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Lie Algebras of Differential Operators and D-Modules by Dmitry Donin A Lie algebras of differential operators and D-modules by Dmitry Donin A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Graduate Department of Mathematics University of Toronto Copyright °c 2008 by Dmitry Donin Abstract 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 differ- ential 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 holo- morphic 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 con- nection 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 diag- onal 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 gener- alizations 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 construc- tion 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) al- gebras. 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 holomor- phic 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 equiva- lent 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 op- erators 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.
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