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Localization of Cohomologically Induced Modules to Partial Flag Varieties LOCALIZATION OF COHOMOLOGICALLY INDUCED MODULES TO PARTIAL FLAG VARIETIES by Sarah Kitchen A dissertation submitted to the faculty of The University of Utah in partial fulfillment of the requirements for the degree of Doctor of Philosophy Department of Mathematics The University of Utah March 2010 Copyright c Sarah Kitchen 2010 All Rights Reserved THE UNIVERSITY OF UTAH GRADUATE SCHOOL SUPERVISORY COMMITTEE APPROVAL of a dissertation submitted by Sarah Kitchen This dissertation has been read by each member of the following supervisory committee and by majority vote has been found to be satisfactory. Chair: Dragan Miliˇci´c Aaron Bertram Peter Trapa Henryk Hecht Y.P. Lee THE UNIVERSITY OF UTAH GRADUATE SCHOOL FINAL READING APPROVAL To the Graduate Council of the University of Utah: I have read the dissertation of Sarah Kitchen in its final form and have found that (1) its format, citations, and bibliographic style are consistent and acceptable; (2) its illustrative materials including figures, tables, and charts are in place; and (3) the final manuscript is satisfactory to the Supervisory Committee and is ready for submission to The Graduate School. Date Dragan Miliˇci´c Chair, Supervisory Committee Approved for the Major Department Aaron Bertram Chair/Dean Approved for the Graduate Council Charles A. Wight Dean of The Graduate School ABSTRACT Abstract goes here. ?? CONTENTS ABSTRACT ...................................................... ii CHAPTERS 1. INTRODUCTION ............................................. 1 1.1 Cohomological Induction and Harish-Chandra Modules . 1 1.2 The Kazhdan-Lusztig Conjectures . 2 1.3 Localization of Harish-Chandra Modules . 3 1.4 The Beilinson-Ginzburg Equivariant Derived Category . 5 1.5 Equivariant Zuckerman Functors . 6 1.6 Localization of the Equivariant Zuckerman Functor . 6 2. HOMOLOGICAL ALGEBRA ................................... 10 2.1 Adjoint Functors . 10 2.2 Composition of Derived Functors . 16 3. D-MODULES ................................................. 17 3.1 Twisted Sheaves of Differential Operators . 17 3.2 Direct and Inverse Image . 19 3.2.1 Inverse Image . 19 3.2.2 Direct Image . 20 3.2.2.1 Properties of f+ ..................................... 21 3.2.2.2 Surjective Submersions . 22 3.2.3 Base Change . 22 3.3 Duality . 23 3.3.1 Verdier Duality . 23 3.3.2 Holonomic Duality . 24 3.4 Homogeneous Twisted Sheaves of Differential Operators . 24 3.4.1 Global Sections of DX,λ ................................... 26 3.4.2 Generalization to Partial Flag Varieties . 27 3.4.3 Anti-dominance and D-affineness . 30 3.5 Parameterization of Connections on K-orbits . 31 4. THE EQUIVARIANT ZUCKERMAN FUNCTOR ................ 32 4.1 (A,K)-Modules . 32 4.2 Equivariant Derived Categories . 33 4.2.1 The Right Adjoint . 36 4.2.2 The Right Adjoint, Again . 38 4.3 Equivariant Zuckerman Functor . 41 5. EQUIVARIANT HARISH-CHANDRA SHEAVES ................ 43 5.1 Equivariant Sheaves . 43 5.2 Harish-Chandra Sheaves . 45 5.3 The Functor Indh ............................................ 48 5.4 Reduction Principle . 52 5.5 Restriction of Group Actions . 53 5.6 The Geometric Zuckerman Functor . 55 geo 5.7 Properties of ΓK,S ............................................ 57 6. MAIN THEOREMS ........................................... 61 6.1 The Embedding Theorem . 61 6.2 Main Theorem . 65 6.3 Duality and Cohomologically Induced Modules . 67 7. KAZHDAN-LUSZTIG COMPUTATIONS ....................... 69 7.1 Derived Cousin Resolution . 69 7.2 Embedding in the Full Flag Variety . 72 7.3 Kazhdan-Lusztig-Vogan Polynomial Coefficients . 73 8. EXAMPLES .................................................. 76 8.1 Resolving the Structure Sheaf of a Partial Flag Variety . 78 8.1.1 Example: Split A2 ........................................ 78 8.1.2 Example: Split B2 ........................................ 78 8.1.3 Example: Split G2 ....................................... 80 8.1.4 Example: Split A3 ........................................ 81 8.2 Other Composition Series Examples . 83 8.2.1 Example: Split B2 ........................................ 83 8.2.2 Example: Split G2 ....................................... 84 REFERENCES ................................................... 86 v CHAPTER 1 INTRODUCTION 1.1 Cohomological Induction and Harish-Chandra Modules The primary objects of focus in this thesis are representations of real reductive Lie groups. Let GR be a such a group, and (V, π) a representation of GR. Let KR be a maximal compact subgroup of GR. To GR we can associate its complex Harish-Chandra pair (g,K), where g is the complexification of the Lie algebra of GR and K is the complexification of KR. The group K is a complex linear algebraic group. Definition 1 For GR, g, and K as above, a (g,K)-module V is a complex vector space with a representation π of g and an algebraic action ν of K such that 1. ν(k)π(X)ν(k−1) = π(Ad(k)X) for any X ∈ g, k ∈ K, and 2. if k = Lie(K), then π|k = dν. V is admissible if each irreducible representation of K has finite multiplicity in V . The definition of admissible also applies to the KR-action on GR-modules. A rep- resentation V of K is algebraic if it decomposes into finite-dimensional subspaces Vi on which the representation K → GL(Vi) is a morphism of algebraic groups. For (V, π) a representation of GR, a vector v ∈ V is K-finite if π(K)v is finite-dimensional. Denote the set of KR-finite vectors in V by VK . The action π of GR induces a (g,K)-module structure on VK . We call this (g,K)-module the Harish-Chandra module for V . Definition 2 Two admissible representations of GR are infinitessimally equivalent if their Harish-Chandra modules are isomorphic. Therefore, there is an injection from the set of infinitessimal equivalence classes of admissible GR-representations to the set of admissible (g,K)-modules. In fact, for irreducible modules, the map is also surjective. 2 Theorem 1 (Lepowsky) For GR a reductive Lie group, KR its maximal compact sub- group, and (g,K) its complex Harish-Chandra pair, let V be an irreducible (g,K)-module. Then, V is the Harish-Chandra module of an irreducible admissible representation of GR. The benefit of this equivalence is that the representation theory of complex Lie algebras and compact real groups is less complicated than the representation theory for real (noncompact) groups. Replacing KR with K in our Harish-Chandra pair additionally allows us to introduce algebro-geometric methods to study representations. For example, if KR is any real compact Lie group, its irreducible representations are completely classi- fied by highest weight. There is a geometric interpretation of this classification, given by the Borel-Weil-Bott theorem. Theorem 2 (Borel-Weil-Bott) Let KR be a compact Lie group, TR a Cartan subgroup, K, T their complexifications, and k, t their complexified Lie algebras. Fix a choice of positive roots ∆+ of t. This corresponds to a Borel subgroup B ⊂ K containing T . Then, the natural inclusion KR/TR→˜ K/B is a diffeomorphism, which fixes a complex structure on KR/TR. Let Cλ be the irreducible representation of T of weight λ and O(λ) the corresponding line bundle on K/B. There is a unique element w ∈ W, the Weyl group of K, for which wλ is antidominant. Let l(w) denote its length. Then, 1. Hi(K/B, O(λ)) = 0 for all i 6= l(w), and l(w) 2. H (K/B, O(λ)) is the irreducible representation of KR with lowest weight wλ. Usually, the Borel-Weil-Bott theorem is given with respect to the opposite complex structure from the one we have given. That is, conventionally, the homogeneous structure for the flag variety is taken to be K/B¯, where B¯ is the Borel for the roots −∆+. In this formulation, the non-vanishing of sheaf cohomology occurs in degree l(w) for wλ l(w) ¯ dominant, and H (K/B, O(λ)) is the KR-module with highest weight wλ. This geometric picture for compact groups serves as major motivation for taking a geometric approach to the study of representations of noncompact real groups. 1.2 The Kazhdan-Lusztig Conjectures An important category of representations for complex Lie algebras is that of highest weight modules, typically referred to in literature as category O. Let V be a highest 3 weight module of g, with respect to a fixed Cartan h and set of positive roots ∆+. There exists a surjection from some Verma module M(λ) = U(g) ⊗U(b) Cλ−2ρ to V , where Cλ−2ρ denotes the one-dimensional h-representation with weight λ − 2ρ, extended trivially to b, a Borel containing h. Therefore, the Verma modules generate the Grothendieck group for O. Alternatively, there is a basis of irreducible highest weight modules. In [10], Kazhdan and Lusztig develop an algorithm which they conjecture determines the multiplicity of an irreducible module in the composition series of the M(λ), and vice versa. In the paper [1], Beilinson and Bernstein established a correspondence between some specific sheaves on the flag variety of g and the Verma modules V (λ). Specifically, let G be a complex Lie group, g its Lie algebra, and X the flag variety for g. Fix a sheaf of ∗ twisted differential operators (tdo) Dλ on X, with λ ∈ h , and a point x ∈ X and let B be its stabilizer in G, so X = G/B is a homogeneous space for G. .Let iw : Qw → X be the inclusion of a B-orbit corresponding to w ∈ W, the Weyl group of g, and define Iw(λ) = iw+OQw . There is an analog to the Borel-Weil-Bott theorem in this case which identifies the sheaf cohomology of Iw(λ) with co-Verma modules. i Theorem 3 For λ anti-dominant and regular, H (X, Iw(λ)) = 0 for all i > 0 and ∗ Γ(X, Iw(λ)) = M(λ) . 1.3 Localization of Harish-Chandra Modules The natural successor of the Kazhdan-Lusztig conjectures is the analogous statement for real groups.
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