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Representation Theory of Real Reductive Groups

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Representation Theory of Real Reductive Groups Representation Theory of Real Reductive Groups Thesis by Zhengyuan Shang In Partial Fulfillment of the Requirements for the Degree of Bachelor of Science in Mathematics CALIFORNIA INSTITUTE OF TECHNOLOGY Pasadena, California 2021 Defended May 14, 2021 ii © 2021 Zhengyuan Shang ORCID: 0000-0002-8823-9483 All rights reserved iii ACKNOWLEDGEMENTS I want to thank my mentor Dr. Justin Campbell for introducing me to this fascinating topic and offering numerous helpful suggestions. I also wish to thank Prof. Xinwen Zhu and Prof. Matthias Flach for their time and patience serving on my committee. iv ABSTRACT The representation theory of Lie groups has connections to various fields in mathe- matics and physics. In this thesis, we are interested in the classification of irreducible admissible complex representations of real reductive groups. We introduce two ap- proaches through the local Langlands correspondence and the Beilinson-Bernstein localization respectively. Then we investigate the connection between these two classifications for the group GL¹2, Rº. v TABLE OF CONTENTS Acknowledgements . iii Abstract . iv Table of Contents . v Chapter I: Introduction . 1 1.1 Background . 1 1.2 Notation . 1 1.3 Preliminaries . 2 Chapter II: Local Langlands Correspondence . 4 2.1 Parabolic Induction . 4 2.2 Representations of the Weil Group ,R ................ 5 2.3 !-function and n-factor . 6 2.4 Main Results . 8 Chapter III: Beilinson–Bernstein Localization . 10 3.1 Representation of Lie Algebras . 10 3.2 Algebraic D-modules . 11 3.3 Main Results . 11 3.4 Induction in the Geometric Context . 12 Chapter IV: An Example of GL¹2, Rº ..................... 14 4.1 Notation . 14 4.2 Langlands Approach . 14 4.3 Geometric Approach . 15 4.4 Parabolic Induction . 16 Bibliography . 17 1 C h a p t e r 1 INTRODUCTION 1.1 Background In the late nineteenth century, motivated by the view of Klein that the geometry of space is determined by its group of symmetries, Lie started to investigate group actions on manifolds, laying the foundation of the theory of Lie groups. Around the same time, while working on algebraic number theory, Dedekind discovered a for- mula involving multiplicative characters of finite abelian groups. Built on this idea, Frobenius and Schur developed the representation theory of finite groups. With the introduction of invariant integration, analogous results for compact groups were also established soon after. Since then, representation of Lie groups has found applica- tions to numerous fields in mathematics and physics, notably number theory (Tate thesis) and quantum mechanics (Heisenberg and Lorentz groups). Progress was made to expand the theory to locally compact groups, nilpotent/solvable Lie groups, and semisimple Lie groups. These efforts culminated in Langlands classification [9] of all irreducible admissible representations of reductive groups as quotients of induced representations. When the field is R or C, this classification takes an alter- native, more concrete form in terms of representation of Langlands groups. From an entirely different viewpoint and as a generalization of the Borel-Weil theorem for compact groups, Beilinson and Bernstein [1] identified representations of Lie algebras with global sections of certain D-modules on the flag variety, paving the way for a geometric classfication. In this thesis, we first introduce the details of these two distinct approaches, following the expositions in [7], [3], and [4]. Then we apply them explicitly to the group GL¹2, Rº and investigate the connection between the results. 1.2 Notation Throughout the thesis, unless indicated otherwise, we adopt the following notations: 퐺0 is a connected real reductive Lie group 퐺 is the complexification of 퐺0 with Lie algebra g 퐵, 퐻 are Borel and Cartan subgroups of 퐺 0 is the maximal compact subgroup of 퐺0 with complexification *¹gº is the universal enveloping algebra of g with center / ¹gº 2 1.3 Preliminaries Definition 1.3.1 A connected linear algebraic group ! over R is reductive if the base change !C is reductive, which is equivalent to the largest smooth connected unipotent normal subgroup of !C being trivial. Definition 1.3.2 A real Lie group 퐺0 is reductive if there exists a linear algebraic group ! over R whose identity component (in the Zariski topology) is reductive and a homomorphism i : 퐺0 ! !¹Rº with ker i finite and im i open in !¹Rº. Example 1.3.3 Every connected semisimple real Lie group (i.e. with semisimple Lie algebra) with finite center is reductive. Definition 1.3.4 A subgroup 퐵 of an algebraic group 퐺 is Borel if it is maximal among all Zariski closed connected solvable subgroups. A subgroup 퐻 ⊂ 퐺 is Cartan if it is the centralizer of a maximal torus. Example 1.3.5 For GL¹=, Cº, the subgroup of upper triangular matrices is a Borel subgroup, while the subgroup of diagonal matrices is a Cartan subgroup. Definition 1.3.6 A continuous representation ¹d, +º of 퐺0 on a complex Hilbert + d space is admissible if j 0 is unitary and each irreducible representation of 0 d + E + occurs with at most finite multiplicity in j 0 . If is admissible, a vector 2 is called 0-finite if it lies in a finite-dimensional space under the action of 0. Definition 1.3.7 A ¹g, º-module is a vector space + with actions of g and such that for any E 2 +, : 2 , - 2 g, and . 2 k (the Lie algebra of ), we have : · ¹- · Eº = ¹Ad¹:º-º · ¹: · Eº E ⊂ + spans a finite-dimensional subspace on which acts continuously 3 C. E . E ¹ 3C exp¹ º · ºjC=0 = · Given an admissible representation ¹d, +º of 퐺0, we can consider the space + of 0-finite vectors. Since the action of 0 on + is locally finite, it extends to an action of . Moreover, it is shown that + is dense in + and inherits a g action. The actions of and g are compatible in the sense that ¹d, + º is a ¹g, º-module. [6, Chapter III]. We also have the following result [6, Chapter VIII]. Proposition 1.3.8 ¹d, +º is irreducible if and only if ¹d, + º is irreducible. Another result by [10] shows that: 3 Proposition 1.3.9 Every irreducible ¹g, º-module is isomorphic to ¹d, + º for some irreducible admissible representation ¹d, +º of 퐺0. Definition 1.3.10 Two admissible representations of 퐺0 are infinitesimally equiva- lent if their underlying ¹g, º-modules are isomorphic. We would like to classify irreducible admissible representations of 퐺0 up to infinites- imal equivalence, or equivalently, irreducible ¹g, º-modules up to isomorphism. 4 C h a p t e r 2 LOCAL LANGLANDS CORRESPONDENCE 2.1 Parabolic Induction Definition 2.1.1 A unitary representation of 퐺0 is a continuous norm-preserving group action of 퐺0 by linear transformations on a Hilbert space. Definition 2.1.2 Let ¹d, +º be an irreducible admissible representation of 퐺0 and 0 E, E be 0-finite vectors. Then a 0-finite matrix coefficient of d is a function 0 2¸n G 7! hd¹GºE, E i. If all 0-finite matrix coefficients of d are in ! ¹퐺º for 8n ¡ 0, we say d is a tempered representation. Proposition 2.1.3 (Iwasawa) Let g0 be the Lie algebra of 퐺0 and g0 = k0 ¸ p0 be the Cartan decomposition. Then there is moreover a decomposition g0 = k0 ¸a0 ¸n0 such that a0 is abelian, n0 is nilpotent, and »a0 ⊕ n0, a0 ⊕ n0¼ = n0. Definition 2.1.4 Let 퐴p,#p be the analytic subgroups of 퐺0 with Lie algebras a0 and n0 respectively. Let "p be the centralizer of a0 in 0. The subgroup "p 퐴p#p is call a minimal parabolic subgroup of 퐺0. Any closed subgroup &0 containing "p 퐴p#p is called a standard parabolic subgroup. Proposition 2.1.5 Any standard parabolic subgroup &0 has a Langlands decom- position &0 = "0 퐴0#0, where "0 is reductive, 퐴0 is abelian, and #0 is nilpotent. Example 2.1.6 For GL¹2, Rº, the only proper parabolic is the subgroup of upper triangular matrices with ( !) ( ! ) ( !) ±1 0 G 0 1 I "0 = 퐴0 = : G, H ¡ 0 #0 = 0 ±1 0 H 0 1 Theorem 2.1.7 (Langlands) [8] The equivalence classes (up to infinitesimal equiv- alence) of irreducible admissible representations of 퐺0 are in bijection with all triples ¹"0 퐴0#0, »f0¼, a0º satisfying a) "0 퐴0#0 is a parabolic subgroup of 퐺0 containing "p 퐴p#p b) f0 is an irreducible tempered (unitary) representation of "0 and »f0¼ its equivalence class 5 a 0 a ,U ¡ c) 0 is a member of a0 such that Reh 0 i 0 for every positive restricted root not vanishing on a0 In particular, the correspondence is that ¹"0 퐴0#0, »f0¼, a0º corresponds to the 퐺0 class of the unique irreducible quotient of Ind ¹f0 ⊗ a0 ⊗ 1º. "0 퐴0#0 퐺0 Specifically, the induced representation Ind ¹f0 ⊗ a0 ⊗ 1º is given by first "0 퐴0#0 considering all 퐹 2 퐶¹퐺0, +º with −¹a0¸dº log 0 −1 퐹¹G<0=º = 4 f0¹<º 퐹¹Gº 퐺0 −1 and defining Ind ¹f0 ⊗ a0 ⊗ 1º¹6º퐹¹Gº = 퐹¹6 Gº. "0 퐴0#0 Definition 2.1.8 The parameters ¹"0 퐴0#0, »f0¼, a0º are called the Langlands pa- rameters of the given irreducible admissible representation. 2.2 Representations of the Weil Group ,R × Definition 2.2.1 The Weil group of R, denoted ,R, is the nonsplit extension of C × × 2 −1 × by Z/2Z given by ,' = C [ 9C , where 9 = −1 and 92 9 = 2 for 82 2 C . For reasons that will become clear, we are interested in classifying =-dimensional complex representations of ,R whose images consist of semisimple elements (i.e.
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