1 This book is an introduction to geometric representation theory. 2 What is geometric representation theory? It is hard to define exactly 3 what it is as this subject is constantly growing in methods and scope. 4 The main aim of this area is to approach representation theory which 5 deals with symmetry and non-commutative structures by geometric 6 methods and also get insights on the geometry from the representa- 7 tion theory!. Here by geometry we mean any local to global situation 8 where one tries to understand complicated global structures by gluing 9 them from simple local structures. The main example is the #eilinson- 10 Bernstein locali$ation theorem. This theorem essentially says that the 11 representation theory of a semi-simple %ie algebra such as sl n,C!! is 12 encoded in the geometry of its &ag variety. This theorem enables the 13 transfer of “hard( global! problems about the universal enveloping al- 14 gebra, to “easy” local! problems in geometry. The #eilinson-Bernstein 15 localization theorem has been extremely useful in solving problems in 16 representation theory of semi-simple %ie algebras and in gaining deeper 17 insight into the structure of representation theory as a whole. There 18 are many more examples of geometric representation theory in action) 19 from *eligne-Lusztig varieties to the geometric %anglands’ program 20 and categorification. 21 The focus of this book is the #eilinson-Bernstein localization theo- 22 rem. It follows the advice of the great mathematician Israel M. Gelfand. 23 we only cover the case of sl2 classical and /uantum!. This approach 24 allows us to introduce many topics in a very concrete way without go- 25 ing into the general theory. Thus we cover the 0eter-Weyl theorem) 26 the #orel-Weil theorem) the Beilinson-Bernstein theorem and much 27 more for both the classical and /uantum case. *ealing with the quan- 28 tum case allows us also to introduce many tools from non-commutative 29 algebraic geometry and /uantum groups. These topics are usually con- 30 sidered very advanced. To have a full understanding of them requires 31 a good grasp of algebraic geometry) *-module theory) category theory) 32 homological algebra and the theory of semi-simple %ie algebras. We 33 think that by focusing on the simplest case of sl2 the student can gain 34 much insight and intuition into the subject. 1 good and deep under- 35 standing of sl2 makes the general theory much simpler to learn and 36 appreciate. 37 This book is based on a graduate lecture course given at MIT by 38 the second author. We are grateful to the students taking that course 39 for sharing their notes with us as we prepared this manuscript. i 40 Contents 41 Introduction 2 42 Chapter 1. The first classical example: sl2. 4 43 2. The %ie algebra sl2 5 44 4. Irreducible finite-dimensional modules 6 45 5. The universal enveloping algebra 7 46 6. 8emisimplicity 9 47 7. Characters : 48 9. The 0#W theorem) and the center ofU sl 2!; 49 Chapter 2. Hopf algebras and tensor categories 24 50 2. Hopf algebras 25 51 4. The first examples of Hopf Algebras 29 52 4.1. The Hopf algebraU sl 2! 29 53 4.2. The Hopf algebra SL ! 29 O 2 54 5. Tensor Categories 2< 55 Chapter 3. Geometric =epresentation Theory forSL 2 4> 56 2. The algebra of matrix coefficients 42 57 4. 0eter-Weyl Theorem for 8% 4! 44 58 5. Reconstructing SL ! fromU sl ! via matrix coe?cients. 45 O 2 2 59 6. @/uivariant vector bundles, and sheaves 46 60 7. Quasi-coherent sheaves on the &ag variety 49 61 9. The #orel-Weil Theorem 4: 62 <. Beilinson-Bernstein %ocalization 5> 1 63 <.1.D-modules onP 5> 64 <.2. The %ocalization Theorem 52 65 Chapter 4. The first /uantum example.U q sl2!. 56 66 2. The /uantum integers 57 67 4. The /uantum enveloping algebraU q sl2! 57 68 5. Representation theory forU q sl2! 5< 69 4.U q is a Hopf algebra 5: 70 7. More representation theory forU q 5; 71 7.1. Auantum Casimir element 6> ii CONTENTS iii 72 9. The locally finite part and the center ofU q sl2! 62 73 Chapter 5. Categorical 3ommutativity in #raided Tensor 74 Categories 65 75 2. Braided and 8ymmetric Tensor 3ategories 66 76 4. R-matrix Preliminaries 69 77 5. *rinfeld+s UniversalR-matrix 6< 78 6. Lusztig’sR-matrices 6< 79 7. Weights of Type I, and bicharacters needs better title) 6: 80 9. The Cang-Baxter @/uation 7> 81 <. The hexagon *iagrams 75 82 Chapter 6. Geometric =epresentation Theory forSL q 4! 76 83 2. The Auantum -roup Algebra 77 84 4. Generators and =elations forO q 77 85 3.O q comodules 7< 86 6. Borel-Weil Correspondences in 3lassical and Auantum 87 Cases 7: 88 6.1. The Quantum case 7; 89 6.2. An equivalence of categories arising from the Hopf pairing 9> 90 6.3. =estriction and Induction Dunctors 92 1 91 6.4. AuantumP 94 92 7. Lecture 26 - Quasi-coherent sheaves 96 93 7.1. 3lassical case 96 94 7.2. Auantum case 99 2 95 7.3. Auantum differential operators onA q 9: 96 9. QuantumD-modules <> 97 Bibliography <7 INTRODUCTION 1 98 Introduction 99 In representation theory) the %ie algebrasl 2 .Fsl 2 C! comprises the 100 first and most important example of a semi-simple %ie algebra. In this 101 introductory text) which grew out of a course taught by the first au- 102 thor, we will walk the reader through important concepts in geometric 103 representation theory) as well as their /uantum group analogues. Gur 104 focus is on developing concrete examples to illustrate the geometric 105 notions discussed in the text. As such) we will restrict our attention 106 almost exclusively to sl2) giving more general definitions only when it 107 is convenient or illustrative. 108 In Chapter 2) we show that the category of finite-dimensional sl2- 109 modules is a semi-simple abelian categoryH we prove this important fact 110 in a way which will generalize most easily to the /uantum setting in 111 later chapters. 112 In 3hapter 4) we introduce the formalisms of Hopf algebras and 113 tensor categories. These capture the essential properties of algebraic 114 groups, their representations, and their coordinate algebras, in a way 115 that can be extended to the quantum setting. 116 In 3hapter 5) we discuss the relation between geometry of various 117 G-varieties and the representation theory ofG. We discuss the 0eter- 118 Weyl theorem) and obtain as a corollary the #orel-Weil theorem. We 1 119 define *-modules onP ) and we relate them to representations of sl2. 120 this is the first instance of the so-called #eilinson-Bernstein localization 121 theorem. 122 In 3hapter 6) we introduce the /uantized universal enveloping al- 123 gebraU q sl2!) and extend the results of Chapter 2 to the /uantum 124 setting. 125 In 3hapter 7) we explain the notion of a braided tensor category) 126 a mild generali$ation of the notion of a symmetric tensor category. 127 Braided tensor categories underlie the representation theory ofU q sl2! 128 in a way analogous to the role of symmetric tensor categories in the 129 representation theory of sl2. 130 In 3hapter 9) we reproduce the results of Chapter 5 in the /uantum 131 setting. We have /uantum analogs of each of the 0eter-Weyl) #orel- 132 Weil) and #eilinson-Bernstein theorems. 133 Throughout the text assume some passing familiarity with the the- 134 ory of %ie algebras. Two excellent introductions are Humphreys I?J 135 and Knapp I?J. 3"10TER 2 136 The first classical example: sl2. 2 1. THE LIE ALGEBRA sl2 3 137 1. The Lie algebra sl2 The Lie algebrasl 2 .Fsl 2 C! consists of the traceless 4 4 matrices, with the standard Lie bracket. × IA, BJ .FAB BA. − 138 1 standard and convenient presentation of sl2 C! is given as follows. 139 We let: > 2 > > 2 > 2!EF ,FF ,HF , > > 2 > > 2 � � � � � − � Then sl2 is spanned byE,F) andH) with commutators: IH,EJF4E,IH,FJF 4F,IE,FJFH. − 140 =ecall that a representation ofg equivalently) ag-module! is a vec- 141 tor spaceV ) together with a %ie algebra homomorphismρ.g End V!. → 142 We will often omitρ from notation) and write simply x.v forρ x!.v. 143 The finite-dimensional representations ofsl 2 are su?ciently compli- 144 cated to be interesting) yet can be completely understood by elemen- 145 tary means. In this chapter) we recall their classification. We begin 146 with some examples. 147 Example 1.2. The defining representation. The Lie algebra sl2 2 148 acts onC by matrix multiplication. 149 Example 2.4. The adjoint representation. Any Lie algebrag acts 150 on itself by x.y.FIx, yJ. 151 Example 2.5. Given any representationV of a %ie algebrag) its 152 dual vector spaceV ∗ carries an action defined by X.f! v!Ff X.v!. − 153 The corresponding representation is also denotedV ∗. 154 Example 1.6. Given two representationsV andW ofg) the vector 155 spaceV W carries an action ofg defined byx v,w!.F xv,xw! for ⊕ 156 v,w! V W) andx g. The corresponding representation is also ∈ ⊕ ∈ 157 denotedV W. ⊕ 158 Example 1.7. Given two representationsV andW ) the vector 159 spaceV W carries an action ofg defined byx v w! Fx v! ⊗ ⊗ ⊗ 160 wLv x w!) forv w V W ) andx g.