 8, 9, 10 we deﬁne all the necessarily ingredi- ents for this, which embrace the following fundamental notions: the Bernstein-Gelfand-

Gelfand category O, the category of Harish-Chandra modules, the Arkhipov-Soergel-

Mazorchuk twisting functors and the classical Jacquet functor. We brieﬂy review all these concepts below in this chapter.

Our geometric discussions lead to the proof of the following result:

Theorem 12.6. The Beilinson-Bernstein localization of the semiregular module is obtained by applying the Emerton-Nadler-Vilonen geometric Jacquet functor to the D- module of distributions on the square of the ﬂag variety of g with support on the diagonal. Chapter 1. Introduction 7

The proof of Theorem 12.6 is based on our algebraic result (Theorem 11.6): we can localize the modules U∆ and S on certain categories of D-modules, whose interaction with the (classical) Jacquet functor is already known. On the way of the proof we discuss the following concepts (which are covered in Chapter 12 and which we briefly review below in this chapter): the De Concini-Procesi compactification (often also called the wonderful compactification), the construction of a degeneration of the diagonal in the square of the flag variety of g (due to Brion), and the geometric Jacquet functor (due to Emerton-Nadler-Vilonen). For completeness, in Appendices B, C of Part II we review the Beilinson-Bernstein localization theory and construction of the nearby cycles functor, which is the main ingredient of the definition of the geometric Jacquet functor.

Let us explain these results in more details.

Let g be a semi-simple Z-graded Lie algebra having triangular decomposition g = n− ⊕ h ⊕ n. Denote by b the Borel subalgebra h ⊕ n of g. Let G, N −,N,T,B be the algebraic groups of g, n−, n, h, b respectively.

The semiregular module, corresponding to the subalgebra n− is by deﬁnition the module

g n− − ∗ S := Indn− CoindC C = U(g) ⊗U(n−) U(n ) .

It is a very non-trivial result ([3, 4]) that S is in fact a g-bimodule.

In our considerations we work with the following categories of g-modules. Let O be the Bernstein-Gelfand-Gelfand category of g-modules. This is a classical category playing the crucial role in representation theory of semi-simple Lie algebras whose properties we thoroughly discuss in Chapter 8 of the thesis. We denote by O0 the principal block of O. Because the semiregular module is a g-bimodule, we mostly work in the category of g-bimodules, namely, in the category HC(g ⊕ g,G∆) of Harish-Chandra modules (here

G∆ stands for the diagonal in G × G).

We single out two subcategories HC0(g,B) ⊂ O0 and HC(0,0)(g ⊕ g,G∆) ⊂ HC(g ⊕ Chapter 1. Introduction 8

g,G∆) which are isomorphic with the help of the Bernstein-Gelfand functor H0 ([11]). The functor H0 assigns to a g-module M the G∆-integrable part of the module HomC(M(0),M), where M(0) is the Verma module of highest weight 0.

We review the Harish-Chandra categories in Section 8.6.

Let w0 be the longest element of the Weyl group associated with g. We denote by U∆ the image of the module M(w0.0) under the functor H0. Thus U∆ is the G∆-integrable part of HomC(M(0),M(w0.0)). Let us describe the classical Jacquet functor J. In pure algebraic terms it can be deﬁned as follows (see e.g. [53]). We can assign to a g-module M the N-integrable part of its dual space M ∗. This way we obtain a functor J† to the category of N-integrable g-modules. Then the Jacquet functor is deﬁned by J(M) := (J†(M))∗.

While this interpretation of the Jacquet functor is the one we mostly deal with in our algebraic considerations, it is important to understand the geometric nature of this functor, which will be useful for our geometric discussions (cf. [25, 21]). To this end consider the cocharacter µ : Gm → G which when paired with roots of g is positive precisely on the roots of n. Let h ∈ h be the image of the vector ﬁeld t∂t on Gm under the map dµ (t denotes a coordinate on Gm). The Jacquet functor J assigns to a g-module M the sum of its generalized h-eigenspaces in the n-adic completion Mc := lim(M/U(n)kM) ←− k of M.

One can observe some similarity of this definition with the definition of the functor of nearby cycles. In fact, it can be shown ([21]) that the Malgrange-Kashiwara filtration on the localization of the module M can be defined purely in terms of the filtration on

M by the eigenvalues of h.

Let us apply the Jacquet functor J to the Harish-Chandra module U∆. The assertion of Theorem 11.6 is that this way we obtain a module isomorphic to the semiregular module S.

Let us turn to the geometric description of the semiregular module S and explain the Chapter 1. Introduction 9 assertion of Theorem 12.6.

Note that this theorem was originally conjectured by S. Arkhipov and R. Bezrukavnikov, inspired by results of M. Brion and P. Polo [13], and its proof constitutes one of the main results of this thesis.

We summarize the basic facts on D-modules in Appendix B as well as the Beilinson-

Bernstein construction of their localization. To a Harish-Chandra module M ∈ HC(0,0)(g⊕ g,G∆) the localization functor Loc assigns the D-module Loc(M) on the square of the ﬂag variety of g, equivariant with respect to the action on G/B × G/B of the diagonal

G∆ ⊂ G × G.

Let us apply the localization to the Harish-Chandra module U∆. It can be shown ([6]) that the resulting module is isomorphic to the module Dist∆(G/B×G/B) of distributions on G/B × G/B with support on the diagonal (G/B)∆:

∼ Loc(U∆) = Dist∆(G/B × G/B).

We now use the construction of the geometric Jacquet functor due to Emerton-Nadler-

Vilonen ([21]). This is the functor Jgeom from the category of G∆-equivariant D-modules

− on G/B × G/B to the category of N n T∆ o N -equivariant D-modules on G/B × G/B, having the following property:

Jgeom ◦ Loc = Loc ◦ J.

Applying this to the Harish-Chandra module U∆ and taking into account Theorem 11.6 we obtain the statement of Theorem 12.6.

To be consistent let us comment on the nature of the geometric Jacquet functor Jgeom (see Section 12.4 for the exact deﬁnitions).

Our considerations rely on the following fundamental construction of a degeneration of the diagonal in the square of the ﬂag variety of g due to Brion ([12]) (Sections 12.1, 12.2 in the text). Denote by X the canonical completion of the group G, constructed by De

Concini and Procesi ([15]). This is a smooth projective variety, containing G as an open Chapter 1. Introduction 10

G × G-orbit and having the following property: X \ G is a union of simple divisors with normal crossing whose intersection Y is isomorphic to the square of the ﬂag variety

G/B × G/B. It is possible to construct a curve γ ⊂ T ⊂ X isomorphic to A1 and transversal to Y at a certain point z, and a ﬂat family π over γ with the following properties: the ﬁber over the point g ∈ γ \{z} is isomorphic to the the diagonal in

(G/B)∆ ⊂ G/B × G/B and the ﬁber over the point z is equal to the union of products S of Schubert varieties w∈W S(w) × w0S(w0w) (recall that w0 is the longest element in the Weyl group W ).

In what follows the family π over γ will play the role of the supporting family for the

D-modules we consider.

Consider the constant family G/B × G/B × γ∗ → γ∗ over γ∗ := γ \{z}. Note that the Brion’s family π, restricted to γ∗, is the non-trivial subfamily in that family. We can extend the action of the diagonal G∆ on G/B × G/B to an action of a certain group

∗ ∗ ∗ scheme G → γ on the constant family G/B × G/B × γ → γ . If M is a G∆-equivariant D-module on G/B × G/B we may apply to it the procedure of equivariantization to obtain a D-module M˜ on the family G/B × G/B × γ∗ → γ∗, equivariant with respect to the action of the group scheme G → γ∗.

The geometric Jacquet functor Jgeom assigns to M the nearby cycles of its equivariantization M˜ along the ﬁber at the point z ∈ γ.

Notice that because the module Dist∆(G/B × G/B) has its support on the diagonal

(G/B)∆ ⊂ G/B × G/B, its equivariantization will have its support on the Brion’s family

∗ π, restricted to γ , while its nearby cycles (i.e. the module Jgeom(Dist∆(G/B × G/B))) S will have its support on the union of products of Schubert varieties w∈W S(w) × w0S(w0w).

Remark. Throughout the thesis we adopt the following agreement. If a theorem can be found in the literature, we give a reference to it before the statement and do not include Chapter 1. Introduction 11 the proof, unless it requires substantial changes. If the theorem does not contain any citation, it belongs to the author of this thesis. Part I

Krichever-Novikov algebras of

diﬀerential operators

12 Chapter 2

History and Context

The algebras of Krichever-Novikov type is a class of infinite-dimensional Lie algebras which naturally generalize the Witt algebra, the Virasoro algebra, affine Lie algebras and the like. Recall that the Witt algebra is the algebra of those meromorphic vector fields on the Riemann sphere P1 which are allowed to have poles only at 0 and ∞. Denote the

n+1 standard basis of the Witt algebra by {en = z ∂/∂z}n∈Z, where z is the coordinate on P1. The Lie bracket of vector ﬁelds yields the commutation relations between the basis elements:

[en, em] = (m − n)en+m.

The Witt algebra becomes graded by setting deg(en) := n. The Virasoro algebra is the one-dimensional central extension of the Witt algebra by means of the Gelfand-Fuchs 2-cocycle given by

1 c(e , e ) := (n3 − n)δ n m 12 n,−m

Another interesting object of study is the loop algebra g⊗C[z, z−1], where g is a semi- simple Lie algebra, C[z, z−1] is the ring of Laurent polynomials and the commutator is given by

[x ⊗ zn, y ⊗ zm] := [x, y] ⊗ zn+m,

13 Chapter 2. History and Context 14 as well as its central extensions, the aﬃne and Kac-Moody algebras.

Algebras of Krichever-Novikov type appear as analogs of the above algebras, in which one replaces the sphere P1 with a Riemann surface Σ of arbitrary genus and instead of poles at two points one allows poles at a set A of ﬁnitely many points, which is divided into the disjoint union of subsets I and O called correspondingly in- and out-points.

For an arbitrary Riemann surface Σ and the two points situation the algebra of vector

ﬁelds was introduced by Krichever and Novikov ([35, 36]) and the corresponding aﬃne algebras by Sheinman ([47]). The Krichever-Novikov type algebras for arbitrary number of punctures were described by Schlichenmaier ([42, 43, 44]).

Although these algebras are not graded, they can be equipped with the so-called generalized grading (see Section 4.5), which is often suﬃcient in order to apply methods of representation theory to these inﬁnite-dimensional Lie algebras.

The passage to central extensions of these algebras is of fundamental importance for many problems in quantum ﬁeld theory. To extend the above-mentioned generalized grading structure on these algebras to their central extensions requires the deﬁning 2- cocycle to be local (with respect to the generalized grading structure), i.e. to vanish on a pair of homogeneous elements of degree n and m if the sum n + m lies outside some

ﬁxed range, independent of n and m ([35, 36, 42]).

The cocycles for these algebras are obtained by integration over cycles on our Riemann surface with punctures. Among those we single out a cocycle obtained by integration over the cycle, which separates the in- and out-points in the set A. This cocycle is local with respect to the almost grading corresponding to the splitting A = I ∪ O and the corresponding extended Lie algebra is also almost graded.

We also note that the cohomology of the Lie algebra of diﬀerential operators (in pure algebraic terms) were computed by Shoikhet [49] and Dzhumadildaev [20].

Krichever-Novikov type algebras appear in conformal field theory and in the theory of integrable models. In particular, in the interpretation of the Riemann surface as a possible Chapter 2. History and Context 15 world sheet of the closed string theory, the points in I correspond to free incoming strings and the points in O to free outgoing strings. One should also mention that these algebras are related to moduli spaces of flat connections, which naturally appear in the coadjoint orbit classification for these algebras (see [47]). Chapter 3

Meromorphic pseudodiﬀerential symbols on Riemann surfaces

3.1 The algebras of meromorphic diﬀerential and pseu-

dodiﬀerential symbols

Let Σ be a compact Riemann surface and M be the space of meromorphic functions on Σ. Fix a meromorphic vector field v on the surface and denote by D (or Dv) the operator of Lie derivative Lv along the field v: if locally, in a neighborhood U with local coordinate x the field v is given by v(x) = f(x)∂/∂x, then

0 Dg = Lv(g(x)) = f(x)g (x), where g0(x) is the derivative of g with respect to x.

Then D sends the space M to itself, and one can consider the operator algebras generated by D.

Deﬁnition 3.1. The associative algebras of meromorphic diﬀerential operators

Xn k MDO := {A = akD | ak ∈ M} k=0

16 Chapter 3. Meromorphic pseudodifferential symbols on Riemann surfaces17 and meromorphic pseudodifferential symbols Xn k MΨDS := {A = akD | ak ∈ M} k=−∞ are the above spaces of formal polynomials and series in D which are equipped with the associative multiplication ◦ defined by µ ¶ X k D ◦ D = D2, a ◦ D = aD, Dk ◦ a = (Dà)Dk−` , (3.1) ` `>0

¡k¢ k(k−1)...(k−`+1) where the binomial coeﬃcient ` = `! makes sense for both positive and negative k. This multiplication law naturally extends the Leibnitz rule D◦a = aD+(Da), i.e. the commutator [D, a] := D ◦ a − a ◦ D is the Lie derivative Da of the function a.

Note that aD is a symbol, which may be regarded as a diﬀerential operator of the ﬁst order, while Da is a function.

The algebras MDO and MΨDS are also Lie algebras with respect to the bracket

[A, B] = A ◦ B − B ◦ A.

Note that the algebra MDO is both an associative and a Lie subalgebra in MΨDS. (In the sequel, we will write simply XY instead of X ◦ Y whenever this does not cause an ambiguity.)

For diﬀerent choices of the meromorphic vector ﬁeld v the corresponding algebras of

(pseudo) differential symbols are isomorphic: any other meromorphic field w on Σ can be presented as w = fv for f ∈ M, and then the relation Dw = fDv delivers both the associative and Lie algebra isomorphism. Meromorphic vector fields on Σ embed both into MDO and MΨDS as Lie subalgebras.

Remark 3.2. Equivalently, one can deﬁne the product of two pseudodiﬀerential symbols

Pm i Pn j by the following formula: if A(D) = i=−∞ aiD and B(D) = j=−∞ bjD , for D := Lv then Ã ! X 1 k k A ◦ B := ∂ξ A(ξ)∂v B(ξ) . (3.2) > k! k 0 ξ=D Chapter 3. Meromorphic pseudodifferential symbols on Riemann surfaces18

Here ∂v is the operator of taking the Lie derivative of coeﬃcients of a symbol (i.e. of functions bj) along v. Note that the right hand side of this formula expresses the commutative multiplication of functions A(z, ξ) and B(z, ξ). Of course, this formula also extends the usual composition of diﬀerential operators.

3.2 Outer derivations of pseudodiﬀerential symbols

It turns out that both the associative and Lie algebras of meromorphic pseudodifferential symbols have many outer derivations (see [24] for all necessary definitions in cohomology of infinite-dimensional Lie algebras).

We remind that a derivation of a Lie algebra is a linear operator on the algebra satisfying the Leibnitz rule relative to the Lie bracket. A derivation is inner if it is of the form [X, ·], where X is an element of the Lie algebra. All the other derivations are called outer.

Definition 3.3. (cf. [34]) Let v be a meromorphic vector field on Σ and set D := Lv. Define the operator log D or, rather, [log D, ·]: MΨDS → MΨDS by

X (−1)k+1 [log D, aDn] := (Dka)Dn−k. (3.3) k k>1

For example, the action of log D on a ﬁeld aD is given by

1 1 [log D, aD] = (Da) − (D2a)D−1 + (D3a)D−2 − ... 2 3

The above formula is consistent with the Leibnitz formula (3.1): it can be obtained from the latter by regarding k as a complex parameter, say, λ and diﬀerentiating in λ

λ at λ = 0: d/dλ|λ=0 D = log D. (Below we will be using the notation log D for the derivation and [log D, ·] for explicit formulas.) Chapter 3. Meromorphic pseudodifferential symbols on Riemann surfaces19

Proposition 3.4. The operator log D : MΨDS → MΨDS deﬁnes a derivation of the

(both associative and Lie) algebra MΨDS of meromorphic pseudodiﬀerential symbols for any choice of the meromorphic vector ﬁeld v.

Proof. One readily veriﬁes (see [34]) that for any two symbols A and B,

[log D, AB] = [log D,A]B + A[log D,B], i.e. log D is a derivation of the associative algebra MΨDS. This also implies that log D is a derivation of the Lie algebra structure.

It turns out that one can describe a whole class of derivations in a similar way.

Definition 3.5. Associate to any meromorphic pseudodifferential symbol X the derivation log X : MΨDS → MΨDS, where the commutator [log X,A] with a symbol A is defined by means of the formula (3.2).

Namely, recall that log(v(z)ξ) can be regarded as a (multivalued) symbol for log Dv, i.e. a multivalued function on T ∗Σ. Indeed, only the derivatives of this function in ξ or along the ﬁeld v appear in the formula for the commutator [log D,A] = log D ◦ A − A ◦ log D, where the products in the right-hand-side are deﬁned by formula (3.2). Similarly, one can regard log X(ξ) as a function on T ∗Σ and only its derivatives appear in the commutators [log X,A] with any meromorphic symbol A ∈ MΨDS.

Remark 3.6. We note that the formula for [log X,A] involves the inverse X−1, which is a well-defined element of MΨDS. Indeed, to find, say, the inverse of a pseudodifferential symbol X we have to solve X ◦ A = 1 with unknown coefficients. If

n n−1 −n −n−1 X = fnD + fn−1D + ...,A = anD + an−1D + ..., we solve recursively the equations

fnan = 1, fnan−1 + fn−1an + nfn(Dan) = 0,.... Chapter 3. Meromorphic pseudodifferential symbols on Riemann surfaces20

Each equation involves only one new unknown aj as compared to preceding ones and hence the series for X−1 = A can be obtained term by term, i.e. its coeﬃcients are meromorphic functions.

Example 3.7. To any meromorphic function f ∈ M on Σ we associate the operator log f : MΨDS → MΨDS given by Df f(D2f) − (Df)2 [log f, aDn] := na Dn−1 + n(n − 1)a Dn−2 + .... f f 2 Note that, while the function log f is not meromorphic and branches at poles and zeros of f, all its derivatives Dk(log f) with k > 1 are meromorphic, and the right hand side of the above expression is a meromorphic pseudodiﬀerential symbol.

We shall show that log X for any symbol X is an outer derivation of the Lie algebra

MΨDS, i.e. it represent a nontrivial element of H1(MΨDS,MΨDS). The latter space is by deﬁnition the space of equivalence classes of all derivations modulo inner ones (cf.

[24]).

Theorem 3.8. All derivations deﬁned by log X for any meromorphic pseudodiﬀeren- tial symbol X are outer and equivalent to a linear combination of derivations given by

logarithms log Dvi of meromorphic vector ﬁelds vi on Σ.

This theorem is implied by the following two properties of the log-map.

Theorem 3.80. The map X 7→ log X ∈ H1(MΨDS,MΨDS) is nonzero and satisﬁes the properties:

a) the derivation log(X ◦Y ) is equivalent to the sum of derivations log X +log Y , and

b) the derivation log(X + Y ) is equivalent to the derivation log X if the degree of the symbol X is greater than the degree of Y .

One can see that any derivation log X is equivalent to a linear combination of log f for some meromorphic function and log Dv for one ﬁxed ﬁeld v. The above properties of derivations log X modulo inner ones are similar to those of tropical calculus. We prove this theorem in Section 3.5. Chapter 3. Meromorphic pseudodifferential symbols on Riemann surfaces21

Conjecture 3.9. All outer derivations of the Lie algebra MΨDS are equivalent to those given by log X for pseudodiﬀerential symbols X.

3.3 The traces

The Lie algebra MΨDS has a trace attached to any choice of the special points on Σ

(see Deﬁnition 3.10 below). All the constructions below will be relying on this choice of the points and we ﬁx such a point (or a collection of points) P ∈ Σ from now on.

Definition 3.10. Define the residue map res from MΨDS to meromorphic 1-forms on D Σ by setting Ã ! Xn k ˜ −1 res akD := a−1D . D k=−∞ Here D˜ −1 in the right-hand-side is understood as a (globally defined) meromorphic differential on Σ, the pointwise inverse of the meromorphic vector field v.

On the algebra MΨDS we deﬁne the trace associated to the point P ∈ Σ by

Tr A := res res (A). P D

Here Z f f 1 f res fD˜ −1 = res = res dz = dz, P P v P h 2πi γ h where v = h(z)∂/∂z is a local representation of the vector ﬁeld v at a neighborhood of the point P , while γ is a suﬃciently small contour on Σ around P which does not contain poles of f/h other than P . (Here and below we omit the index P in the notation of the trace Tr P whenever this is unambiguous.)

Proposition 3.11. Both the residue and trace are well-deﬁned operations on the algebra

MΨDS, i.e. they do not depend on the choice of the ﬁeld v. Furthermore, for any choice of the point(s) P , Tr is an algebraic trace, i.e. Tr [A, B] = 0 for any two pseudo- diﬀerential symbols A, B ∈ MΨDS. Chapter 3. Meromorphic pseudodifferential symbols on Riemann surfaces22

In particular, this property allows us to use the notation res or Tr without mentioning D a speciﬁc ﬁeld v.

−1 Proof. Under the change of a vector ﬁeld v 7→ w = gv only the terms Dv contribute to −1 ˜ −1 Dw , which implies that the corresponding 1-form a−1Dv and hence the residue operator are well-deﬁned.

The algebraic property of the trace is a local assertion, since Tr is deﬁned locally near P . One can show that for any two X,Y ∈ MΨDS the residue of the commutator is a full derivative, i.e.

res [X,Y ] = (Df)D−1 D for some function f deﬁned in a neighborhood of P (cf. [17], p.11). Then the proposition follows from the fact that a complete derivative has zero residue: Z Z Z L f (Df)D−1 = v = df = 0 γ γ v γ for a contour γ around P .

This proposition allows one to deﬁne the pairing ( , ) on MΨDS, associated with the chosen point P ∈ Σ:

(A, B) := Tr (AB). (3.4)

This pairing is symmetric, non-degenerate (cf. the proof of Theorem 3.22 below), and invariant due to Proposition 3.11. The pairings associated to diﬀerent choices of the point(s) P ∈ Σ are, in general, not related by an algebra automorphism (unless there exists a holomorphic automorphism of the surface Σ sending one choice to the other).

Remark 3.12. The existence of the invariant trace(s) on MΨDS allows one to identify this Lie algebra with (the regular part of) its dual space. This identiﬁcation relies on the choice of the point P .

We also note that both res and Tr vanish on the subalgebra MDO of meromorphic D purely diﬀerential operators. In particular, this subalgebra is isotropic with respect to Chapter 3. Meromorphic pseudodifferential symbols on Riemann surfaces23

the above pairing, i.e. ( , ) |MDO = 0. The complementary to MDO subalgebra is the

P−1 k Lie algebra MIS of meromorphic integral symbols { k=−∞ akD }, which is isotropic with respect to this pairing as well.

3.4 The logarithmic 2-cocycles

Being in the possession of a variety of outer derivations, as well as of the invariant trace(s), we can now construct many central extensions of the Lie algebra ΨDS. The simple form of the invariant trace allows us to follow the analogous formalism for pseudodiﬀerential symbols on the circle [22, 32, 34].

Recall that the 2-cocycle on a Lie algebra g is a skew-symmetric bilinear map c satisfying the cocycle identity:

c([X,Y ],Z) + c([Y,Z],X) + c([Z,X],Y ) = 0, for any three elements X,Y,Z of the Lie algebra g. A 2-cocycle c is a 1-coboundary (or simply a coboundary) if there is a linear functional f on g satisfying c(X,Y ) = f([X,Y ]).

The linear space of all 2-cocycles on g modulo the coboundaries is denoted by H2(g, C) (or simply H2(g)). This is the second cohomology space of the Lie algebra g with coefficients in the trivial g-module. The space H2(g) can be identified with the space of 1-dimensional central extensions of g. Namely, every non-trivial 2-cocycle c on g (i.e. the one which is not a coboundary) gives rise to the Lie algebra structure on the linear space bg := g ⊕ C such that the image of C under the natural inclusion C → bg lies in the center of bg, and equivalent 2-cocycles (i.e. those which differ by a coboundary) give rise to isomorphic

Lie algebra structures on bg. For further references on this subject see [24].

We start by deﬁning a logarithmic 2-cocycle attached to the given choice of the point

P and a meromorphic ﬁeld v on Σ: Chapter 3. Meromorphic pseudodifferential symbols on Riemann surfaces24

Theorem 3.13. (cf. [34]) The bilinear functional

cv(A, B) := Tr ([log Dv,A] ◦ B) (3.5) is a nontrivial 2-cocycle on MΨDS and on its subalgebra MDO for any meromorphic

ﬁeld v and any choice of the point P on Σ, where the trace is taken.

Proof. The non-triviality of the cocycle follows from its non-triviality on the subalgebra of vector ﬁelds (see Remark 3.14).

To shorten notations let us denote the commutation operator with log Dv by φ. Recall that Tr deﬁnes a non-degenerate symmetric pairing by (3.4). We have to show the skew- symmetry and the cocycle identity for

cv(A, B) = (φ(A),B).

Since the trace functional has the property Tr ([A, B]) = 0 for any symbols A, B, it follows that

(φ(A),B) + (φ(B),A) = 0, i.e. cv is skew-symmetric. Now, since φ is a derivation of the Lie and the associative structures on HΨDS

(Proposition 3.4), we have the following identities for any symbols A, B, C:

(φ([A, B]),C) = ([φ(A),B],C) + ([A, φ(B)],C),

([A, B],C) = (A, [B,C]).

Using these identities we verify the cocycle identity:

(φ([A, B]),C) + (φ([B,C]),A) + (φ([C,A]),B) = ([φ(A),B],C) + ([A, φ(B)],C)

+ ([φ(B),C],A) + ([B, φ(C)],A) + ([φ(C),A],B) + ([C, φ(A)],B) ³ ´ = 2 (φ(A), [B,C]) + (φ(B), [C,A]) + (φ(C), [A, B]) ³ ´ = −2 (φ([B,C],A) + (φ([C,A],B) + (φ([A, B],C) . Chapter 3. Meromorphic pseudodifferential symbols on Riemann surfaces25

We conclude that

(φ([A, B]),C) + (φ([B,C]),A) + (φ([C,A]),B) = 0, which completes the proof.

Remark 3.14. The restriction of this 2-cocycle to the algebra of vector ﬁelds is the

Gelfand-Fuchs 2-cocycle

2 1 (Dva)(Dvb) cv(aDv, bDv) = res 6 P Dv on the algebra of meromorphic vector ﬁelds on Σ. Indeed, one has

cv(aDv, bDv) = Tr ([log Dv, aDv] ◦ bDv) ³ ´ 1 2 −1 1 3 −2 = res res ((Dva) − (Dva)Dv + (Dva)Dv − ...) ◦ bDv P Dv 2 3 ³ ´ 1 2 −1 1 3 −1 = res (Dva)(Dvb)Dv + (Dva)bDv P 2 3 ³ ´ 2 1 2 −1 1 2 −1 1 (Dva)(Dvb) = res (Dva)(Dvb)Dv − (Dva)(Dvb)Dv = res . P 2 3 6 P Dv

(Here we have used integration by parts formula res (Da)bD−1 = − res (Db)aD−1.) P P The restriction of the cocycle (3.5) to the algebra MDO gives the Kac-Peterson

2-cocycle

n+1 m m n m!n! (Dv a)(Dv b) cv(aDv , bDv ) = res , m, n > 0, (m + n + 1)! P Dv on meromorphic diﬀerential operators on Σ (see [39]).

The above construction can be generalized in the following way.

Deﬁnition 3.15. Associate the logarithmic 2-cocycle

cX (A, B) := Tr ([log X,A] ◦ B) to a meromorphic pseudodiﬀerential symbol X and a point P ∈ Σ (where the trace Tr is taken). Chapter 3. Meromorphic pseudodifferential symbols on Riemann surfaces26

Theorem 3.800. For any meromorphic pseudodiﬀerential symbols X and Y

a) the logarithmic 2-cocycle cXY is equivalent to the sum of the 2-cocycles cX + cY , and

b) the logarithmic 2-cocycle cX+Y is equivalent to the logarithmic 2-cocycle cX provided the degree of the symbol X is greater than the degree of Y .

Proof. This follows from Theorem 3.80 thanks to the following claim.

Lemma 3.16. (see e.g. [20]). Let g be a Lie algebra with a symmetric invariant non- degenerate pairing ( , ). Consider a derivation φ : g → g preserving the pairing, i.e. satisfying (φ(a), b) + (φ(b), a) = 0 for any a, b ∈ g, and associate to it the 2-cocycle c(a, b) := (φ(a), b) ∈ H2(g) on g. Then the subspace in H1(g, g) consisting of invariant derivations (and understood up to coboundary) is isomorphic to the space H2(g): the

2-cocycle c is cohomologically nontrivial if and only if the derivation φ is outer.

Since the outer derivation log X preserves the pairing (3.4), it deﬁnes a nontrivial

2-cocycle (cf. the course of the proof of Theorem 3.13). The properties of the outer derivations in Theorem 3.80 are equivalent to those of the logarithmic 2-cocycles in The- orem 3.800.

Corollary 3.17. (i) For any meromorphic pseudodiﬀerential symbol X the logarith-

mic 2-cocycle cX is equivalent to a linear combination of the 2-cocycles cvi (A, B) :=

Tr ([log Dvi ,A] ◦ B) associated to meromorphic vector ﬁelds vi.

(ii) For two meromorphic vector ﬁelds v and w related by w = fv the cocycles cv and cw are related by

cw = cv + cf ,

where cf (A, B) := Tr ([log f, A] ◦ B) is the 2-cocycle associated to the meromorphic function f ∈ M, and the equality is understood in H2(MΨDS, C), i.e. modulo a 2- coboundary. Chapter 3. Meromorphic pseudodifferential symbols on Riemann surfaces27

Theorem 3.18. All the 2-cocycles cv are nontrivial and non-cohomologous to each other on the algebra MΨDS for diﬀerent choices of meromorphic ﬁelds v 6= 0. Equivalently, cocycles cf are all nontrivial for non-constant functions f.

Proof. Note that the 2-cocycle cv is nontrivial, since its restriction to the subalgebra of vector fields holomorphic in a punctured neighborhood of the point P already gives the nontrivial Gelfand-Fuchs 2-cocycle. (In other words, the nontriviality of the cocycle cv for a meromorphic vector field v follows from its nontriviality on the Krichever-Novikov subalgebra L of holomorphic vector fields on Σ \{P,Q}, where Q is any other point on

Σ, see the next chapter.)

To show the nontriviality of the cocycle cf for a non-constant function f we use the existence of many traces on MΨDS. First we choose the point P (and the corresponding trace TrP ) at a zero of the function f. The corresponding 2-cocycle cf is non-trivial, since so is its restriction to pseudodiﬀerential symbols holomorphic in a punctured neighborhood of P . The latter is obtained by exploiting the nontriviality of the

2-cocycle c0(A, B) = Tr ([log z, A] ◦ B) on holomorphic pseudodiﬀerential symbols on C∗, see [32, 20].

Now, by applying the above-mentioned equivalence between derivations and cocycles

(see Lemma 3.16), we conclude that log f deﬁnes an outer derivation of the algebra

MΨDS. Once we know that the derivation is outer, we can use the same equivalence “in the opposite direction” for the point P anywhere on Σ to obtain a nontrivial 2-cocycle from any other invariant trace.

Remark 3.19. Note that the cocycle cf , for a meromorphic function f vanishes on the subalgebra MDO of meromorphic differential operators: for any purely differential operators X and Y , the expression [log f, X]◦Y is also a meromorphic differential operator

−1 (see Example 3.7) and hence its coefficient at Dv is 0. This shows that all the 2-cocycles cv for the same point P ∈ Σ, but for different choices of the meromorphic field v are cohomologous when restricted to the algebra MDO. The choice of a different point P to Chapter 3. Meromorphic pseudodifferential symbols on Riemann surfaces28

deﬁne the trace may lead to a non-cohomologous 2-cocycle cv.

3.5 Proof of the theorem on outer derivations

In this section we will prove Theorem 3.80 on properties of the derivation log X : MΨDS →

MΨDS.

Proof. For the part a) we rewrite the product XY of two symbols as XY = exp(log X) ◦ exp(log Y ) and use the Campbell-Hausdorﬀ formula:

XY = exp(log X + log Y + R(log X, log Y )).

Note that the remainder term R(log X, log Y ) is a pseudodiﬀerential symbol, since only

(iterated) commutators of log X and log Y appear in it, but not the logarithms them- selves. (In particular, the commutator [log X, log Y ] deﬁned by the formula (3.2) is a pseudodiﬀerential symbol from MΨDS.) Hence

log XY = log X + log Y + R(log X, log Y ).

Thus the derivation log XY is cohomologous to the sum of derivations log X + log Y , since the commutation with the symbol R(log X, log Y ) deﬁnes an inner derivation of the algebra MΨDS. This proves a).

Pn i To prove the part b) we will show that the derivation log X for X = i=−∞ aiDv is

n deﬁned by anDv , the principal term of X. Indeed, rewrite X as

n X = (anDv ) ◦ (1 + Y ),

P−1 j where Y = j=−∞ bjDv is a meromorphic integral symbol. Then due to a), log X is

n cohomologous to log(anDv ) + log(1 + Y ). However, the derivation log(1 + Y ) is inner, i.e. log(1 + Y ) is itself a meromorphic pseudodiﬀerential symbol. Indeed, expand log(1 + Y ) in the series: log(1 + Y ) = Y − Y 2/2 + Y 3/3 − .... The right hand side is a well-deﬁned Chapter 3. Meromorphic pseudodifferential symbols on Riemann surfaces29

n meromorphic integral symbol, since so is Y . Thus log X is cohomologous to log(anDv ), which proves b).

3.6 The double extension of the meromorphic sym-

bols and Manin triples

Definition 3.20. Consider the following double extension of the Lie algebra MΨDS by means of both the central term and the outer derivation for a fixed meromorphic field v: Xn k M^ΨDS = C · log D ⊕ MΨDS ⊕ C · I = {λ log D + akD + µ · I}, k=−∞ where the commutator of a pseudodifferential symbol with another one or with log D for

D = Dv was deﬁned above, while the cocycle direction I commutes with everything else.

There is a natural invariant pairing on the Lie algebra M^ΨDS, which extends the pairing Tr (A ◦ B) on the non-extended algebra MΨDS. Namely,

h(λ1 log D + A1 + µ1 · I, λ2 log D + A2 + µ2 · I)i = Tr (A1 ◦ A2) + λ1 · µ2 + λ2 · µ1 .

Recall the following general notion:

Deﬁnition 3.21. A Manin triple (g, g+, g−) is a Lie algebra g along with two Lie subalgebras g± ⊂ g and a nondegenerate invariant symmetric form ( , ) on g, such that

(a) g = g+ ⊕ g− as a vector space and

(b) g+ and g− are isotropic subspaces of g with respect to the inner product ( , ).

Now consider also two subalgebras of the Lie algebra M^ΨDS: the subalgebra of centrally extended meromorphic diﬀerential operators Xn k MDO\ = { akD + µ · I} k=0 Chapter 3. Meromorphic pseudodifferential symbols on Riemann surfaces30 and the subalgebra of co-centrally extended meromorphic integral symbols X−1 k MIS] = C · log D ⊕ MIS = {λ log D + akD }. k=−∞ Similarly to the case of smooth coeﬃcients, one proves the following

Theorem 3.22. Both the triples of algebras (MΨDS,MDO,MIS) and

(M^ΨDS, MDO,\ MIS] ) are Manin triples.

Proof. Conditions (a) and (b) in the deﬁnition of a Manin triple are clearly satisﬁed for both triples (MΨDS,MDO,MIS) and (M^ΨDS, MDO,\ MIS] ). The only thing we have to check is that the pairing is nondegenerate. But this is evident since for any symbol

k A ∈ MΨDS with highest coefficient ak before Dv one can find a symbol B with highest coefficient b−k−1, so that the pairing of ak and b−k−1 at the point P is nonzero. This implies that Tr (A ◦ B) will also be nonzero, and hence the corresponding pairing is non-degenerate on MΨDS.

The existence of a Manin triple means the existence of a Lie bialgebra structure on both g± and allows one to regard each of the subalgebras as dual to the other with respect to the pairing (see the appendix for the deﬁnitions).

Corollary 3.23. Both the Lie algebras MIS and MIS] are Lie bialgebras, while the groups corresponding to them are Poisson-Lie groups.

Deﬁnition 3.24. A group G equipped with a Poisson structure η is a Poisson–Lie group if the group product G × G → G is a Poisson morphism (i.e., it takes the natural Poisson structure on the product G × G into the Poisson structure on G itself) and if the map

G → G of taking the group inverse is an anti-Poisson morphism (i.e. it changes the sign of the Poisson bracket).

This makes the meromorphic consideration parallel to the case of smooth pseudo- diﬀerential symbols developed in [22, 32]. For holomorphic symbols, however, such a

Manin triple exists only in special cases, as we discuss below. Chapter 4

Holomorphic pseudodiﬀerential symbols on Riemann surfaces

4.1 The Krichever-Novikov algebra

Let Σ be a Riemann surface of genus g. Fix two generic points P+ and P− on the surface. Consider the Lie algebra L of meromorphic vector ﬁelds on Σ, holomorphic on ◦ ◦ Σ := Σ \{P±}. We will call such ﬁelds simply holomorphic (on Σ).

◦ Deﬁnition 4.1. The Lie algebra L of holomorphic on Σ vector ﬁelds is called the Krichever-Novikov (KN) algebra.

A special basis in L, called the Krichever-Novikov (KN) basis, is formed by vector

ﬁelds ek with a pole of order k at P+ and a pole of order 3g−k−2 at P− (as usual we refer to a pole of negative order k as to a zero of order −k). (More precisely, this prescription of basis elements works for surfaces Σ of genus g ≥ 2, while for g = 1 one has to alter it for certain small values of k, see [35].) Note that each ﬁeld ek has g additional zeros elsewhere on Σ \{P±}, since the degree of the tangent bundle of Σ is 2 − 2g. Because the Krichever-Novikov basis plays a crucial role in what follows, we present in Appendix its detailed construction (by help of the Riemann-Roch theorem).

31 Chapter 4. Holomorphic pseudodifferential symbols on Riemann surfaces32

The KN algebra was introduced and studied in [35, 36] along with its central exten- ◦ sions. It generalizes the Virasoro algebra, which corresponds to the case Σ = CP 1 \

{0, ∞}. Below we will be concerned with the case of two punctures P± on Σ, although most of the results below hold for the case of many punctures as well, cf.

[42, 43, 44, 48, 52].

4.2 Holomorphic diﬀerential operators and pseudod-

iﬀerential symbols

◦ Denote the sheaf of holomorphic functions on Σ by O.

Definition 4.2. The sheaves of holomorphic differential operators and pseudodifferential symbols on Σ \{P±} are defined by assigning to each open set U ⊂ Σ an abelian group (a vector space) of sections

n Xn o i HDO(U) := uiD | ui ∈ OU and i=0

n Xn o i HΨDS(U) := uiD | ui ∈ OU , i=−∞ respectively, where D := Dv stands for some holomorphic non-vanishing vector ﬁeld v in U. (Any other choice of a non-vanishing ﬁeld v gives the same spaces of the operators and symbols.)

The Lie algebras of global sections of the sheaves HDO and HΨDS are called the

Lie algebras of holomorphic diﬀerential operators and of pseudodiﬀerential symbols, respectively. We denote these algebras of global sections by HDO and HΨDS.

Note that the definitions of the residue and trace of symbols are local and hence can be defined on holomorphic symbols in the same way as they were defined for meromorphic Chapter 4. Holomorphic pseudodifferential symbols on Riemann surfaces33

ones: resD X is a globally deﬁned holomorphic 1-form, given in local coordinates as ˜ −1 u−1D , while

˜ −1 Tr X := res res X = res (u−1D ) P+ D P+

Pn i for a section X given by X = i=−∞ uiD in a (punctured) neighborhood of P+.

The algebra HΨDS has two subalgebras: that of holomorphic diﬀerential operators

(HDO) and of holomorphic integral symbols (HIS). They consist of those symbols whose restriction to any open subset U ⊂ Σ are, respectively, holomorphic purely diﬀerential operators or holomorphic purely integral symbols. As in the meromorphic case, these holomorphic subalgebras are isotropic with respect to the natural pairing.

◦ The KN-algebra L of holomorphic vector ﬁelds on Σ can be naturally viewed as a subalgebra of the algebras of holomorphic diﬀerential operators (HDO) and pseudodifferential symbols (HΨDS). In turn, the algebra HΨDS is a subalgebra in the algebra of meromorphic symbols MΨDS.

Remark 4.3. The Lie algebra HDO can be alternatively deﬁned as the universal enveloping algebra HDO = U(O o L)/J of the Lie algebra O o L quotiened over the ideal

J, generated by the elements f ◦ g − fg, f ◦ v − fv, and 1 − 1, where ◦ denotes the multiplication in HDO, 1 is the unit of U, while f, g, 1 ∈ O and v ∈ L, see e.g. [43]. It is easy to see that this deﬁnition matches the one above.

A convenient way to write some global sections of the above sheaves is by ﬁxing a

Pn i holomorphic field v ∈ L. Then the symbols X = i=−∞ uiDv with any holomorphic coefficients ui ∈ O define global sections of HΨDS, provided that for every i < 0 the coefficient ui has zero of order at least i at zeros of v (this way we compensate for all the

i poles of the negative powers Dv by appropriate zeros of the corresponding coeﬃcients). Chapter 4. Holomorphic pseudodifferential symbols on Riemann surfaces34

4.3 Holomorphic pseudodiﬀerential symbols and the

spaces of densities

One can think of holomorphic (pseudo)differential symbols as sequences of holomorphic ◦ densities on Σ. Namely, let K be the canonical line bundle over Σ and consider the tensor power Kn of K for any n ∈ Z. Denote by F n the space of holomorphic n-densities on ◦ ◦ Σ, i.e. the space of global meromorphic sections of Kn which are holomorphic on Σ. Note that F 0 is the ring O of holomorphic functions, F 1 is the space of holomorphic ◦ differentials on Σ, and F −1 is the space L of holomorphic vector fields. Holomorphic vector fields act on holomorphic n-densities by the Lie derivative: to

n n v ∈ L and ω ∈ F one associates Lvω ∈ F . Explicitly, in local coordinates for v = h(z)∂/∂z and ω = f(z)(dz)n one has µ ¶ ∂f ∂h L ω = h(z) + nf(z) (dz)n. v ∂z ∂z

This action turns the space F n of n-densities into an L-module. The following proposition is well-known:

Proposition 4.4. Each graded space for the ﬁltration of pseudodiﬀerential symbols by degree is naturally, as an L-module, isomorphic to the corresponding space of holomorphic densities. Namely,

HΨDSn/HΨDSn−1 ≈ F −n , where HΨDSn is the space of pseudodiﬀerential symbols of degree n and the isomorphism is given by taking the principal symbol of the pseudodiﬀerential operator.

Proof. The action of vector ﬁelds from L on pseudodiﬀerential operators of degree n is explicitly given by:

³ ´ [hD, fDn] = h(Df) − nf(Dh) Dn + (terms of degree < n in D). Chapter 4. Holomorphic pseudodifferential symbols on Riemann surfaces35

Thus the action on their principal symbols coincides with the above L-action on (−n)- densities F −n, i.e. they satisfy the same change of coordinate rule.

Furthermore, taking the principal symbols of the operators of given degree is a surjec- tive map onto F −n. One can see this first for differential operators, i.e for n ≥ 0, where it follows from their description as HDO = U(O o L)/J and the Poincaré-Birkhoff-Witt theorem. Indeed, one can form a basis of differential operators of degree n from the products fD ...D , where f ∈ F 0 and e form the KN-basis in L = F −1. Their principal ei1 ein i symbols will be the (commutative) products of the principal symbols of the basis elements, which, by definition, span the space of meromorphic sections of Kn, holomorphic ◦ on Σ, i.e. the space F −n. The surjectivity for negative n, i.e. for principal symbols of integral operators, can be derived by considering natural pairing on densities (F n × F −n−1 → C) and on pseu-

−n n−1 dodiﬀerential symbols (HΨDS × HΨDS → C) given by taking at the point P+ the residue for densities and the trace for symbols, respectively.

The whole vector space HΨDS can be treated as the direct limit of the semi-inﬁnite

k −n products of the spaces of holomorphic n-densities: HΨDS ≈ lim Πn=−∞F as k → ∞, −→ on which one has a Lie algebra structure.

4.4 The density of holomorphic symbols in the smooth

ones

◦ The algebra HΨDS of holomorphic symbols on Σ, as well as the KN-algebra L of holomorphic vector ﬁelds, can be regarded as subalgebras of smooth symbols on the circle S1 in the following way.

In [35] a family of special contours Cτ , τ ∈ R on Σ was constructed as level sets of ◦ some harmonic function on Σ. These contours separate the points P±, and as τ → ±∞ Chapter 4. Holomorphic pseudodifferential symbols on Riemann surfaces36

1 the contours Cτ become circles shrinking to P±. Denote by S ≈ Cτ an arbitrary contour from this family for a suﬃciently large τ, thought of as a small circle around P+. Consider ◦ ◦ the natural restriction homomorphism from Σ to S1 ⊂ Σ.

Theorem 4.5. ([35, 36]) The restrictions of holomorphic functions, vector fields, and ◦ 1 differentials on Σ to the contour S ≈ Cτ are dense among, respectively, smooth functions, vector fields, and differentials on the circle S1.

Now we consider the algebra of all smooth pseudodiﬀerential symbols on the circle:

n Xn d o ΨDS(S1) = f (x)∂i | ∂ := , f ∈ C∞(S1) . i dx i i=−∞ The latter is a topological space under the natural topology (on the Laurent series) while the sum, multiplication, and taking the inverse (for a nowhere vanishing highest coeﬃcient fn) are continuous operations. The following theorem is a natural extension of the one above. Consider the restriction homomorphism HΨDS → ΨDS(S1) of holomorphic symbols to the smooth ones for the

1 contour S ≈ Cτ ⊂ Σ and denote by HΨDS |S1 the corresponding image.

Theorem 4.6. The restrictions HΨDS |S1 of holomorphic symbols are dense in the smooth ones ΨDS(S1).

Proof. It suﬃces to prove that the monomials of the form f(x)∂i, i ∈ Z, f ∈ C∞(S1) can be approximated by holomorphic ones. Write out such a monomial as a product of

−1 d f(x), ∂ and ∂ . Since the smooth function f(x), the vector ﬁeld ∂ := dx and the 1-form ∂−1 := dx on the circle can be approximated by the restrictions of holomorphic ones [36], the result follows by the continuity of the multiplication in ΨDS(S1).

Note that the original density result in [35] for a pair of points P± extends to a collection of points by representing functions, ﬁelds, etc. with many poles as sums of the ones with two poles only. Chapter 4. Holomorphic pseudodifferential symbols on Riemann surfaces37

4.5 The property of generalized grading

Deﬁnition 4.7. An associative or Lie algebra A is generalized graded (or N-graded) if it admits a decomposition A = ⊕n∈ZAn into the ﬁnite-dimensional subspaces, with the property that there is a constant N such that MN Ai Aj ⊂ Ai+j+s, s=−N for all i, j.

Similarly, a module M over a generalized graded algebra A is generalized graded, if

M = ⊕n∈ZMn, and there is a constant L such that ML Ai Mj ⊂ Mi+j+s, s=−L for all i, j.

◦ Theorem 4.8. ([35, 36]) The KN-algebra L of vector ﬁelds holomorphic on Σ is generalized graded. The space F n of holomorphic n-densities for any n is a generalized graded module over L.

(n) n (n) The generalized graded components Mj for the module F are the spaces C · fj , (n) where the forms fj are uniquely determined by the pole orders at both points P± (as before, we assume the points to be generic): for n 6= 0 they have the following expansions

(n) ± ±j−g/2+n(g−1) n fj := aj z± (1 + O(z±))(dz±) in local coordinates z± of neighborhoods of the points P±. Here the index j runs over integers Z if g is even and over half-integers Z + 1/2 if g is odd. (The formulas diﬀer slightly for F 0 = O, see details in [35]. For the space of vector ﬁelds L ≈ F −1 this basis

(−1) {fj } diﬀers by an index shift from the basis {ej}.)

Now we consider the spaces HDO and HΨDS of holomorphic (pseudo)differential symbols not only as modules over vector fields L, but as Lie algebras. These Lie algebras are not generalized graded, but naturally filtered by the degree of D = Dv. Con-

(n) sider a basis {Fj } (which we construct below) in pseudodiﬀerential symbols HΨDS, Chapter 4. Holomorphic pseudodifferential symbols on Riemann surfaces38 which is compatible with the basis in the forms: the principal symbol of the operator

(n) (−n) Fj of degree n is the (−n)-form fj . It turns out that the algebras of holomorphic (pseudo)diﬀerential symbols have the following analogue of the generalized grading:

Theorem 4.9. The Lie algebras HDO and HΨDS are ﬁltered generalized graded: the

(n) pseudodiﬀerential symbols of an appropriate basis {Fj } in HΨDS satisfy

X∞ NX(k) (n) (m) s (n+m−k) [Fi ,Fj ] = αijFi+j+s , k=1 s=−N(k)

r for some constants αij ∈ C, where n, m ∈ Z, the indices i, j, and s are either integers of half-integers according to parity of the genus g, and N(k) is a linear function of k.

Proof. First we deﬁne a basis for diﬀerential operators from HDO ⊂ HΨDS recursively in degree n (cf. [43], where a similar basis was constructed for HDO). Assume that the genus g is even, so that all the indices are integers (the case of an odd g is similar).

(0) Consider the KN-basis {Fj } in diﬀerential operators of degree 0, which constitute 0-

0 (1) densities F , and the KN-basis {Fj } in holomorphic vector fields, which are differential operators of degree 1. ˜(n) For a degree n ≥ 2 consider a differential operator Fj whose principal symbol is the (−n) (−n)-density fj , and which exists due to surjectivity discussed in Proposition 4.4. One ˜(n) can “kill the lower order terms” of the operator Fj by adding a linear combination of the basis elements in HDOn−1 constructed at the preceding step. (More precisely, due to ˜(n) the filtered structure of HDO only the cone of lower order terms for Fj is well-defined by the orders of poles of the coefficients of the differential operators at the points P±. By ˜(n) “killing the terms” above we mean confining Fj to this cone.) We call these differential

(n) (k) n operators Fj . Along with {Fj } for 0 ≤ k ≤ n − 1 they constitute a basis in HDO , holomorphic diﬀerential operators of degree ≤ n.

Finally, for integral symbols (of negative degrees) we choose the basis dual to the one chosen in diﬀerential operators, by using the nondegenerate pairing: HIS can be thought Chapter 4. Holomorphic pseudodifferential symbols on Riemann surfaces39 of as the dual space HDO∗ (the duality being given by the residue). It is easy to see that this basis in integral symbols of degree −n is also given by the orders of zeros and poles at P± and hence compatible with the KN-basis in n-forms.

(n) Once the basis {Fj } is constructed, a straightforward substitution of these symbols into the formula for the symbol commutator and the calculation of orders of poles/zeros at the points P± yield the result.

This theorem implies the property of generalized grading for modules of holomorphic densities, established in [35, 43, 48], as for principal symbols of holomorphic pseudodifferential operators.

4.6 Cocycles and extensions

Recall first the cocycle construction for the KN-algebra L of holomorphic vector fields ◦ on Σ. A closed contour γ on Σ, not passing through the marked points P±, defines the Gelfand-Fuchs 2-cocycle on the algebra L. Namely, in a fixed projective structure

(where admissible coordinates diﬀer by projective transformations) it is deﬁned by the

Gelfand-Fuchs integral Z c(f, g) = f 00(z)g0(z)dz γ ∂ ∂ for vector fields f = f(z) ∂z and g = g(z) ∂z given in such a coordinate system. One can check that this cocycle is well-defined, nontrivial, and represents every cohomology class in the space H2(L, C) of 2-cocycles on the algebra L for different contours γ, see [36, 44].

In this variety of 2-cocycles there is a subset of those satisfying the following property of locality.

Deﬁnition 4.10. ([35]) Let A = ⊕n∈ZAn be a generalized graded Lie algebra. A 2- cocycle c on A is called local if there is a nonnegative constant K ∈ Z such that c(Am,An) = 0 for all |m + n| > K. Chapter 4. Holomorphic pseudodifferential symbols on Riemann surfaces40

The reason for introducing local cocycles is the following: the central extensions of generalized graded Lie algebras deﬁned by local 2-cocycles are also generalized graded

Lie algebras.

Theorem 4.11. ([35]) The cohomology space of local 2-cocycles of the Krichever-Novikov algebra L is one-dimensional. It is generated by the Gelfand-Fuchs 2-cocycle on any separating contour Cτ on Σ.

As we discussed above, for large τ one can think of Cτ as a small circle around P+. Thus the local cocycles on L are deﬁned by the restrictions of vector ﬁelds to a small neighborhood of P+.

To describe the logarithmic cocycles on the algebras HDO and HΨDS of holomorphic ◦ (pseudo)diﬀerential symbols on Σ we adapt the notion of the cocycle locality to the ﬁltered generalized grading.

First we recall the corresponding results for the algebras DO(S1) and ΨDS(S1) of smooth operators and symbols of the circle.

Theorem 4.12. (i) The cohomology space of 2-cocycles on the algebra DO(S1) of differential operators on the circle is one-dimensional ([23, 37]). A non-trivial 2-cocycle is deﬁned by the restriction of the logarithmic cocycle Tr ([log ∂, A] ◦ B) to diﬀerential

∂ 1 operators, ∂ := ∂x , and A, B ∈ DO(S ) ([34]). (ii) The cohomology space of 2-cocycles of the algebra ΨDS(S1) of pseudodiﬀerential symbols is two-dimensional ([23, 20]). It is generated by the logarithmic cocycle above and the 2-cocycle Tr ([x, A] ◦ B), where x is the coordinate on the universal covering of

S1, and A, B ∈ ΨDS(S1) ([32, 34]). R Here the trace is Tr A := S1 res ∂ A for smooth pseudodiﬀerential symbols, which replaces Tr A := res P+ res D A for holomorphic ones.

Example 4.13. Consider the Lie algebra of holomorphic symbols on C∗ = CP 1 \{0, ∞}, whose elements are allowed to have poles at 0 and ∞ only, and where we take Dv := Chapter 4. Holomorphic pseudodifferential symbols on Riemann surfaces41 z∂/∂z. Two independent outer derivations of the latter algebra are log(z∂/∂z) and log z, the logarithms of a vector ﬁeld and a function, respectively [34, 32]. The corresponding

2-cocycles are ∂ Tr ([log z ,A] ◦ B) and Tr ([log z, A] ◦ B). ∂z This algebra can be thought of as a graded version of smooth complex-valued symbols

ΨDS(S1) on the circle S1 = {|z| = 1}: the change of variable z = exp(ix) sends

∂ := ∂/∂x to iDv:

∂/∂x = ∂z/∂x · ∂/∂z = i exp(ix)∂/∂z = iz∂/∂z = iDv.

Under this change of variables (and upon restricting the symbols to the circle S1), the

∗ derivations [log(iDv),.] and −i[log z, .] for holomorphic symbols in ΨDS(C ) become the derivations [log ∂, .] and [x, .] for smooth symbols in ΨDS(S1). The above theorem describes the 2-cocycles on ΨDS(S1) constructed with the help of the latter outer derivations and the corresponding change in the notion of trace.

Note that the restriction of the cocycle Tr ([x, A]B) to the algebra of diﬀerential operators DO vanishes (and this is why only one out of two cocycles survives this restriction).

After having described the smooth case, we adapt the definition of the local 2-cocycle to the filtered generalized graded case of the algebra HΨDS by allowing the constant K in Definition 4.10 to depend on the filtered component.

Deﬁnition 4.14. A 2-cocycle on the ﬁltered generalized graded algebra HΨDS is called

(n) (m) local if for any integers i, j there is a number N = N(n + m) such that c(Fi ,Fj ) = 0 (n) (m) for basis pseudodiﬀerential operators Fi and Fj as soon as |i + j| > N.

One can see that such a cocycle preserves the property of ﬁltered generalized grading when passing to the corresponding central extension. ◦ ◦ Consider a holomorphic vector ﬁeld v on Σ and a smooth path γ on Σ \ (div v), i.e. a smooth path γ avoiding P±, as well as zeros and poles of v. Associate to v and γ the Chapter 4. Holomorphic pseudodifferential symbols on Riemann surfaces42

2-cocycle cv,γ deﬁned as the following bilinear functional on HΨDS and HDO: Z

cv,γ(A, B) := res ([log Dv,A] ◦ B), γ Dv

◦ where we integrate over γ the residue, which is a meromorphic 1-form on Σ (with possible poles at the divisor of v, and hence oﬀ γ). We conﬁne ourselves to considering cocycles of the form cv,γ.

Theorem 4.15. (i) The cohomology space of local 2-cocycles of the form cv,γ on the Lie ◦ algebra HDO on Σ is one-dimensional and it is generated by the 2-cocycle cv,P+ (A, B) :=

Tr ([log Dv,A] ◦ B) for a holomorphic vector ﬁeld v. The cocycles cv,P+ are local for any ◦ choice of a holomorphic ﬁeld v on Σ.

(ii) The cohomology space of local 2-cocycles cv,γ on the algebra HΨDS is two-

dimensional. It is generated by the 2-cocycles cvi,P+ for two holomorphic vector ﬁelds v1 and v2 with diﬀerent orders of poles/zeros at P+.

Remark 4.16. Alternatively, one can generate the 2-dimensional space of local cocycles for HΨDS by considering

Tr ([log Dv,A] ◦ B) and Tr ([log f, A] ◦ B) where v is any holomorphic vector ﬁeld, while f is a function with a zero or pole (of any non-zero order) at P+. The restriction to HDO of the latter 2-cocycle vanishes.

Proof of Theorem 4.15. We have adapted the deﬁnitions of grading and locality in such a way that the locality of a 2-cocycle on the ﬁltered algebras HDO and HΨDS implies its locality on the subalgebra L. According to the Krichever-Novikov Theorem 4.11 local cocycles on L are given by the integrals over contours Cτ . In turn, the cocycles cv,γ for

γ = Cτ for large τ correspond to integration over a simple contour around P+, and hence reduce to

cv,P+ (A, B) := Tr P+ ([log Dv,A] ◦ B) = res res ([log Dv,A] ◦ B). P+ Dv Chapter 4. Holomorphic pseudodifferential symbols on Riemann surfaces43

To ﬁnd the dimension of the cohomology space of such cocycles for HDO and HΨDS we consider the restriction homomorphism to the smooth operators and symbols on the contour. In both cases the image is dense in the latter due to Theorem 4.6.

One can see that the cocycle cv,P+ for any v is nontrivial in both HDO and HΨDS,

1 since it is nontrivial on the subalgebra L. Indeed, upon restriction to the contour S ≈ Cτ it gives the (nontrivial) Gelfand-Fuchs 2-cocycle on V ect(S1). Furthermore, the cohomology space of 2-cocycles for smooth differential operators DO(S1) is 1-dimensional, and hence so is the cohomology space of local 2-cocycles for holomorphic differential operators HDO, due to the density result and continuity of the cocycles. Verification of

locality of cv,P+ for any holomorphic ﬁeld v is a straightforward calculation. This proves part (i).

For part (ii) we note that the algebra ΨDS(S1) admits exactly two independent nontrivial 2-cocycles up to equivalence. The density theorem will imply the required statement, once we show that there are two linearly independent cocycles of the type

cv,P+ . Take 2-cocycles cv,P+ and cw,P+ for two fields v and w of different orders of pole/zero at P+. Then v = fw for a meromorphic function f on Σ, which is either zero or infinity

at P+. The same argument as in Corollary 3.17 (ii) gives that cv,P+ = cw,P+ + cf,P+ ,

where cf,P+ (A, B) := Tr P+ ([log f, A] ◦ B).

In order to show that the cocycle cf,P+ is nontrivial and independent of cv,P+ , provided that f has a zero or pole at P+, we again consider the restriction homomorphism to the

1 1 smooth symbols ΨDS(S ) on the contour S ≈ Cτ . In a local coordinate system around

k P+ we have f(z) = z g(z) with holomorphic g(z) such that g(0) 6= 0 and k 6= 0. Since log f(z) = k log z + log g(z), the corresponding logarithmic cocycles are related in the

same way. Note that Tr P+ ([log g, A] ◦ B) deﬁnes a trivial cocycle (i.e. a 2-coboundary)

1 upon restriction to S . Indeed, the function log g(z) is holomorphic at P+ = 0 since g(0) 6= 0, and its restriction to a small contour around P+ = 0 is univalued. Hence, it deﬁnes a smooth univalued function on the contour, and therefore [log g, A] is an inner Chapter 4. Holomorphic pseudodifferential symbols on Riemann surfaces44 derivation of the corresponding algebra ΨDS(S1).

1 On the other hand, Tr P+ ([log z, A]◦B) upon restriction to S ≈ Cτ deﬁnes the second

1 non-trivial cocycle of the algebra ΨDS(S ), see Example 4.13. Hence the cocycle cf,P+

is nontrivial and deﬁnes the same cohomology class as k · cz,P+ . This completes the proof of (ii).

Conjecture 4.17. Every continuous 2-cocycle on the Lie algebras HΨDS and HDO is cohomologous to a linear combination of regular 2-cocycles cv,γ for some holomorphic ◦ ﬁelds v and a contour γ on Σ.

Remark 4.18. The latter is closely related to Conjecture 3.9. Presumably, all the contin- R uous 2-cocycles on HΨDS and HDO have the form cX,γ(A, B) := γ res D([log X,A]◦B) for holomorphic pseudodiﬀerential symbols X and cycles γ on the surface Σ. In turn, one can reduce the cocycles cX,γ for an arbitrary symbol X to cocycles cv,γ with X = Dv, similarly to the proof of Theorem 3.8.

4.7 Manin triples for holomorphic pseudodiﬀerential

symbols

Given the point P+ ∈ Σ and the invariant pairing on HΨDS associated to the trace at

P+ it is straightforward to verify the following proposition.

Proposition 4.19. The (non-extended) algebras (HΨDS,HDO,HIS) form a Manin triple.

Although to any holomorphic ﬁeld v with poles at P± one can associate the central extension of the Lie algebra HΨDS by the local 2-cocycle cv(A, B) = Tr ([log Dv,A]◦B), the double extension of HΨDS does not necessarily exist. Chapter 4. Holomorphic pseudodifferential symbols on Riemann surfaces45

◦ Confine first to the special case, in which on Σ there exists a holomorphic field v ◦ ◦ without zeros, i.e. to Σ with a trivialized tangent bundle. Such a surface Σ can be obtained from any Σ and any field v on it by choosing the collections of points P± to include all zeros and poles of v. (Example: v = z ∂/∂z in C∗.)

In this case the operator log Dv maps HΨDS to itself, i.e it is an outer derivation of the latter. Then the construction of the co-central extension HIS] = C · log Dv ⊕ HIS and the double extension H^ΨDS goes through in the same way as for the meromorphic or smooth cases.

◦ Proposition 4.20. If the holomorphic ﬁeld v has no zeros on Σ, the Lie algebras (H^ΨDS, HDO,\ HIS]) form a Manin triple. Equivalently, HIS] is a Lie bialgebra.

In this case the Lie bialgebra on HIS] defines a Poisson-Lie structure on the corresponding pseudodifferential symbols of complex degree, just like in the case of C∗ or in the smooth case on the circle. ◦ Now let v have zeros in Σ. Consider the central extension HDO\ of the algebra of holomorphic differential operators HDO by the 2-cocycle cv. One can see that now the “regular dual” space to HDO\ cannot be naturally identified with the vector space

C · log Dv ⊕ HIS. Indeed, the coadjoint action of HDO\ is uniquely deﬁned by the commutator [log Dv,A] wherever v 6= 0. However, this commutator may have poles at zeros of v, i.e. the space C · log Dv ⊕ HIS does not form a Lie algebra, as it is not closed under commutation. This does not allow one to deﬁne a natural Lie bialgebra structure on HDO\ or HIS]. The same type of obstruction arises for the existence of a formal group of symbols of complex degrees on the surface Σ. Chapter 5

Appendix A: Poisson–Lie groups,

Lie bialgebras, and Manin triples

Deﬁnition 5.1. A group G equipped with a Poisson structure η is a Poisson–Lie group if the group product G × G → G is a Poisson morphism (i.e., it takes the natural Poisson structure on the product G × G into the Poisson structure on G itself) and if the map

G → G of taking the group inverse is an anti-Poisson morphism (i.e. it changes the sign of the Poisson bracket).

Theorem 5.2. ([38]) For any connected and simply connected group G with Lie algebra g there is a one-to-one correspondence between Lie bialgebra structures on g and Poisson-

Lie structures η on G. This correspondence sends a Poisson-Lie group (G, η) into the

Lie bialgebra g tangent to (G, η).

By deﬁnition, a Lie algebra g is a Lie bialgebra if its dual space g∗ is equipped with a

Lie algebra structure such that the map g → g∧g dual to the Lie bracket map g∗∧g∗ → g∗ on g∗ is a 1-cocycle on g relative to the adjoint representation of g on g ∧ g.

Theorem 5.3. ([46]) Consider a Manin triple (g, g+, g−). Then g+ is naturally dual to g− and each of g− and g+ is a Lie bialgebra. Conversely, for any Lie bialgebra g one

∗ can ﬁnd a unique Lie algebra structure on g = g ⊕ g such that the triple (g, g+, g−) is a

46 Chapter 5. Appendix A: Poisson–Lie groups, Lie bialgebras, and Manin triples47

Manin triple with respect to the natural pairing on g and the corresponding Lie bialgebra structure on g is the given one. Chapter 6

Appendix B: The Krichever-Novikov basis

6.1 The Riemann-Roch theorem for line bundles and

the Krichever-Novikov basis

Let L be a holomorphic line bundle over a compact Riemann surface X and s a nonzero meromorphic section of L. Recall that a divisor of s is the formal sum

X (s) := ordp(s) · p. p∈X

Here ordp(s) is the order of pole of any (meromorphic) function representing s at a neighborhood of the point p. Note that this is a ﬁnite sum. It is known that the divisors of any two sections s1 and s2 of L belong to the same linear equivalence class, i.e. there is a meromorphic function h such that (s1) = (s2)+(h). This way we may associate to every holomorphic line bundle L a divisor class of its arbitrary meromorphic section. It can be shown that this provides an injective homomorphism from the group of isomorphism classes of line bundles (the group operation being the tensor product and inverse being the dual bundle) to the group of divisor classes. For a smooth projective variety X this

48 Chapter 6. Appendix B: The Krichever-Novikov basis 49 homomorphism is actually an isomorphism of abelian groups.

Now let (sL) be a divisor class associated to L:

X (sL) = np · p. p∈X

Deﬁne the degree of L to be the number

X degL := deg(sL) = np = (number of zeros of s) − (number of poles of s). p∈X

Recall that the space of meromorphic functions f on X with

(f) > −(sL) is naturally isomorphic to the space H0(X,L) of holomorphic sections of the bundle L.

The Riemann-Roch theorem implies the following:    = 0, if degL < 0  dim H0(X,L) = > 1 − g + degL, if 0 6 degL < 2g − 1   = 1 − g + degL, if degL > 2g − 1, see e.g. [45].

Examples. Let X = P1 (i.e. the genus g = 0). Holomorphic vector ﬁelds are sections of the holomorphic tangent bundle T , which is dual to the cotangent bundle ω. We have

deg T = −(2g − 2) = 2 − 2g ⇒ deg T = 2 > 2 · 0 − 1 = 1 and by the Riemann-Roch theorem

dim H0(X,T ) = deg T − 0 + 1 = 3,

1 ∂ ∂ 2 ∂ (in coordinates of the standard covering of P the basis is given by ∂z , z ∂z , z ∂z ). In the case g = 1 the spaces of holomorphic differentials and holomorphic vector fields are 1-dimensional (the latter is spanned by the vector field ∂/∂z). Chapter 6. Appendix B: The Krichever-Novikov basis 50

For X = Σ a Riemann surface of genus g > 2 we have deg T = −2g + 2 6 0. Hence there are no holomorphic sections of T . Note that the space of holomorphic diﬀerentials on X has dimension g.

For integers ni, i = 1, . . . , k, we set O ⊗ni R = Lpi , i where Lpi is the line bundle which has a section spi with exactly one zero at the point pi and non-vanishing otherwise [45]. Set M = L ⊗ R. If s is a section of M, then

s Q t = ni i spi is a section of L. This way the space H0(X,M) of holomorphic sections of M is isomorphic to the space of meromorphic sections of the bundle L which are holomorphic outside the points pi and have a pole of order at most ni at the points pi.

6.2 Construction of the Krichever-Novikov basis

Let Σ be a Riemann surface of genus g > 2. We ﬁx points P+ and P− on Σ and consider the Krichever-Novikov algebra associated with these two points.

Set

M = T ⊗ L−k ⊗ L3g+k−2, k ∈ Z. k P+ P−

Here, as in the previous section, LP± is the line bundle having a section sP± with exactly one zero at the point P± and non-vanishing otherwise. Then we have:

degMk = (2 − 2g) − k + (3g + k − 2) = g.

0 By the Riemann-Roch theorem we get that the dimension of the space H (Σ,Mk) of meromorphic vector ﬁelds which can have poles only at P+ and P− of orders −k and 3g + k − 2 respectively is equal to

0 dim H (Σ,Mk) > degMk − g + 1 = 1. Chapter 6. Appendix B: The Krichever-Novikov basis 51

0 In fact, one can show that dim H (Σ,Mk) = 1 (see e.g. [45]). The vector ﬁelds ek of the

0 KN-basis are the generators of H (Σ,Mk). Each ﬁeld ek has zero of the order k at P+ and pole of the order 3g + k − 2 at P−. Since

2 − 2g = deg T = (number of zeros of ek) − (number of poles of ek), while (the number of poles of ek) = 3g − 2, we conclude that each ek has exactly g zeros

at points Qk1 ,...,Qkg , diﬀerent from the points P±, which we will suppose generically diﬀerent.

6.3 Relation to the Lie algebra of vector ﬁelds on the

circle

It can be shown that there is a meromorphic differential ρ which has poles of order 1 at the points P± with residues ±1 at P± respectively (see e.g. [45]). By adding a suitable global holomorphic differential we obtain differential with only imaginary periods. Such a differential is uniquely determined.

Fix a point Q0 6= P± on X. Then the function Z Q u(Q) = Re ρ Q0 is well-deﬁned because the periods of ρ are pure imaginary. Deﬁne

Cτ := {Q ∈ Σ| u(Q) = −τ}.

Due to the prescribed residues 1 and −1 at points P+ and P− respectively, for τ → ±∞ we have that Cτ is a small circle around P±. Denote by LΣ the Lie algebra of meromorphic vector ﬁelds on Σ, holomorphic outside

Σ the points P±. If we restrict L to Cτ for |τ| À 0, we obtain a homomorphism

Σ i : L → L(Cτ ), Chapter 6. Appendix B: The Krichever-Novikov basis 52

Σ of the algebra L to the algebra L(Cτ ) of vector ﬁelds on the circle.

Σ Theorem. ([35, 36]). The image i(L ) is dense in L(Cτ ). Part II

A geometric realization of

semiregular modules

53 Chapter 7

Notations and Preliminaries

Throughout the rest of the thesis we ﬁx the following notations: g – semi-simple complex Lie algebra; h ⊂ g – the Cartan subalgebra;

U := U(g) – the universal enveloping algebra of g;

Z = Z(g) – the center of U(g);

R ⊂ h∗ – the set of roots of g;

R+ (respectively R−) – the set of positive (respectively negative) roots in R, so that

R = R+ t R−;

∆ – the set of simple roots;

Λ – the integral weight lattice, associated to R;

Λ+ ⊂ Λ – the set of dominant integral weights;

Λr ⊂ Λ – the root lattice;

W – the Weyl group of g generated by all reﬂections sα, α ∈ R;

− L − g = n ⊕ h ⊕ n – the corresponding Cartan decomposition, where n = gα, n = α∈R+ L + gα. For α ∈ R we denote by eα a generator of gα, by fα a generator of g−α, and α∈R− by hα := [eα, fα]. This is the standard basis of g; ht : W → Z – the height function, ` : W → Z+ – the length function (deﬁned to be

54 Chapter 7. Notations and Preliminaries 55

`(w) := n if w = s1 . . . sn, si simple reﬂections and n is as small as possible, i.e. this is the reduced expression). w0 will stand for the unique element in W of maximal length |R+| (sending R+ to R−); b := h ⊕ n – the corresponding standard Borel subalgebra, b− := h ⊕ n− – opposite Borel subalgebra; w.α – the dot-action of W on R deﬁned by

w.α := w(α + ρ) − ρ

(w ∈ W , α ∈ R), where ρ is the half sum of positive roots (= sum of fundamental weights).

We denote by G, T , N −, N, and B the algebraic groups corresponding to g, h, n−, n, and b respectively. Chapter 8

Categories of g-modules

In this chapter we introduce the main objects of our study: the Bernstein-Gelfand-

Gelfand category O and the category HC of Harish-Chandra modules. Both these cat-

egories are the certain subcategories in the category of g-modules. We list the basic

properties of the objects of these categories and give some statements we will use in

the sequel. As a reference we mostly use [30], but one can also ﬁnd all the results we

reproduce here in [27] and in the original paper [9].

In what follows we will interchangeably use the notions of g-modules and U(g)-

modules as well as the notions of representations over g and representations over U(g).

8.1 Category O

Deﬁnition 8.1. ([9]) Let g be a semi-simple Lie algebra. The BGG category O = O(g)

(named after B. Bernstein, I. Gelfand, and S. Gelfand) is the full subcategory of g-

modules, consisting of the modules satisfying the following three conditions:

(O1) M is ﬁnitely generated U(g)-module;

(O2) M is locally n-ﬁnite, that is, for each v ∈ M, the subspace U(n) · v of M is ﬁnite

dimensional;

56 Chapter 8. Categories of g-modules 57

(O3) M is locally h-ﬁnite and h-semi-simple, that is, M is a direct sum of its h-weight

spaces: M = ⊕λ∈h∗ Mλ.

From deﬁnition it follows that all ﬁnite dimensional modules lie in O. Indeed, such

modules clearly satisfy axioms (O1) and (O2), while the axiom (O3) follows from the

fact that for ﬁnite-dimensional representation of a semi-simple Lie algebra the abstract

and the usual Jordan decompositions coincide (see e.g. [29]).

Lemma 8.2. (cf. [30], Chapter 1.1) Let M ∈ O. Then

(i) All the weight spaces of M are ﬁnite dimensional;

(ii) The set Π(M) of all weights of M is contained in the union of ﬁnitely many sets of

∗ + the form λ − Γ, where λ ∈ h and Γ is the semigroup in Λr, generated by Φ .

Below we list some basic properties of the category O (see [30] Chapters 1.1, 1.2 and

[27]).

(a) O is an abelian category;

(b) O is a noetherian category;

M 7→ L ⊗ M

is an exact functor from O to O.

(d) If M ∈ O, then M is Z(g)-ﬁnite.

(e) The action of n on every object of O is locally nilpotent.

Note that because of (e) and because h acts in the semi-simple way, the modules

from O are B-integrable (recall that B is the algebraic group of the Borel subalgebra

b = h ⊕ n ⊂ g). Chapter 8. Categories of g-modules 58

Deﬁnition 8.3. A vector v+ in a g-module M is called a highest weight vector if n·v+ = 0.

A g-module is the highest weight module of weight λ ∈ h∗ if there is a highest weight

+ + vector v ∈ Mλ such that M = U(g) · v .

Remark 8.4. Notice that the Poincar´e-Birkhoﬀ-Witt(PBW) theorem implies that M =

U(n−) · v+.

Proposition 8.5. (cf. [30], Theorem 1.2) Let M be a highest weight module of weight λ ∈ h∗. Then M lies in O. M has a unique maximal submodule and unique simple quotient.

All simple highest weight modules of weight λ are isomorphic and dim EndOM = 1.

Proposition 8.6. (cf. [30], Corollary 1.2) Let M ∈ O be nonzero. Then M has a ﬁnite

ﬁltration

0 = M1 ⊂ M2 ⊂ · · · ⊂ Mn = M with nonzero quotients each of which is a highest weight module.

8.2 Verma modules

∗ Deﬁnition 8.7. Any λ ∈ h deﬁnes a 1-dimensional b-module Cλ on which b acts through λ the character b → h → C. Denote by

g M(λ) := IndbCλ = U(g) ⊗U(b) Cλ the Verma module corresponding to λ with the natural left U(g)-module structure.

Remark 8.8. By the PBW theorem

∼ − M(λ) = U(n ) ⊗C Cλ.

Thus M(λ) is free as a left U(n−)-module, of rank one. The vector v+ := 1⊗1 is nonzero and is acted on freely by U(n−), while n · v+ = 0 and h · v+ = λ(h)v+ for all h ∈ h. Thus v+ is a highest weight vector and generates the g-module M(λ). Chapter 8. Categories of g-modules 59

We conclude that M(λ) lies in O.

Note that the Verma module M(λ) is uniquely determined by the following condition: for any g-module M,

Homg(M(λ),M) = Homb(Cλ,M).

Remark 8.9. If M is an arbitrary highest weight module of weight λ, then M(λ) maps naturally onto M. Indeed, M(λ) is isomorphic to the module U(g)/I(λ), where I(λ) is a left ideal in U(g), generated by n and h−λ(h)·1, h ∈ h (i.e. by the elements annihilating the highest weight vector v+ of M(λ)). But I(λ) obviously annihilates highest weight vector of weight λ, generating the module M. We get the desired projection. Therefore

M(λ) plays the role of universal highest weight module of weight λ.

Denote by L(λ) the unique simple quotient of M(λ). Since every nonzero module in

O has a highest weight vector, we conclude:

Theorem 8.10. (cf. [30], Theorem 1.3) Every simple module in O is isomorphic to

L(λ) with λ ∈ h∗ and is therefore determined uniquely up to isomorphism by its highest weight.

We would like to know which of the simple modules L(λ), λ ∈ h∗, are ﬁnite dimensional. Recall that an integral weight λ ∈ Λ is called dominant if it belongs to the closure of the fundamental Weyl chamber.

Theorem 8.11. (cf. [30], Theorem 1.6) The simple module L(λ) in O is ﬁnite-dimensional if and only if λ is a dominant weight.

Remark 8.12. Let λ ∈ Λ+ be a dominant weight. Consider the dual space L(λ)∗ with the action

(x · f)(v) := −f(x · v),

∗ ∗ ∼ for x ∈ g, v ∈ L(λ), f ∈ L(λ) . It can be shown that L(λ) = L(−w0λ) (recall that w0 is the longest element of the Weyl group). Note also that L(λ)∗ is again simple. Its weights relative to h are negative of those of L(λ). Chapter 8. Categories of g-modules 60

8.3 Action of the Center and Harish-Chandra’s The-

orem

Let M be a highest weight module with highest weight vector v+. If z ∈ Z(g) and h ∈ h, then

h · (z · v+) = z · (h · v+) = λ(h)z · v+.

+ + Since dim Mλ = 1, we conclude that z · v = χλ(z)v for some χλ(z) ∈ C.

Definition 8.13. For fixed λ the function z 7→ χλ(z) defines an algebra homomorphism

χλ : Z(g) → C, called the central character, associated with λ. Thus the set of all the central characters is in natural bijection with the set SpecZ(g).

Let pr : U(g) ∼= U(n−) ⊗ U(h) ⊗ U(n) → U(h) be the projection, mapping the monomials from 1 ⊗ U(h) ⊗ 1 identically to U(h) and mapping all other monomials to zero. Then

χλ(z) = λ(pr(z)), for all z ∈ Z(g). (Here the linear functional λ extends canonically to the algebra homomorphism λ : U(h) → C.) It follows that the restriction of pr to Z(g) is an algebra homomorphism. Denote this restriction by ξ. The map ξ composed with the automorphism of U(h) = S(h), p(λ) 7→ p(λ − ρ), is called the (twisted) Harish-Chandra homomorphism and is denoted by ψ.

Theorem 8.14 (Harish-Chandra). (cf. [30], Theorem 1.10) (i) The homomorphism

ψ : Z(g) → S(h) is an isomorphism of Z(g) onto S(h)W (the W -invariant polynomials with respect to the dot-action).

∗ (ii) For all λ, µ ∈ h , we have χλ = χµ if and only if µ = w.λ for some w ∈ W .

∗ (iii) Every central character χ : Z(g) → C is of the form χλ for some λ ∈ h . Chapter 8. Categories of g-modules 61

8.4 Decomposition of the category O with respect to

the center and duality

Let M ∈ O. For each central character χ denote

M χ := {v ∈ M | (z − χ(z))n · v = 0 for some n > 0 depending on z}.

Then M χ is the submodule in M.

Deﬁnition 8.15. Denote by Oχ the full subcategory of O consisting of the modules M for which M = M χ.

Proposition 8.16. (cf. [30], Proposition 1.12 or [27]) Category O is the direct sum of the subcategories Oχ as χ ranges over the central characters of the form χλ: M O = Oχ. (8.1) χ∈SpecZ(g)

Each indecomposable module lies in a unique Oχ. In particular, each highest weight

module of weight λ lies in Oχλ .

Remark 8.17. Note that each of the categories Oχλ contains only ﬁnitely many simple

modules: L(µ) belongs to Oχλ if and only if µ = w.λ for some w ∈ W .

Deﬁnition 8.18. The subcategory Oχ0 is called the principal block of O and is denoted by O0.

Note (cf. Remark 8.17) that L(λ) ∈ O0 if and only if λ = w.0 for some w ∈ W .

Remark 8.19. It can be shown (cf. [30] Section 4.9) that the category O can be decomposed as a direct sum of its blocks. Two simple modules M1,M2 of O belong to the same block, if they can be extended non-trivially, i.e. there is a non-split short exact sequence

0 → M1 → M → M2 → 0 Chapter 8. Categories of g-modules 62

(in other words, Ext(M2,M1) is non-trivial). An arbitrary module M belongs to a certain block if all its composition factors do. It can be shown that each module from O decomposes in a direct sum of submodules, each lying in a single block. In particular, each indecomposable module belongs to a single block.

In the decomposition (8.1) each block of O will lie in some Oχ. When the weights

are integral, i.e. λ ∈ Λ, the subcategories Oχλ are precisely the blocks of O.

For any U(g)-module M, the dual vector space M ∗ is a U(g)-module by

(x · f)(v) := −f(x · v), for all v ∈ M, f ∈ M ∗, x ∈ g. In general it is not true that if M lies in O then M ∗ also does. The reason for this is the following. If f belongs to the ﬁnite-dimensional space

∗ ∗ Mλ = (Mλ) , then an element of h acts on by the weight −λ. Now eα · f has the weight

∗ −λ + α, forcing f ∈ Mλ−α. Thus if dim M = ∞, this prevents n from acting locally ﬁnitely on weight vectors.

We use the Chevalley (or transpose) anti-involution τ to deﬁne the new action of g

∗ on M . Recall that τ ﬁxes h pointwise and sends gα to g−α for all roots α. Deﬁne

(x · f)(v) := f(τ(x) · v).

Consider now a weight module M (i.e. M is the direct sum of its weight spaces) with

ﬁnite dimensional weight spaces. Deﬁne (cf. [30]) the contragredient dual of M by

M ∨ ∗ M := Mλ . λ∈h∗

∗ ∗ Here Mλ is the dual space of Mλ and it consists of all f ∈ M which vanish on all Mµ for µ 6= λ.

Proposition 8.20. (cf. [30], Theorem 3.2) The contragredient duality functor on O sending M to M ∨ is exact and contravariant with the following properties: Chapter 8. Categories of g-modules 63

(a) The square of the contragredient duality functor is naturally isomorphic to the identity functor on O;

(b) For any M ∈ O and any central character χ, (M ∨)χ ∼= (M χ)∨; in particular, if

∨ M ∈ Oχ, then M ∈ Oχ; (c) M and M ∨ have the same composition factor multiplicities. In particular, L(λ)∨ ∼= L(λ);

(d) If M,N ∈ O, then

(M ⊕ N)∨ ∼= M ∨ ⊕ N ∨;

(e) If M,N ∈ O, then

∼ ∨ ∨ ExtO(M,N) = ExtO(N ,M ).

8.5 Projective objects in O, standard ﬁltrations, and

BGG reciprocity

The category O is artinian and noetherian: each M ∈ O is both artinian and noetherian.

This implies that M possesses a composition series, i.e. a series of submodules

0 = M1 ⊂ M2 ⊂ · · · ⊂ Mn = M

∗ with simple quotients Mi+1/Mi isomorphic to L(λ) for diﬀerent λ ∈ h . The multiplicity of L(λ) is independent of the choice of composition series and is denoted by [M : L(λ)].

The common length of all such composition series is called the Jordan-Ho¨lder length of

Theorem 8.21. (cf. [30], Theorem 1.11) If M is indecomposable, all of its composition factors L(λ) have linked highest weights (i.e. they lie in the same orbit of W relative to the dot-action). Chapter 8. Categories of g-modules 64

Deﬁnition 8.22. An object P in an abelian category is called projective if the left exact functor Hom(P, −) is also right exact. Dually, an object Q is injective if the right exact functor Hom(−,Q) is also left exact. We say that a category C has enough projectives (correspondingly enough injectives) if for each M ∈ C, there is a projective object P ∈ C (correspondingly an injective object Q ∈ C) and an epimorphism P → M

(correspondingly a monomorphism M → Q).

Remark 8.23. The duality functor on O sends projective P to an injective P ∨. So the existence of enough projectives in O will imply the existence of enough injectives.

Proposition 8.24. (cf. [30], Proposition 3.8) Let λ ∈ h∗ be dominant. Then M(λ) is projective in O. If P ∈ O is projective and L is ﬁnite dimensional, then P ⊗ L is also projective in O.

Theorem 8.25. (cf. [30], Theorem 3.8) Category O has enough projectives.

Deﬁnition 8.26. For each M ∈ O a projective cover is an essential epimorphism

π : PM → M of the projective module PM ∈ O to M (i.e. no proper submodule of the projective module PM is mapped onto M).

Since the category O is artinian, from the existence of enough projectives it follows that each of its objects has a projective cover.

For each λ ∈ h∗ denote by

πλ : P (λ) −→ L(λ) a projective cover. From deﬁnition it follows that P (λ) has a unique maximal submodule

(isomorphic to ker πλ and hence indecomposable) and one has the sequence of epimor- phisms

P (λ) −→ M(λ) −→ L(λ).

Below are the two properties of the modules P (λ): Chapter 8. Categories of g-modules 65

Proposition 8.27. (cf. [30] Theorem 3.9) (i) In the category O, every indecomposable projective module is isomorphic to a projective cover P (λ) for some λ ∈ h∗,

(ii) For all M ∈ O,

dim HomO(P (λ),M) = [M : L(λ)].

Definition 8.28. We say that M ∈ O has a standard filtration or a Verma flag if there is a sequence of submodules

0 = M0 ⊂ M1 ⊂ M2 ⊂ · · · ⊂ Mn = M, for which Mi/Mi−1 is isomorphic to a Verma module. The number n is called the ﬁltration length. Denote by (M : M(λ)) the multiplicity with which each Verma module M(λ) occurs as a subquotient in standard ﬁltration of M.

An intrinsic characterization of objects which admit standard ﬁltration gives the following propositions.

Proposition 8.29. (cf. [27]) A module M ∈ O admits a standard ﬁltration if and only

1 ∨ ∗ if ExtO(M,M(λ) ) = 0 for any λ ∈ h .

Theorem 8.30. (cf. [30], Theorem 3.10) Each projective module in O has a standard

ﬁltration. In a standard ﬁltration of P (λ), the multiplicity (P (λ): M(µ)) is nonzero only if µ > λ while (P (λ): M(λ)) = 1.

Theorem 8.31 (BGG reciprocity). (cf. [30], Theorem 3.11) Let λ, µ ∈ h∗. Then

(P (λ): M(µ)) = [M(µ): L(λ)].

8.6 Harish-Chandra modules and the functor H

In present section we recall the well-known notion of the category of Harish-Chandra modules along with the functor H which establishes an isomorphism of its certain subcategory with the corresponding block of the BGG-category O. As the source of the material we mostly use [11]. Chapter 8. Categories of g-modules 66

Consider the category of U(g) ⊗ U(g)opp-modules (that is the category of U(g)- bimodules). Let

∼ g∆ := {(g, −g) ∈ g ⊕ g | g ∈ g} = g,

be the anti-diagonal in g ⊕ g and let G∆ be the algebraic group of g∆.

Deﬁnition 8.32. The category HC = HC(g ⊕ g,G∆) of Harish-Chandra modules is the subcategory of the category of U(g)⊗U(g)opp-modules, consisting of the modules M such that:

(HC1) M is U(g) ⊗ U(g)opp-ﬁnitely generated,

(HC2) M is G∆-integrable.

Remark 8.33. Note that (HC2) is equivalent to the following condition: considered as a g∆-module, M can be decomposed into a direct sum of ﬁnite-dimensional irreducible g∆-modules, each of them entering with ﬁnite multiplicity.

We have the restricted duality functor ∗ : HC → HC on the category of Harish-

Chandra modules deﬁned as follows. If M = ⊕Mλ is the decomposition of M ∈ HC relative to g∆ into the direct sum of ﬁnite-dimensional irreducible modules Mλ, then

∗ ∗ opp ∗ M = ⊕(Mλ) as a vector space. The actions of U(g) and U(g) on M are deﬁned correspondingly by

(uf)(m) = f(mu), (fu)(m) = f(um), u ∈ U, m ∈ M. We have M ∗∗ ∼= M.

opp Let M,N ∈ O. Endow the space HomC(M,N) with the U(g) ⊗ U(g) -module structure by

((u ⊗ u0)ϕ)(m) := uϕ(u0m),

0 opp for m ∈ M, ϕ ∈ HomC(M,N), u ⊗ u ∈ U(g) ⊗ U(g) . Chapter 8. Categories of g-modules 67

Deﬁnition 8.34. ([11]) For each M,N ∈ O denote by H(M,N) the G∆-integrable part of the space HomC(M,N):

G −int H(M,N) := (HomC(M,N)) ∆ .

opp (In other words, H(M,N) is U(g) ⊗ U(g) -submodule in HomC(M,N), consisting of g∆-ﬁnite vectors.)

Then H is a bifunctor. For each M ∈ O denote by HM the functor deﬁned by

HM (N) := H(M,N).

Proposition 8.35. (cf. [11], Lemma 6.1) (i) H is a bifunctor O × O → HC, contravariant in the ﬁrst variable and covariant in the second one;

(ii) If M ∈ O is projective, then the functor HM is exact.

Let HC0(g,B) be the subcategory in O0(g), consisting of the modules, on which the center acts by zero. Thus, on the modules from HC0(g,B) the center acts by zero and h acts diagonally.

We can decompose the category HC into the sum of categories corresponding to diﬀerent characters of Z2 := Z(g) ⊗ Z(g) ⊂ U(g) ⊗ U(g)opp. Each character of Z2 is

l 2 a pair (θ1, θ2), where θ1 is a character of Z := Z ⊗ 1 ⊂ Z and θ2 is a character of

r 2 r l Z := 1 ⊗ Z ⊂ Z . Let Iθ := 1 ⊗ ker θ ⊂ 1 ⊗ Z, Iθ := ker θ ⊗ 1 ⊂ Z ⊗ 1 Let now θ = χ0 be the zero central character of Z. We set

l r HC(0,0)(g ⊕ g,G∆) := {M ∈ HC | IθM = Iθ M = 0}. (8.2)

Consider now the functor H0 := HM(0), corresponding to the module M(0) ∈ O0(g). Chapter 8. Categories of g-modules 68

Theorem 8.36. (cf. [11], Proposition 6.3) The functor H0 establishes an equivalence of the categories HC0(g,B) and HC(0,0)(g ⊕ g,G∆).

Remark 8.37. The inverse functor to the functor H0 is the functor deﬁned by

N 7→ N ⊗U(g) M(0)

(cf. [11]). In particular, its composition with H0 is isomorphic to the identity functor.

Deﬁnition 8.38. For each w ∈ W deﬁne the Harish-Chandra module

G∆−int Uw := H0(M(w.0)) = (HomC(M(0),M(w.0)) . (8.3)

Note that Uw lies in HC(0,0)(g ⊕ g,G∆).

Set

U0 = U0(g) := C ⊗Z(g) U(g).

Lemma 8.39. ([7]) One has: Ue = U0(g).

We have the following braid relations for the modules Uw:

Theorem 8.40. ([33]) If w1, w2 ∈ W are such that `(w1w2) = `(w1) + `(w2), then

∼ Uw1 ⊗U0(g) Uw2 = Uw1w2 .

Let us associate with any Harish-Chandra module P the functor TP deﬁned by

TP : M 7→ P ⊗U(g) M

(the systematic study of functors of this nature is done in [11]). Since P is a quotient of a module V ⊗ U for some ﬁnite dimensional module V , it follows that TP preserves the categories HC, HC(0,0), O and O0.

In what follows we will deal only with the functors TP : M 7→ P ⊗U0(g) M deﬁned on the subcategories HC(0,0) and O0.

Set Tw := TUw . Note that Tw : O0 → O0. As a corollary from Theorem 8.40 we get the following braid relations for the functors Tw: Chapter 8. Categories of g-modules 69

Corollary 8.41. If w1, w2 ∈ W are such that `(w1w2) = `(w1) + `(w2), then

∼ Tw1 ◦ Tw2 = Tw1w2 .

∼ In particular, Te = IdO0 . Chapter 9

Semiregular modules

9.1 Semiregular module

Let us recall that we are working with a semi-simple complex Lie algebra g, having triangular decomposition

g = n− ⊕ h ⊕ n, and

b := h ⊕ n is the Borel subalgebra in g corresponding to the set R+ of positive roots.

We have the natural ZR-grading on g, where the elements of h have degree 0 and elements of the root space gα have degree α, α ∈ R. This induces the grading of the universal enveloping algebra of g:

M U(g) = Uλ(g), λ∈ZR and then the Z-grading: U(g) = ⊕n∈ZUn(g), where

M Un(g) = Uλ(g). ht(λ)=n

70 Chapter 9. Semiregular modules 71

Deﬁnition 9.1. ([3]) The left g-module

U(g) − ∗ Sn− := IndU(n−)U(n ) is called the left semiregular module with respect to the subalgebra n−.

Here Ind is the induction functor and U(n−)∗ denotes the left coregular n−-module:

M − ∗ − U(n ) := HomC(U(n )m, C). m>0

The left action of n ∈ n− on f ∈ U(n−)∗ is deﬁned by

(n · f)(u) := f(un), u ∈ U(n−).

Proposition 9.2. ([3, 4]) (i) There exists an inclusion of algebras U(g) ,→ Endg(Sn− ) whose image is a dense subalgebra in Endg(Sn− ).

(ii) One has: ∼ − ∗ EndU(g)(Sn− ) = HomC(U(n ) ,U(b)),

U(g) where U(b) denotes the g-module IndU(n−)C.

Corollary 9.3. ([3]) The semiregular module Sn− is a g-bimodule.

− −1 − Let w ∈ W . Consider the subalgebra nw := n ∩ w (n) of n . The corresponding enveloping algebra U(nw) is then the negatively graded subalgebra in U(g). Note that

− U(ne) = C and U(nw0 ) = U(n ).

The graded dual of U(nw), M ∗ U(nw) = HomC(U(nw)m, C), m>0 is the Z-graded nw-bimodule with

∗ (U(nw) )m = HomC((U(nw)−m, C). Chapter 9. Semiregular modules 72

Consider the semiregular g-module with respect to nw:

U(g) U(nw) ∗ S = Ind Coind C = U(g) ⊗ n U(n ) . w U(nw) C U( w) w

It is obviously a left U(g)-module and right U(nw)-module.

Theorem 9.4. ([3]) For each w ∈ W , Sw is a g-bimodule.

Remark 9.5. For w = e, the identity in group W , we have: Se = U(g); for w = w0, the longest element in W , we have

− ∗ U(g) − ∗ − − Sw0 = U(g) ⊗U(n ) U(n ) = IndU(n−)U(n ) = Sn .

− Let eα ∈ n \{0}. Deﬁne the Ore localization of U(g) at eα by

−1 U(eα)(g) := U(g) ⊗C[eα] C[eα, eα ].

Set Sα := U(eα)(g)/U(g).

Theorem 9.6. ([3] Lemma 3.2.6, cf. also [1, 4]) (i) U(eα)(g) is an associative algebra which contains U(g) as a subalgebra. ∼ (ii) For each simple root α and the simple reﬂection σα ∈ W , corresponding to α, Sα =

Sσα as left g-modules.

(iii) Let w = σα1 . . . σαk be the reduced expression. Then

∼ Sw = Sα1 ⊗U(g) · · · ⊗U(g) Sαk as g-bimodules.

− Remark 9.7. (cf. [4]) Let eα ∈ n \{0}. We have

∼ −1 Sα = U(g) ⊗C[eα] C[eα, eα ]/C[eα].

−1 Here C[eα, eα ]/C[eα] is the restricted dual for C[eα]. The non-degenerate pairing h, i

−1 between C[eα, eα ]/C[eα] and C[eα] is deﬁned by

hx, yi := res (xy) 0

−1 −1 (here x ∈ C[eα, eα ]/C[eα], y ∈ C[eα], so xy ∈ C[eα, eα ] and we take its coeﬃcient before the (−1)-th power). Chapter 9. Semiregular modules 73

9.2 Twisting functors

Deﬁnition 9.8. ([3]) Let w ∈ W . The corresponding automorphism of g deﬁnes the functor on the category of g-modules by conjugating the action of g by this automorphism.

∗ Denote this functor by Φw. Note that Φw(M)λ = Mw(λ), for λ ∈ h and a g-module M. Deﬁne the twisting functor on the category of g-modules by

Sw(M) := Φw(Sw ⊗U(g) M).

Theorem 9.6 implies:

Proposition 9.9. (cf. [3, 1]) The functor Sw is a functor from O to O, which preserves each of the categories Oχ, χ ∈ SpecZ(g).

Let us describe the twisting functor in slightly other terms.

Definition 9.10. For each simple reflection σα ∈ W the modified semiregular module

e σα σα corresponding to σα is deﬁned by Sα := (Sα) , where ( ) means that we twist the left action of g by this automorphism. We will usually refer to the modiﬁed semiregular module simply as to the semiregular module if no ambiguity arises.

Deﬁnition 9.11. Let w = σα1 . . . σαk be the reduced expression for w ∈ W . We deﬁne

e e e Sw := Sα1 ⊗U(g) · · · ⊗U(g) Sαk .

e0 e For w ∈ W denote by Sw the maximal quotient of Sw, belonging to the regular block

O0(g) as a left g-module.

Let h∆ denote the anti-diagonal {(h, −h) | h ∈ h} ⊂ g∆, and let T∆ be the algebraic group of h∆.

+− − Definition 9.12. Let O (g⊕g) denote the category of finitely generated N nT∆ oN - integrable g ⊕ g-modules. (That is h∆-semi-simple g ⊕ g-modules, which are n-locally finite as left g-modules and n−-locally finite as right g-modules.) We also denote by Chapter 9. Semiregular modules 74

+− +− O(0,0)(g ⊕ g) the subcategory in O (g ⊕ g) consisting of the modules, on which the center acts by zero (from both sides).

Remark 9.13. Notice a slight ambiguity in notation: we may treat the category O+−(g⊕ g) as the appropriate BGG category of g-bimodules and then repeat all its properties similar to those of the usual BGG category O (like we did in Chapter 8). In particular, we may decompose this category into the direct sum of its subcategories (blocks) with respect to the action of the center (cf. Section 8.4). Then it would be tempting to denote

+− +− by O(0,0)(g ⊕ g) the regular block of O (g ⊕ g). This is not the case in our notations +− as we require on the modules of O(0,0)(g ⊕ g) the center to act by zero and h∆ to act diagonally, i.e. this is the subcategory in the regular block of O+−(g ⊕ g). (A similar subcategory of the regular block O0 of O was denoted by HC0(g,B).)

Proposition 9.14. ([3, 4]) Let w = w0 be the longest element of the Weyl group. Then e0 +− Sw0 belongs to O(0,0)(g ⊕ g).

Deﬁnition 9.15. Let w ∈ W . Deﬁne the twisting functor Sw : O0(g) → O0(g) by

e0 Sw : M 7→ Sw ⊗U0(g) M.

The following two theorems are the main results of [33]:

Theorem 9.16. ([33], Corollary 7) We have the following braid relations for the twisting ∼ functors: if w1, w2 ∈ W are such that `(w1w2) = `(w1) + `(w2), then Sw1 ◦ Sw2 = Sw1w2 .

Recall the functors

Tw : O0(g) −→ O0(g),

(cf. Section 8.6) assigning to a module M ∈ O0(g) the module H0(M(w.0)) ⊗U0(g) M.

∼ Theorem 9.17. ([33], Corollary 10) Sw = Tw as functors on O0(g). Chapter 9. Semiregular modules 75

9.3 Example

As an explanatory example let us describe the semiregular module in details for the case of the Lie algebra sl2(C).

We ﬁx the standard basis of sl2(C):       0 0 1 0  0 1 f :=   , h :=   , e :=   , 1 0 0 −1 0 0 so that [h, e] = 2e,[h, f] = −2f,[e, f] = h. We ﬁx this basis ordering in the universal enveloping algebra U(g).

− Denote by n the one-dimensional Lie subalgebra of sl2(C) generated by the basis element f and consider the restricted dual of its universal enveloping algebra, U(n−)∗. It is a U(n−)-bimodule: if ϕ ∈ U(n−)∗, u ∈ U(n−), n ∈ n− then

uϕ(n) := ϕ(nu), ϕu(n) := ϕ(un). (9.1)

Consider the g-bimodule

− ∗ U(g) ⊗U(n−) U(n ) .

One has: U(n−) = C[f] (polynomials in f). The restricted dual of U(n−) is, by deﬁnition, the space U(n−)∗ = C[f, f −1]/C[f] of Laurent tails. The pairing between

U(n−) and U(n) is given by the residue at 0 (see Remark 9.7). Set

−1 S := U(g) ⊗C[f] C[f, f ]/C[f].

The left C[f]-module structure on C[f, f −1]/C[f] (deﬁned by (9.1)) obeys the following relations:   0 if k − m > 0, f k · f −m =  f k−m otherwise

(here for convenience k > 0, m > 0). Chapter 9. Semiregular modules 76

S is naturally a left g-module. It can also be equipped with the structure of a right g-module by

(u ⊗ f −1) · f := 0,

(u ⊗ f −1) · h := u(h − 2) ⊗ f −1,

(u ⊗ f −1) · e := ue ⊗ f −1 + u(h − 2) ⊗ f −2.

These formulas can be gotten by the following considerations.

0 = [h, f −1f] = [h, f −1]f + f −1[h, f] = [h, f −1]f − f −12f = [h, f −1]f − 2.

Therefore [h, f −1] = 2f −1, whence

f −1h = hf −1 − [h, f −1] = hf −1 − 2f −1 = (h − 2)f −1.

In the same fashion we get the formula for the right action by e.

By this construction we get a new algebra

−1 U(g)(f) := U(g) ⊗C[f] C[f, f ] with the multiplication deﬁned by the above formulas, which is called the non-commutative localization of U(g) at f.

∼ 1 2 Consider now the center Z of U(g). It is known that Z = C[c], where c = ef +fe+ 2 h is the Casimir element. Consider the g-bimodule

0 −1 S := C ⊗Z S = C ⊗Z U(f)/U = C ⊗Z U(g) ⊗C[f] C[f, f ]/C[f].

− This is the semiregular representation of sl2. Notice that the left and the right n -actions on S0 are locally nilpotent. Therefore the semiregular module S0 belongs to the regular block of the appropriate BGG category (which we could denote by O−−(g ⊕ g) reﬂecting the fact that both the left and the right actions of g are n−-locally ﬁnite).

If we twist the left g-action on S0, we will get the module Se0, lying in the category

+− O(0,0)(g ⊕ g). Chapter 10

The Jacquet functor

In this section we describe the construction of the (algebraic) Jacquet functor. The main references we follow are [25, 21]. A more classical description of this functor can be found in [53].

Let I+ be the associative subalgebra of U(g) without unit, generated by n. Thus I+ is a quotient of an augmentation ideal in the enveloping algebra U(n).

? Let M be a left g-module. We equip its linear dual M := HomC(M, C) with the following left g-module structure: (xf)(m) := f(τ(x)m), x ∈ g, f ∈ M ?, m ∈ M. Here

τ is the Chevalley involution on g. Deﬁne the following g-module:

† k ? ? k J+(M) := lim(M/I+M) = {f ∈ M | I+f = 0, for k = k(f) À 0}. −→

This way we get a functor

† J+ : left g-modules −→ left g-modules locally nilpotent relative to n.

Let HC(g,T ) be the category of ﬁnitely generated T -integrable left g-modules (i.e. the modules which satisfy the axioms (O1) and (O3) of the category O, see Deﬁnition 8.1).

† Lemma 10.1. ([25]) The functor J+ is an exact functor from the category HC(g,T ) to the BGG category O(g).

77 Chapter 10. The Jacquet functor 78

We deﬁne the Jacquet functor J on the category HC(g,T ) by taking the contragredient

† dual of J+:

† ∨ J(M) := (J+(M)) .

According to Lemma 10.1 and the property of contragredient duality the functor J is the exact functor from the category HC(g,T ) to the category O.

In the sequel we will need the notion of the Jacquet functor on the category of g- bimodules. Its construction is parallel to the one above and it is as follows.

Let I+− be the associative subalgebra in U(g ⊕ g) = U(g) ⊗ U(g) without unit, generated by n ⊕ n−. Consider the functor

† − J+− : g ⊕ g-modules −→ g ⊕ g-modules locally nilpotent relative to n ⊕ n deﬁned by

† k ? ? k J+−(M) := lim(M/I+−M) = {f ∈ M | I+−f = 0, for k = k(f) À 0}. −→

Similarly to the previous lemma it can be shown (cf. [25]) that the image of the functor

† J+− on ﬁnitely generated T∆-integrable g-bimodules (i.e. h∆-locally ﬁnite, h∆-semi-simple

+− − g-bimodules) lies in the category O (g⊕g) of ﬁnitely generated N nT∆ oN -integrable g ⊕ g-modules (Deﬁnition 9.12).

Definition 10.2. We define the Jacquet functor J on the category of finitely generated

T∆-integrable g-bimodules by:

† ∨ J(M) := (J+−(M)) .

(Here ( )∨ means taking the contragredient duality with respect to the action of the subalgebra h∆, cf. Section 8.4.)

Note that by the very construction the image of the functor J lies in the category

O+−(g ⊕ g). (Here we use the fact that the contragredient duality functor preserves the category O.) It can be shown ([25]) that this functor is exact and faithful. Chapter 10. The Jacquet functor 79

In what follows we will need to consider the Jacquet functor J only on the subcategory

HC(0,0)(g ⊕ g,G∆):

Lemma 10.3. (cf. [21]) The Jacquet functor J is an exact covariant functor from the

+− category HC(0,0)(g ⊕ g,G∆) to the category O(0,0)(g ⊕ g).

Let us describe a more direct construction of the Jacquet functor (cf. [21, 25]), which will be more useful for our geometric discussions. For simplicity we describe it for g- modules, but it can be naturally adapted to the case of g-bimodules.

Let µ : Gm → G be a cocharacter which when paired with the roots of g is positive precisely on the roots of n. (In fact µ corresponds to the sum of all positive coroots of g). Let h ∈ h denote the image of the Euler vector ﬁeld t∂t under the map dµ:

h := dµ(t∂t).

(It is known that the Euler vector field does not depend on the initial choice of the coordinate t on Gm.) Let now M be a module from HC(g,T ), i.e. a finitely generated T -integrable g- module. The action of h on M induces an action on each of the finite-dimensional vector

k k spaces M/I+M, k > 1. It follows that each of the spaces M/I+M can be decomposed into the direct sum of generalized eigenspaces relative to the action of h. The projec-

k+1 k tion M/I+ M → M/I+M is h-equivariant and so induces surjection on generalized eigenspaces relative to h.

Denote by

c k M := lim(M/I+M) ←− k the n-adic completion of M. It can be shown ([21], Lemma 2.3) that the generalized eigenspaces in Mc relative to h are ﬁnite dimensional and Mc is equal to their direct product. Chapter 10. The Jacquet functor 80

Proposition 10.4. ([21], Proposition 2.4 and cf. [25]) Let M ∈ HC(g,T ). Then J(M) is isomorphic to the sum of generalized h-eigenspaces in the n-adic completion Mc of M. Chapter 11

Algebraic results

11.1 Semiregular module as the image of the Jacquet

functor

† Let M be a g ⊕ g-module. Recall the Jacquet functor J+ assigning to the module M

† ? a g-module J+(M) equal to the n-locally nilpotent part of the linear dual M , and the

† † − Jacquet functor J+− assigning to M a g-module J+−(M) equal to the n ⊕ n -locally nilpotent part of M ?.

† † Note that both J+(M) and J+−(M) possess a natural structure of g ⊕ g-module. It turns out that in case when M is a Harish-Chandra module these structures coincide:

Theorem 11.1. Let P ∈ HC(g ⊕ g,G∆) be a Harish-Chandra module. Then

† ∼ † J+(P ) = J+−(P ) as g ⊕ g-modules.

To prove this theorem we need the following three lemmas.

Denote by O− = O−(g) the category opposite to O, consisting of U(g)-ﬁnitely generated, h-semisimple, n−-locally ﬁnite modules.

81 Chapter 11. Algebraic results 82

T −int − Lemma 1. Let M ∈ O, P ∈ HC. Then HomU (M,P ) belongs to O .

T −int Recall that HomU (M,P ) denotes the T -integrable part of HomU (M,P ), i.e. all h-ﬁnite vectors in HomU (M,P ).

∗ ∗ Proof of Lemma 1. It is known (cf. [11]) that P ⊗U M lies in O (P is the restricted dual of P in HC). We have:

T −int ∗ ∗ T −int HomU (M,P ) = HomU (M, (P ) )

∼ ∗ ∗ ∗ ∗ ∗ = ⊕λHomC(P ⊗U M, C)λ = ⊕λ(P ⊗U M)λ = (P ⊗U M)

∗ – the restricted dual of P ⊗U M in O. Then all we have to show is that if M ∈ O then its restricted dual M ∗ belongs to

O−. Obviously it is U-ﬁnitely generated and h-semisimple. We have to prove that it is n−-locally ﬁnite.

∗ ∗ Let M = ⊕λMλ be the direct sum of h-eigenspaces of M, so that M = ⊕λ(Mλ) .

∗ Let f ∈ (Mλ) . Then for any h ∈ h, x ∈ Mλ,

(hf)(x) = −f(hx) = −f(λ(h)x) = −λ(h)f(x).

∗ ∗ − we conclude that (Mλ) = (M )−λ. Let now g ∈ gα, α ∈ R . Then g : Mλ+α → Mλ, so

∗ ∗ g :(Mλ) → (Mλ+α) , i.e.

∗ ∗ g :(M )−λ → (M )−(λ+α).

Since the weights of M ∈ O are bounded from above we conclude that the weights of M ∗ are bounded from below. It then follows that the space U(n−)f is ﬁnite dimensional, i.e.

M ∗ is n−-ﬁnite.

Lemma 2. Let F : O → O− be a contravariant functor. Assume F is representable by an object X. Then on X there is a natural g-module structure, commuting with the original one, which is n−-locally finite. Thus X becomes a g-bimodule, n-locally finite with respect to the original action and n−-locally finite with respect to the new action. Chapter 11. Algebraic results 83

Proof. By Yoneda’s lemma, the second action of U on X (which we write from the right) is given by (x)u := uX (1X )(x), x ∈ X, u ∈ U. Here u is realized as an endomorphism of the functor F given by the morphisms uM : F (M) → F (M), which are the actions of u on the modules F (M). Note that since F (X) belongs to O−, n− acts on it locally

− ﬁnite. That is to say, for each y ∈ F (X), the space U(n )X y is ﬁnite dimensional.

Letting y to be the element of F (X) corresponding to 1X ∈ EndU (X) (under the equality

− F (X) = EndU (X)) we see that the space U(n )X (1X )(x) is ﬁnite-dimensional for any x ∈ X. But then the right action of n− on X is locally ﬁnite.

Lemma 3. Assume for each n ∈ Z we are given contravariant functors Fn on the category of U-ﬁnitely-generated g-modules, which are represented by the objects Xn. Then ⊕nFn is represented by ⊕nXn.

Proof of Lemma 3. Indeed, one has M M HomU(g)(M, ⊕nXn) = HomU(g)(M,Xn) = Fn(M). n n Here we use that M is ﬁnitely generated over U(g).

Now we are ready to give the proof of Theorem 11.1:

Proof of Theorem 11.1. Consider the (contravariant) functor FP from the category O to the category of vector spaces deﬁned by

∗ FP,λ(M) := HomU(g)(M,P )λ

∗ – the λ-eigenspace of HomU(g)(M,P ) relative to the (left) action of h. Since M is n-locally ﬁnite and hence N-integrable we may write:

∗ ∗ N−int FP,λ(M) = HomU(g)(M,P )λ = HomU(g)(M, (P ) )λ

† † = HomU(g)(M, J+(P ))λ = HomU(g)(M, J+(P )λ).

† We get that the functor FP,λ is representable by the (left) module J+(P )λ ∈ O(g). Chapter 11. Algebraic results 84

† By Lemma 2 there is another structure of g-module on J+(P )λ, commuting with the

† − original one, relative to which J+(P )λ is n -locally ﬁnite. By Lemma 1 the functor

M ∗ T −int FP := FP,λ : M 7→ HomU (M,P ) λ acts from O to O−. Using Lemma 3 we get that this functor is representable by the

† † object J+(P ) = ⊕λJ+(P )λ. Note that we automatically got the decomposition of the † module J+(P ) ∈ O(g) into the direct sum of its ﬁnite dimensional λ-eigenspaces relative † to the second action of h, i.e. J+(P ) becomes T -integrable with respect to the second † g-action. Therefore the bimodule J+(P ) (relative to the original action and the action gotten by Lemma 2) belongs to O+−(g ⊕ g) (see Deﬁnition 9.12).

† † +− We conclude that both J+(P ) and J+−(P ) lie in the same category O (g ⊕ g). † Since J+−(P ) obviously represents FP , from the uniqueness of the representing object we † ∼ † conclude that J+(P ) = J+−(P ).

For our purposes it is convenient to make the following deﬁnition:

Deﬁnition 11.2. A functor F : O0 → O0 is called ⊗-representable by an object X ∈ +− ∼ O(0,0)(g ⊕ g) if for any M ∈ O0, F(M) = X ⊗U0(g) M.

Proposition 11.3. The ⊗-representing object is unique up to isomorphism.

+− Proof. Let X ∈ O(0,0)(g ⊕ g) ⊗-represent a functor F : O0 → O0, i.e. for every M ∈ O0,

F(M) = X ⊗U0(g) M ∈ O0. One has:

∗ ∗ ∼ ∗ T −int (F(M)) = (X ⊗U0(g) M) = HomU0(g)(M,X ) .

Denote the functor

∗ T −int − M 7→ HomU0(g)(M,X ) , O0 → O0

by GX∗ . Chapter 11. Algebraic results 85

+− Assume now, that Y ∈ O(0,0)(g ⊕ g) is another object, ⊗-representing F. Then we get ∗ that Y represents (in the usual sense) the functor GX∗ . Since the representing (in the usual sense) object is unique up to isomorphism, we conclude that X∗ ∼= Y ∗ and then

X ∼= Y by the property of restricted duality.

Recall the functor TP : O0 → O deﬁned by TP (M) := P ⊗U0 M.

Theorem 11.4. Let P ∈ HC(0,0)(g ⊕ g,G∆) be such that TP is ⊗-representable by X ∈ +− ∼ O(0,0)(g ⊕ g). Then J(P ) = X as g-bimodules.

Proof. From the conditions of the theorem it follows that TP : O0 → O0.

Since X ⊗-represents the functor TP , one has:

∨ ∨ ∼ ∨ T −int (TP (M)) = (X ⊗U0(g) M) = HomU0(g)(M,X ) .

On the other hand,

∨ ∨ ∼ ? T −int (TP (M)) = (P ⊗U0(g) M) = HomU0(g)(M,P ) .

(Recall that P ? denotes the linear dual of P with the twisted (left) g-action xf(m) := f(τ(x)m).)

Since M lies in O0 and hence is N-integrable, we conclude that the last expression is equal to

? N−int T −int † T −int HomU0(g)(M, (P ) ) = HomU0(g)(M, J+(P )) .

Now using Theorem 11.1 we see that

∨ ∼ † T −int (TP (M)) = HomU0(g)(M, J+−(P )) .

† ∨ We conclude that both J+−(P ) and X represent the functor GP ∗ : O0 → O0 deﬁned by

? T −int ? GP (M) := HomU0(g)(M,P ) .

† ∼ ∨ +− Therefore J+−(P ) = X as bimodules (which belong to the category O(0,0)(g ⊕ g)). But † ∨ ∼ then (J+−(P )) = J(P ) = X by the property of contragredient duality. Chapter 11. Algebraic results 86

Remark 11.5. Notice that in the conclusion of the proof of the last theorem, both the

+− modules J(P ) and X belong to the category O(0,0)(g ⊕ g).

As a corollary of this theorem we obtain the main statement of our algebraic part:

e0 Theorem 11.6. (i) The modules Sw ⊗U0 J(Ue) and J(Uw) are isomorphic as the modules

+− from O(0,0)(g ⊕ g). ∼ e0 (ii) Let w = w0 be the longest element in the Weyl group W . Then J(Uw0 ) = Sw0 as the +− modules from O(0,0)(g ⊕ g).

Proof. Because the functors Sw and Tw = TUw are isomorphic (Theorem 9.17), their

⊗-representing objects are isomorphic (Proposition 11.3). The twisting functor Sw is obviously ⊗-representable by the module

e0 ∼ e0 +− Sw ⊗U0 J(U0) = Sw ⊗U0 J(Ue) ∈ O(0,0)(g ⊕ g).

Hence this module also ⊗-represents the functor Tw. Now we apply Theorem 11.4 and get (i).

e0 +− Note that for w = w0 the module Sw0 already belongs to O(0,0)(g ⊕ g). Since the twisting functor Sw0 is ⊗-representable by this module, we again apply Theorem 11.4 to obtain (ii).

? ∼ e0 ∨ ∼ e0 Corollary 11.7. One has Uw0 = Uw0 , (Sw0 ) = Sw0 .

∨ Proof. This is because M(w0.0) = L(w0.0) = M (w0.0).

11.2 Filtrations

For the modules M,N ∈ O(g) we denote by M £ N their exterior tensor product (i.e.

M £N = M ⊗N as vectors spaces and the action of (g1, g2) ∈ g⊕ g on (m ⊗ n) ∈ M £ N is given by (g1m ⊗ g2n)).

If M ∈ O admits a ﬁltration with subquotient modules isomorphic to Mn ∈ O, denote by [M : Mn] the number of times Mn appears as a subquotient in this ﬁltration. Chapter 11. Algebraic results 87

Lemma 11.8. (i) The Verma modules in O(g ⊕ g) have the form M(λ, µ) := M(λ) £

M(µ), where M(λ),M(µ) ∈ O(g).

(ii) A module M ∈ O(g⊕g) admits a ﬁltration with the subquotients isomorphic to Verma

1 modules M(λ, µ) ∈ O(g ⊕ g) if and only if ExtO(g⊕g)(M(λ, µ),M) = 0, for all λ and µ in X.

(iii) Let M ∈ O(g) have a ﬁltration with the subquotients isomorphic to contragredient

∨ [M:M(λ)∨] ∼ dual Verma modules M(λ) . Then as vector spaces C = HomO(M(λ),M).

1 ∨ (iv) ExtO(g)(M(λ),M(µ) ) = 0, for any Verma modules M(λ),M(µ) ∈ O(g).

Proof of (iii). One has:

g HomO(M(λ),M) = HomO(Indb+ C(λ),M) = Homb+ (C(λ),M) M ³ ´ ∗ [M:M(µ)∨] ∼= Homb+ (C(λ),M (µ)|b+ ) ⊗ C . µ

The last isomorphism is the isomorphism of vector spaces. Now

∨ b+ Homb+ (C(λ),M(µ) |b+ ) = Homb+ (C(λ), Coindh C(µ))   C, if λ = µ, = Homh(C(λ), C(µ)) =  0, if λ 6= µ.

We conclude that as vector spaces

[M:M(λ)∨] Homb+ (C(λ),M) ∼= C .

− g Let M (λ) := Indb− C(λ) be the module opposite to the Verma module M(λ) ∈ O(g).

− g Lemma 11.9. One has M(λ) ⊗ M (µ) = Indh C(λ + µ).

+ − − − + Proof. Let vλ generate M(λ) over b and vµ generate M (µ) over b . If h ∈ h then

+ − + − + − + − h(vλ ⊗ vµ ) = hvλ ⊗ vµ + vλ ⊗ hvµ = (λ + µ)(h)(vλ ⊗ vµ ). Chapter 11. Algebraic results 88

On the other hand,

+ + − + − − + − − n (vλ ⊗ vµ ) = vλ ⊗ M (µ), n (vλ ⊗ vµ ) = M(λ) ⊗ vµ .

− g Thus M(λ) ⊗ M (µ) = U(g) ⊗h C(λ + µ) = Indh C(λ + µ).

† +− Theorem 11.10. There is a bimodule ﬁltration of J+−(Ue) (as of an object from O (g⊕ g)) with subquotients isomorphic to M(w.0)∨ £ M −(−w.0)∨,(w ∈ W ) and for each

† ∨ − ∨ w ∈ W , [J+−(Ue): M(w.0) £ M (−w.0) ] = 1.

Proof. One has for any weights λ and µ:

1 − † ExtO+−(g⊕g)(M(λ) £ M (µ), J+−(Ue))

1 − g⊕g 1 − = Ext +− (M(λ) £ M (µ), Coind C) = Ext (M(λ) £ M (µ), C) O (g⊕g) g∆ g

1 ∨ = Extg(M(λ),M(−µ) ) = 0.

† By Lemma 11.8 (ii) we conclude that there is a ﬁltration of J+−(Ue) with subquotients isomorphic to M(λ)∨ £ M −(µ)∨.

By Lemma 11.8 (iii) and Lemma 11.9 one has:

† ∨ − ∨ − † [J+−(Ue): M(λ) £ M (µ) ] = HomO+−(g⊕g)(M(λ) £ M (µ), J+−(Ue))

− g = Homg(M(λ) £ M (µ), C) = Homg(Indh C(λ + µ), C)   C, if λ = −µ, = Homh(C(λ + µ), C) =  0, if λ 6= −µ.

Since the Verma modules M(λ),M(µ) both belong to O0(g), we have that λ and µ must be of the form w.0 for some w ∈ W , whence we get the desired.

Corollary 11.11. There is a bimodule ﬁltration of J(Ue) with the subquotients isomorphic to M(w.0) £ M −(−w.0), w ∈ W .

−1 ∨ Proposition 11.12. ([1]) The twisting functor Sw0 takes M(w.0) to M(w0w .0) . Chapter 11. Algebraic results 89

Corollary 11.13. If M ∈ O0 has a ﬁltration with subquotients isomorphic to Verma

modules, then Sw0 (M) has a ﬁltration with subquotients isomorphic to the contragredient Verma modules.

∼ Proof. Let M ⊃ M1 ⊃ · · · ⊃ Mk = 0 be a ﬁltration of M ∈ O0 with Mi/Mi+1 = M(wi.0), wi ∈ W for every i = 1, . . . , k − 1. Then

e0 e0 e0 Sw0 ⊗U0 M ⊃ Sw0 ⊗U0 M1 ⊃ · · · ⊃ Sw0 ⊗U0 Mk = 0, i.e.

Sw0 (M) ⊃ Sw0 (M1) ⊃ · · · ⊃ Sw0 (Mk) = 0

is a ﬁltration of Sw0 (M). By Proposition 11.12, for every i = 1, . . . k − 1

e0 e0 ∼ e0 Sw0 (Mi)/Sw0 (Mi+1) = Sw0 ⊗U0 Mi/Sw0 ⊗U0 Mi+1 = Sw0 ⊗U0 (Mi/Mi+1) ∼ e0 ∼ −1 ∨ = Sw0 ⊗U0 M(wi.0) = Sw0 (M(wi.0)) = M(w0wi .0) , what proves the assertion.

+− e0 Theorem 11.14. As an object of O(0,0)(g ⊕ g) the module Sw0 has a ﬁltration with −1 ∨ − subquotients isomorphic to M(w0w .0) £ M (−w.0), w ∈ W .

Proof. By Theorem 11.6 one has

e0 ∼ ∼ e0 Sw0 ⊗U0 J(Ue) = J(Uw0 ) = Sw0 .

+− Therefore, considering J(Ue) as an element of O(0,0)(g ⊕ g) we may write:

e0 ∼ Sw0 = Sw0 (J(Ue)).

Now we use Corollary 11.11. As a left g-module J(Ue) has a ﬁltration with subquotients ∼ e0 isomorphic to the Verma modules M(w.0), hence by Corollary 11.13, Sw0 (J(Ue)) = Sw0

−1 ∨ (as an object of O0(g)) has a ﬁltration with subquotients isomorphic to M(w0w .0) , e0 +− w ∈ W . Therefore Sw0 , as a bimodule (i.e. as an object of O(0,0)(g ⊕ g)), has the desired ﬁltration. Chapter 12

Geometric results

12.1 The De Concini-Procesi compactiﬁcation

In this section we recall the construction and the main properties of the canonical com- pactiﬁcation of an algebraic group G, due to De Concini and Procesi ([15]). For the references we mostly use [13, 12].

Let dim g = n. Consider the diagonal

eg∆ := {(x, x) ∈ g ⊕ g | x ∈ g} ⊂ g ⊕ g.

Then eg∆ can be regarded as a point in the Grassmanian Gr(g ⊕ g, n). The group G × G acts on Gr(g ⊕ g, n) in the natural way and the orbit of the point eg∆ is isomorphic to

G × G modulo the stabilizer of eg∆, which is equal to the diagonal G∆ ⊂ G × G:

∼ ∼ Orb(eg∆) = (G × G)/G∆ = G.

Deﬁnition 12.1. The closure X := Orb(eg∆) is called the DeConcini-Procesi or the wonderful compactiﬁcation of the group G.

The variety X is known to have the following properties ([15]):

(1) X is a smooth projective subvariety in Gr(g⊕g, n), containing G as an open orbit.

90 Chapter 12. Geometric results 91

(2) For any y ∈ X \ G, the corresponding point gy in the Grassmannian Gr(g ⊕ g, n) is a Lie subalgebra in g ⊕ g. If y1, y2 ∈ X belong to the same G × G-orbit, then the

subalgebras gy1 and gy2 corresponding to them are isomorphic. S (3) X\G = i∈I Di – the union of simple divisors (called the boundary divisors) with normal crossings and I is enumerated by simple roots of g. Each G × G-orbit closure in

X is a transversal intersection of the boundary divisors which contain it. Each Di is a

closure of an open G × G-orbit in X and Di1 ∩ · · · ∩ Dir is a closure of a G × G-orbit in X of codimension r. T (4) i∈I Di := Y is the unique closed G × G-orbit in X; it is isomorphic to the square of the ﬂag variety G/B × G/B.

(5) The closure T of T in X meets Y transversally at a discrete set of points, enumerated by the Weyl group W : T ∩ Y = {(w, w0w) | w ∈ W }.

12.2 Degeneration of the diagonal in the square of

the ﬂag variety

In this chapter we reproduce the results of M. Brion [12] (see also [13]) and construct a degeneration of the diagonal in the square of the ﬂag variety G/B × G/B into a union of Schubert varieties. Recall that the Schubert variety S(w) associated to w ∈ W is the closure of BwB/B in the ﬂag variety G/B.

Let X be the De Concini-Procesi compactiﬁcation of G. Consider the action of B ×B on B (by left and right multiplication). Then this action extends to an action of B × B on the completion B ⊂ X of B. Consider the G × G-equivariant map

π : G × G ×B×B B → X, deﬁned by (g, h, x) 7→ (g, h)x = gxh−1. One has the following diagram of families: Chapter 12. Geometric results 92

i / G × G ×B×B B G/B × G/B × X

π p X id / X Here i is the natural imbedding (g, h, x) 7→ (gB, hB, gxh−1) and p is the projection.

−1 π (1) consists of all triples (g, h, x) ∈ G × G ×B×B B such that gx = h, i.e. coincides

−1 −1 with the diagonal (G/B)∆ ⊂ G/B × G/B (the same is true for p (1)). The ﬁber π (y) at the point y ∈ Y ∼= G/B × G/B corresponding to the identity in G/B × G/B is equal to [ −1 π (y) = S(w) × S(w0w). w∈W It is possible to ﬁnd a smooth curve γ ⊂ T isomorphic to A1, containing 1 ∈ G and transversal to Y at z := (1, w0)y. Note that γ \{z} ⊂ T .

Proposition 12.2. ([15]) The Lie group of the Lie algebra corresponding to the point z

− in Gr(g ⊕ g, n) is isomorphic to N n T∆ o N .

Theorem 12.3. (cf. [13], Theorem 16) The morphism π : G × G ×B×B B → X is flat. Its restriction to γ, π−1(γ) → γ, is flat with fibers over g ∈ γ \{z} equal to ∼ {(x, gx) | x ∈ G/B} = (G/B)∆ and with fiber over z equal to [ −1 π (z) = S(w) × w0S(w0w). w∈W

12.3 Example

3 Let G := P GL2(C). Consider the natural imbedding P GL2(C) ,→ CP . Then the complement CP 3 \ G is deﬁned by the equation det A = 0 and so is isomorphic to

CP 1 × CP 1 ⊂ CP 3. In this case the De Concini - Procesi compactiﬁcation X of G is equal to CP 3, while the only boundary divisor is Y = CP 1 × CP 1 ∼= G/B × G/B. We identify the Cartan subgroup T in G with C∗. Let f, h, e be the standard basis of the Lie algebra g = sl2(C) (cf. Section 9.3): [h, f] = −2f,[h, e] = 2e,[e, f] = h. The Chapter 12. Geometric results 93 adjoint representation Ad gives:

−2 2 Adt(f) = t f, Adt(h) = h, Adt(e) = t e.

Consider the diagonal g∆ ⊂ g ⊕ g. Taking the 3-dimensional space containing the lines C · (f, f), C · (h, h), C · (e, e) we identify it with the point in the Grassmanian

Gr(6, 3).

−1 Consider the adjoint action of the anti-diagonal Cartan subgroup T∆ = {(t, t ) | t ∈

∗ T = C } ⊂ G × G on g∆:

−2 2 2 −2 Ad(t,t−1)(f, f) = (t f, t f), Ad(t,t−1)(h, h) = (h, h), Ad(t,t−1)(e, e) = (t e, t e).

The three lines passing through these elements deﬁne a 3-dimensional space that is a point in Gr(6, 3). This is the corresponding Lie subalgebra in g ⊕ g. Note that to 1 ∈ C∗ corresponds the algebra g∆. If we let now t go to ∞ we would get the Lie algebra corresponding to the boundary point: as a vector space it is spanned by the elements (0, f), (h, h), (e, 0). The corre-

− sponding Lie group is then N × T∆ × N .

12.4 The geometric Jacquet functor

Consider the curve γ in the De Concini-Procesi compactiﬁcation X of the group G, deﬁned in Theorem 12.3. Consider the constant family

p : G/B × G/B × γ → γ over γ and set γ∗ := γ \{z}.

On the ﬁber G/B × G/B over the point g ∈ γ \{z} we have an action of the

−1 −1 corresponding group Gg := (1, g)G∆(1, g) = {(y, gyg ) | y ∈ G} ⊂ G × G, while over

− z we have an action of the group Gz = N n T∆ o N . Note that Ge = G∆ – the diagonal Chapter 12. Geometric results 94 in G × G. We denote the obtained group scheme over γ by the same letter G, so that

G → γ is the group scheme acting on the constant family p : G/B × G/B × γ → γ.

The restriction to γ of the family π (deﬁned in Theorem 12.3) is a non-trivial subfamily in p.

Consider a DG/B×G/B-module M on the fiber of the family p at the identity e, which is equivariant with respect to the action of the group Ge = G∆ (see Appendix B for definitions). Thus M belongs to the category (D(G/B × G/B),G∆)-mod. In order to extend the D-module M to the D-module on the whole family G/B × G/B × γ∗ → γ∗, equivariant with respect to the group scheme G → γ, we consider an automorphism of this family acting on the fiber G/B × G/B over g ∈ γ∗ by (1, g):

(1, g)(h1B, h2B) := (h1B, gh2B).

∗ ˜ ∗ For each g ∈ γ we consider the D-modules Mg := (1, g) M (the inverse image of M under the action of the element (1, g) ∈ G × G). This way we get a D-module M˜ on the family G/B × G/B × γ∗ → γ∗, equivariant with respect to the action of the group

∗ ˜ ∗ scheme G → γ : each Mg, g ∈ γ , is a module from (D(G/B × G/B),Gg)-mod. The module M˜ is called the equivariantization of M.

Consider the natural imbedding j of the family G/B ×G/B ×γ∗ → γ∗ into the family

G/B × G/B × γ → γ.

Definition 12.4. (cf. [21]) Let M be a G∆-equivariant DG/B×G/B-module. We define ˜ the geometric Jacquet functor Jgeom by assigning to M the nearby cycles of j?(M) along the singular fiber p−1(z), where M˜ is the equivariantization of M.

The main result of Emerton-Nadler-Vilonen [21] is the following theorem:

Theorem 12.5. ([21]) (i) The Beilinson-Bernstein localization functor Loc (see Ap- pendix B) takes the Jacquet functor J to the geometric Jacquet functor Jgeom:

Loc ◦ J = Jgeom ◦ Loc. Chapter 12. Geometric results 95

(ii) The global sections functor Γ takes the geometric Jacquet functor Jgeom to the Jacquet functor J:

Γ ◦ Jgeom = J ◦ Γ.

12.5 A geometric realization of semiregular modules

According to Beilinson-Bernstein [6] (see Appendix B), the functor Γ of global sections maps the category (D(G/B × G/B),G∆)-mod of G∆-equivariant DG/B×G/B-modules to the category HC(0,0)(g ⊕ g,G∆) of Harish-Chandra modules,

Γ(G/B × G/B, Dist∆(G/B × G/B)) = Uw and

Loc(Uw) = Dist∆(G/B × G/B), where Dist∆(G/B × G/B) is the module of distributions on G/B × G/B with support on the diagonal (G/B)∆ (see Appendix B for the deﬁnition).

Let us recall the functor H0 assigning to a module M from the category O0 the

G∆-integrable part of the space HomC(M(0),M). In Theorem 11.6 we proved that the

Jacquet functor J applied to the module Uw0 := H0(M(w0.0)) is isomorphic to the semireg- e0 ular module Sw0 (as usual, w0 stands for the longest element of the Weyl group). Consider the geometric Jacquet functor (Deﬁnition 12.4):

− Jgeom :(D(G/B × G/B),G∆)-mod −→ (D(G/B × G/B),N × T∆ × N )-mod.

We apply the Emerton-Nadler-Vilonen construction (Theorem 12.5) to the D-module

Dist∆(G/B × G/B). Notice that this module is supported on the diagonal (G/B)∆ ⊂ G/B × G/B, so its equivariantization will be supported on the Brion’s subfamily π ⊂ p of “twisted diagonals” constructed in Theorem 12.3, and the nearby cycles of this Chapter 12. Geometric results 96 equivariantization will be supported on the ﬁber π−1(z) which is (by the same theorem) S the union of products of Schubert varieties w∈W S(w) × w0S(w0w). This way we obtain the following diagram of modules:

H0 / J / 0 M(w.0) Uw Se _ 0 _w0 Loc Loc J Dist (G/B × G/B) geom / e0 ∆ Loc(Sw0 )

We have proved the following statement, which was originally formulated as a conjecture by S. Arkhipov and R. Bezrukavnikov, inspired by results of M. Brion and P. Polo

[13]:

Theorem 12.6. The Beilinson-Bernstein localization of the semiregular module is isomorphic to the module obtained by applying the geometric Jacquet functor to the D-module

Dist∆(G/B × G/B) of distributions on the ﬂag variety G/B × G/B with support in the diagonal (G/B)∆: e0 ∼ Loc(Sw0 ) = Jgeom(Dist∆(G/B × G/B)).

∼ e0 In other direction: Γ(Jgeom(Dist∆(G/B × G/B))) = Sw0 as g ⊕ g-modules. Chapter 13

Appendix A: Representable functors and Yoneda lemma

Let C be a category and let Set be the category of sets. For each object A of C let

Hom(A, −) be the functor which maps objects X to the set Hom(A, X).

Deﬁnition 13.1. A functor F : C → Set is called representable if it is naturally isomorphic to the functor Hom(A, −) for some object A ∈ C, called the representing object of functor F.A representation of F is a pair (A, Φ), where

Φ : Hom(A, −) → F is a natural isomorphism.

A contravariant functor G : C → Set is representable if it is naturally isomorphic to the contravariant functor Hom(−,A) for some object A ∈ C.

Theorem 13.2 (Yoneda lemma). Let F be a functor from C to Set. Then for each object A ∈ C the natural transformations from the functor hA := Hom(A, −) to F are in one-to-one correspondence with the elements of F(A):

∼ Nat(hA,F ) = F (A).

97 Chapter 13. Appendix A: Representable functors and Yoneda lemma 98

Given a natural transformation Φ from hA to F, the corresponding element of F (A) is u = ΦA(IdA).

There is a contravariant version of Yoneda’s lemma which concerns contravariant functors from C → Set. This version involves the contravariant hom-functor Hom(−,A).

Given an arbitrary contravariant functor G from C → Set, Yoneda’s lemma asserts that

0 ∼ Nat(hA, G) = G(A).

From Yoneda’s lemma follows the uniqueness of the representing object:

Proposition 13.3. Representations of functors are unique up to a unique isomorphism.

That is, if (A1, Φ1) and (A2, Φ2) represent the same functor, then there exists a unique isomorphism ϕ : A1 → A2 such that

−1 Φ1 ◦ Φ2 = Hom(ϕ, −)

as natural isomorphisms from Hom(A2, −) to Hom(A1, −). Chapter 14

Appendix B: D-modules and the

Beilinson-Bernstein localization

In this chapter we brieﬂy recall the notion of a D-module along with the Beilinson-

Bernstein localization theory. The proofs of all the assertions cited here can be found in

[27] and [16].

14.1 Basics on D-modules

Let X be a smooth aﬃne algebraic variety over C. Denote by OX the structure sheaf of

X. A linear map D : OX → OX is a differential operator of order k if its commutator with the operation of multiplication by any function f ∈ OX ,[D, f], is a differential operator of order k − 1. The space of differential operators of order k is denoted by

D(X)k and the union ∪D(X)k is denoted by D(X).

Proposition 14.1. The space DX is generated by the elements f ∈ OX , ξ ∈ TX , subject to the following relations:

f1 ? f2 = f1f2, f ? ξ = fξ, ξ1 ? ξ2 − ξ2 ? ξ1 = [ξ1, ξ2], ξ ? f = fξ + ξ(f), where ? denotes the multiplication in D(X).

99 Chapter 14. Appendix B: D-modules and the Beilinson-Bernstein localization100

n n Example. Let X be an aﬃne variety A = SpecC[x1, . . . , xn], Then D(A ) is generated by x1, . . . , xn, ∂1, . . . , ∂n with the relations

[xi, xj] = 0 = [∂i, ∂j], [∂i, xj] = δij.

This algebra is called the Weyl algebra and is denoted by Wn.

Remark 14.2. Note that in general we have the following isomorphism of left OX - modules: ∼ D(X) = OX ⊗ C[∂1, . . . , ∂n]

∗ (where we have fixed a system of functions on X whose differentials span the space Tx X for every x ∈ X and ∂i are the vector fields for which [∂i, xj] = δij).

If X is an arbitrary (not necessarily affine) algebraic variety, we may define a sheaf of algebras D(X) by setting for an affine U ⊂ X,

Γ(U, D(X)) := D(U).

It is quasi-coherent with respect to the structure of sheaf of OX -module on D(X), i.e. for each basic open subset Xf in X

D(Xf ) = OXf ⊗OX D(X).

Deﬁnition 14.3. A module M over D(X) is called a D-module on X (we write DX - module, when we need to specify the variety). It can be thought of as an OX -module, endowed with the action of the Lie algebra of vector ﬁelds, and such that

f · (ξ · m) = (f · ξ) · m, ξ · (f · m) = ξ(f) · m − f · (ξ · m),

for m ∈ M, f ∈ OX , ξ ∈ T (X).

Example. 1) D(X) is both a left and a right D-module.

2) OX is a left D-module by D · f := D(f). Chapter 14. Appendix B: D-modules and the Beilinson-Bernstein localization101

n 3) The space ΩX of top diﬀerential forms (n = dim X) is a right D-module by

ω · f := fω, ω · ξ := −Lie ξ(ω)

(where Lie ξ(ω) denotes the Lie derivative of ω along ξ).

r n 4) For a left D-module M on X we can deﬁne a right D-module M := M ⊗OX ΩX by

(m ⊗ ω) · f := m ⊗ fω, (m ⊗ ω) · ξ := −(ξ · m) ⊗ ω + m ⊗ ω · ξ.

This deﬁnes an equivalence between the categories of left and right D-modules on X.

Denote by D(X)-mod (resp. by D(X)r-mod) the category of quasi-coherent sheaves of left (resp. right) D(X)-modules.

Let φ : Y → X be a map of algebraic varieties. Let M ∈ D(X)-mod. Deﬁne

∗ φ (M) := OX ⊗OY M ∈ D(Y )-mod with the following action of vector ﬁelds on it:

ξ · (f ⊗ m) := ξ(f) ⊗ m + f · dφ(ξ)(m),

where ξ 7→ dφ(ξ) is the diﬀerential of φ, thought of as a map TY → OY ⊗OX TY . The functor φ∗ is called the inverse image functor.

Remark 14.4. The functor φ∗ : D(X)-mod→ D(Y )-mod is only right exact and needs to be derived by replacing M by a complex of D(X)-modules, which are ﬂat as OX - modules.

Consider now the left D(Y )-module

∗ ∼ φ (D(X)) = OY ⊗OX D(X).

The right multiplication on D(X) acts by endomorphisms of the left D-module structure.

Hence φ∗(D(X)) carries a right D(X)-module structure. Therefore given a right D(Y )- module N we can consider the right D(X)-module

∗ φ?(N) := N ⊗D(Y ) φ (D(X)). Chapter 14. Appendix B: D-modules and the Beilinson-Bernstein localization102

This is the direct image functor.

14.2 Equivariant D-modules and the Beilinson-Bernstein

localization

Now we give a construction of the Beilinson-Bernstein localization functor.

Let X be a smooth algebraic variety, and let G be a group acting on it. We have a homomorphism of Lie algebras a : g → Γ(X,T (X)), and hence a homomorphism of associative algebras

a : U(g) → Γ(X, D(X)).

Hence, if M is a D-module on X, its space of global sections is naturally a g-module.

Thus we obtain a functor

Γ : D(X)-mod → g-mod.

The left adjoint to the functor Γ is called the localization functor. If M ∈ g-mod, then it is deﬁned by

Loc : M → D(X) ⊗U(g) M.

(Here U(g) denotes the constant sheaf of algebras over X and M denotes the corresponding sheaf of modules over it.)

When G is a semi-simple aﬃne algebraic group, and X is its ﬂag variety: X = G/B, where B is the Borel subgroup in G then we have the following result:

Theorem 14.5 (Beilinson-Bernstein [6]). (i) The homomorphism

Z(g) → Γ(G/B, D(G/B))

factors through the character χ0, corresponding to the trivial g-module.

(ii) The resulting homomorphism U(g)0 → Γ(G/B, D(G/B)) is an isomorphism.

(iii) The functor Γ : D(G/B)-mod → U(g)0-mod is exact and faithful. Chapter 14. Appendix B: D-modules and the Beilinson-Bernstein localization103

(iv) The functor Γ and its adjoint Loc : U(g)0-mod → D(G/B)-mod are mutually quasi- inverse equivalences of categories.

Example 14.6. (Localization of Verma modules) Let iw : Xw ,→ X, w ∈ W be the

imbedding of the Schubert cell Xw into X and let OXw be the sheaf of functions on Xw. It can be shown (see e.g. [27]) that

∨ ∼ Loc(M(−w(ρ) − ρ) ) = (iw)?(OXw ),

and the support of this D-module is the closure of the Schubert cell Xw.

Consider now the case when X is the square of the ﬂag variety X = G/B × G/B on F which the diagonal G∆ ⊂ G × G acts with |W | orbits Cw: X = w∈W Cw. Denote the category of quasi-coherent sheaves of G∆-equivariant DG/B×G/B-modules by (D(G/B ×

G/B),G∆)-mod. For completeness we give the deﬁnition of equivariant sheaves of D- modules below after the formulation of Theorem 14.7.

w Let OCw be the sheaf of functions on the orbit Cw and denote by i the inclusion of

Cw into X. The following result is similar to the one above:

Theorem 14.7. ([6]) (i) The localization functor Loc establishes an equivalence of the category HC(0,0)(g ⊕ g,G∆) (see (8.2)) and the category (D(G/B × G/B),G∆)-mod with the quasi-inverse functor of global sections Γ.

(ii) Let w := w0 be the longest element of the Weyl group W . The image of the module Uw

w (see (8.3)) under Loc is isomorphic to the module i? OCw of distributions on the square of ∼ the ﬂag variety G/B × G/B with support on the diagonal (G/B)∆ = Cw ⊂ G/B × G/B.

Let us denote the module iw0 O by Dist (G/B × G/B). ? Cw0 ∆ We now give the deﬁnition of G-equivariant D-modules (cf. [27]). Let X be a scheme endowed with an action of an algebraic group G:

act : G × X → X. Chapter 14. Appendix B: D-modules and the Beilinson-Bernstein localization104

We have a natural map a : g = Lie(G) → Vect(X). A quasi-coherent sheaf M of

DX -modules is called G-equivariant, if it is given an isomorphism

∗ ∼ ∗ φM : act (M) = p (M), (14.1) where p : G × X → X is a projection. The following axioms, where all the morphisms are the morphisms of D-modules, must hold:

(1) the restriction of φM on 1 × X is the identity map M → M; (2) the following diagram must commute:

(Id×act)∗(φ ) Id×φ (Id × act)∗ ◦ act∗(M) M / (Id × act)∗ ◦ p∗(M) M / q∗(M)

e = φ e (m × Id)∗ ◦ act∗(M) M / (m × Id)∗p∗(M) / q∗(M) where q : G × G × X → X is the projection and m the multiplication map on G.

We can naturally extend the above construction considering a group scheme G → S over the base S acting on a smooth family X → S. We come to the notion of a D-module on the family X → S equivariant with respect to the action of the group scheme G → S.

In particular, we have the morphisms

act, p : G ×S X → X given by the action map and the projection, and the condition of equivariance means that it is given an isomorphism between act∗(M) and p∗(M), satisfying the conditions

(1)-(3) above, appropriately adapted. Chapter 15

Appendix C: The nearby cycles functor

Here we give a construction of the nearby cycles functor following the lecture notes of Claude Sabbah ([41, 40]) and [14]. All the D-modules under the consideration are supposed to be regular holonomic.

15.1 The Kashiwara-Malgrange ﬁltration

Let f : X → A1 be a smooth function on a complex algebraic variety X. We replace X

1 1 1 by X × A = {(x, f(x))} and f by projection to A . Let if : X,→ X × A denotes the graph inclusion. Denote by t the coordinate on A1.

Denote by V•DX the following increasing filtration (called the V -filtration or the Kashiwara-Malgrange filtration) indexed by Z associated with t: if we denote by I the ideal (t), then

+ l l−k P ∈ VkDX ⇔ for any l ∈ Z , P ·I ⊂ I .

0 In any local coordinate system (t, x2, . . . , xn) = (t, x ) on X, the germ P ∈ DX is in

VkDX if

105 Chapter 15. Appendix C: The nearby cycles functor 106

P 0 0 j1 j (a) P = 0 a (t, x )(t∂ ) ∂ 0 , if k = 0; j=(j1,j ) j t x |k| + (b) P = t Q with Q ∈ V0DX , if k ∈ {−Z − 0};

P j + (c) P = 06j6k Qj∂t with Qj ∈ V0DX , if k ∈ {Z − 0}. Let Y := f −1(0). We have the following properties of the V -ﬁltration:

• VkDX · VlDX ⊂ Vk+lDX with equality for k, l 6 0 or k, l > 0;

• VkDX\Y = DX\Y for any k ∈ Z; ³ ´ • ∩k VkDX = {0}. |Y

Let M be a left DX -module (here we replace M by its direct image (if )?M) equipped with an exhaustive increasing ﬁltration U•M indexed by Z, which is compatible with the

V -ﬁltration on DX : VkDX · UlM ⊂ Uk+lM. The ﬁltration is said to be good if for any compact set K ⊂ X, there exists k0 > 0 such that in a neighborhood of K we have for all k > k0: X k−k0 j U−kM = t U−k0 M and UkM = ∂t Uk0 M,

06j6k−k0 and each UlM is V0DX -coherent.

15.2 Specializable modules and nearby cycles

A coherent DX -module is called specializable along {t = 0} if locally there exist a good

V -ﬁltration U•M and a monic polynomial (called the Bernstein polynomial) b(s) ∈ C[s] such that

U b(−(∂tt + k)) · Grk M = 0, for all k ∈ Z.

If M is specializable along {t = 0}, then there exists a good V -ﬁltration on M,

V indexed by a discrete subset of C, such that each Grα M = VαM/V<αM is a coherent

DY -module, on which the endomorphism N induced by −∂tt + α is nilpotent.

Deﬁnition 15.1. We denote

V ψαM := Grα M Chapter 15. Appendix C: The nearby cycles functor 107

P and call the sum Ψ := α∈C ψα the nearby cycles functor (along X × {0}).

An important property of the functor of nearby cycles Ψ is that it commutes with the inverse images under smooth morphisms:

Theorem 15.2. Let α : X → Y be a smooth morphism of varieties, f : Y → C be a ˜ smooth function, and f be the lift of f to X. Consider the DY -module M and its nearby cycles Ψ(M) along f −1(0). Then the module α∗(Ψ(M)) is naturally isomorphic to the module Ψ(α∗(M)) of nearby cycle of α∗(M) along f˜−1(0).

Assume that a group scheme G → C∗ acts on a smooth ﬂat family X → C∗. Let M be a D-module on X → C∗ equivariant with respect to the action of the group scheme

∗ ∗ G → C . Denote by Mt the Gt-equivariant module over the point t ∈ C (i.e. the ﬁber of M at t). In particular this means that it is given an isomorphism of the corresponding

DGt×Xt -modules: ∗ ∼ ∗ φt : act (Mt) = p (Mt).

Let now G˜ be the group scheme over C whose restriction to C∗ is equal to G, and

X˜ → C be a ﬂat family over C, on which G˜ acts. Denote by i the open imbedding

X,→ X˜.

Consider the direct image i?M. This is a DX˜ -module. Let us check that it is equivariant with respect to the action of the group scheme G˜. We have to check that for each t ∈ C there is an isomorphism

˜ ∗ ∼ ∗ φt : act (i?Mt) = p (i?Mt).

(We denote by the same symbols the corresponding action map and the projection G˜ ×

X˜ → X˜.) But this follows from the fact that i is an open imbedding of ﬂat families and so commutes with inverse images:

∗ ∼ ∗ ∼ ∗ ∼ ∗ act (i?Mt) = i?(act Mt) = i?(p Mt) = p (i?Mt). Chapter 15. Appendix C: The nearby cycles functor 108

˜ Therefore i?M is G-equivariant. ˜ ˜ Consider now the nearby cycles functor Ψ along the ﬁber X0 of the family X over t = 0.

Proposition 15.3. The D -module Ψ(i M) is equivariant with respect to the action of X˜0 ? ˜ the group G0.

Proof. This follows from the fact that the nearby cycles functor commutes with inverse images under smooth morphisms (Theorem 15.2): we construct the desired automorphism by ∗ ∼ ∗ ∼ ∗ ∼ ∗ act0(Ψ(i?M)) = Ψ(act (i?M)) = Ψ(p (i?M)) = p0(Ψ(i?M)). Bibliography

[1] H. H. Andersen and N. Lauritzen, Twisted Verma modules, in: Studies in Memory

of Issai Schur, 1-26, v.210, Progress in Math., Birkh¨auser,Basel, 2002

[2] Henning Haahr Andersen and Catharina Stroppel, Twisting functors on O, Rep-

resent. Theory 7 (2003), 681-699.

[3] S. M. Arkhipov, A new construction of the semi-inﬁnite BGG resolution, arXiv:q-

alg/9605043 v3.

[4] Sergey Arkhipov, Algebraic construction of contragradient quasi-Verma modules

in positive characteristic, Adv. Stud. Pure Math. 40, Math. Soc. Japan, Tokyo

(2004), 27-68.

[5] Erik Backelin, The Hom-spaces between projective functors, Representation The-

ory, Volume 5 (2001) 267-283.

[6] A. Beilinson and J. Bernstein, Localisations de g-modules, C. R. Acad. Sci. Paris,

vol 292 (1981), 15-18.

[7] Alexander Beilinson and Victor Ginzburg, Wall-Crossing Functors and D-

modules, Represent. Theory 3 (1999), 1-31.

[8] I. N. Bernstein, I. M. Gel’fand, and S. I. Gel’fand, Structure of representations

generated by highest weights, Funct. Anal. Appl. 5 (1971), 1-9.

109 Bibliography 110

[9] I. N. Bernstein, I. M. Gel’fand, and S. I. Gel’fand, Category of g modules, Funct.

Anal. and Appl. 10, (1976), 87-92.

[10] I. N. Bernstein, I. M. Gel’fand, and S. I. Gel’fand, On a category of g-modules,

Funct. Anal. Appl. 10 (1976), 1-8.

[11] J. M. Bernstein, S. I. Gelfand, Tensor products and inﬁnite dimensional represen-

tations of semi-simpleLie algebras, Compositio Mathematica, tome 41, n2 (1980),

245-285.

[12] M. Brion, The behaviour at inﬁnity of the Bruhat decomposition, Comment.

Math. Hev. 73 (1998), 137-174.

[13] Michel Brion, Patrick Polo, Large Schubert varieties, Represent. Theory 4 (2000),

97-126.

[14] Nero Budur, On the V -ﬁltration of D-modules, arXiv:math.AG/0409123v1.

[15] C. De Concini, C. Procesi, Complete symmetric varieties, In: Invariant Theory,

Lecture Notes in Math. 996, Springer-Verlag 1983.

[16] S. C. Coutinho, A Primer of Algebraic D-modules, Cambridge University Press

1995.

[17] L. A. Dickey, Soliton equations and Hamiltonian systems, Adv. Series in Math.

Phys., 12 (1991), 310pp.

[18] J. Dixmier, Algebres enveloppantes, Paris, Gauthier-Villars, 1974.

[19] Dmitry Donin, Boris Khesin, Pseudodiﬀerential symbols on Riemann surfaces and

Krichever-Novikov algebras, Comm. Math. Phys., 272:2 (2007), 507-527. Bibliography 111

[20] A. S. Dzhumadildaev, Derivations and central extensions of the Lie algebra of

formal pseudodiﬀerential operators, St. Petersburg Math. J. 6, no. 1 (1995), 121–

136.

[21] M. Emerton, D. Nadler, and K. Vilonen, A geometric Jacquet functor, Duke Math.

J. Volume 125, Number 2 (2004), 267-278.

[22] B. Enriquez, S. Khoroshkin, A. Radul, A. Rosly, and V. Rubtsov, Poisson-Lie

aspects of classical W-algebras, AMS Translations (2) 167, AMS Providence, RI,

1995, 37–59.

[23] B. L. Feigin, gl(λ) and cohomology of a Lie algebra of diﬀerential operators,

Russian Math. Surveys 43, no. 2 (1988), 169–170.

[24] D. B. Fuchs, Cohomology of inﬁnite-dimensional Lie algebras, Consultants Bu-

reau, New-York - London, 1986.

[25] Victor Ginzburg, On primitive ideals, Selecta Math., 9 (2003), 379-407.

[26] V. Ginzburg, Characteristic varieties and vanishing cycles, Invent. math. 84

(1986), 327-402.

[27] Dennis Gaitsgory, Geometric representation theory, Lecture notes, (2005).

[28] Sergei I. Gelfand, Yuri I. Manin, Methods of Homological Algebra, Springer 2nd

edition, (2006).

[29] J. E. Humphreys, Introduction to Lie Algebras and Representation Theory,

Springer-Verlag (1978).

[30] J. E. Humphreys, Representations of Semisimple Lie Algebras in the BGG Cate-

gory O, Grad. Studies Math., Amer. Math. Soc., Providence, RI, 2008. Bibliography 112

[31] M. Kashiwara, Vanishing cycle sheaves and holonomic systems of diﬀerential equa-

tions, In Algebraic Geometry (Tokyo/Kyoto, 1982), volume 1016 of Lecture Notes

in Math., Springer, Berlin (1983), 134-142.

[32] B. Khesin, I. Zakharevich, Poisson-Lie group of pseudodiﬀerential symbols, Com-

mun. Math. Phys. 171 (1995), 475–530.

[33] Oleksandr Khomenko, Volodymyr Mazorchuk, On Arkhipov’s and Enright’s func-

tors, Math.Z. 249 (2005), 357-386.

[34] O. S. Kravchenko, B. A. Khesin, Central extension of the algebra of pseudodiﬀer-

ential symbols, Funct. Anal. Appl. 25, no. 2 (1991), 83–85.

[35] I. M. Krichever, S. P. Novikov, Algebras of Virasoro type, Riemann surfaces and

structure of the theory of solitons, Funct. Anal. Appl. 21, no. 2 (1987), 126–142.

[36] I. M. Krichever, S. P. Novikov, Virasoro type algebras, Riemann surfaces and

strings in Minkowski space, Funct. Anal. Appl. 21, no. 4 (1987), 294-307.

[37] W. L. Li, 2-cocyles on the algebra of diﬀerential operators, J. Algebra 122, no. 1

(1989), 64–80.

[38] J.-H. Lu, A. Weinstein, Poisson Lie groups, dressing transformations, and Bruhat

decompositions. J. Diﬀ. Geom. 31, no.2 (1990), 501–526.

[39] A. O. Radul, A central extension of the Lie algebra of diﬀerential operators on a

circle and W-algebras, JETP Letters 50, no. 8 (1989), 371–373.

[40] Claude Sabbah, Hodge theory, singularities and D-modules, Lecture notes.

[41] Claude Sabbah, Specialization of D-modules, Application to pure D-modules,

Expos´es`aPadoue, Mai 2001. Bibliography 113

[42] M. Schlichenmaier, Krichever-Novikov algebras for more than two points, Letters

in Math. Phys. 19 (1990), 151–165.

[43] M. Schlichenmaier, Verallgemeinerte Krichever-Novikov Algebren und deren

Darstellungen, PhD-thesis, Universit¨atMannheim, Juni 1990; Diﬀerential oper-

ator algebras on compact Riemann surfaces, Generalized symmetries in physics

(Clausthal, 1993), 425–434, World Sci. Publ., River Edge, NJ, 1994.

[44] M. Schlichenmaier, Local cocycles and central extensions for multipoint algebras

of Krichever-Novikov type, J. reine angew. Math. 559 (2003), 53-94.

[45] Martin Schlichenmaier, An Introduction to Riemann Surfaces, Algebraic Curves

and Moduli Spaces, Springer-Verlag Berlin Heidelberg (1989).

[46] M. A. Semenov-Tyan-Shanskii, Dressing transformations and Poisson group ac-

tions. Kyoto University, RIMS Publications 21, no. 6 (1985), 1237–1260.

[47] O. K. Sheinman, Aﬃne Lie algebras on Riemann surfaces, Funct. Anal. Appl. 27,

no. 4 (1993), 266–272.

[48] O. K. Sheinman, Aﬃne Krichever-Novikov algebras, their representations and

applications, In: Geometry, Topology and Mathematical Physics. S.P.Novikov’s

Seminar 2002-2003, V.M.Buchstaber, I.M.Krichever, eds. AMS Translations (2)

212, AMS Providence, RI, 2004, 297–316.

[49] Boris Shoikhet, Cohomology of the Lie algebras of diﬀerential operators: lifting

formulas, Amer. Math. Soc. Transl. Ser. 2, 185, Amer. Math. Soc., Providence,

RI (1998), 95-110.

[50] Wolfgand Soergel, Equivalences de certainces categories de g-modules, C.R. Acad.

Sc. Paris, t.303, Serie I, n15, (1986). Bibliography 114

[51] A. Voronov, Semi-inﬁnite homological algebra, Inventiones Math. 113 no. 1

(1993), 103-146.

[52] F. Wagemann, Some remarks on the cohomology of Krichever-Novikov algebras,

Letters in Math. Phys. 47 (1999), 173–177.

[53] N. Wallach, Real reductive groups. I, Pure and Applied Math., vol. 132, Academic

Press (1988).