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

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 mathematics and physics. In this thesis, we are interested in the classification of irreducible admissible complex representations of real reductive groups. We introduce two approaches through the local Langlands correspondence and the Beilinson-Bernstein localization respectively. Then we investigate the connection between these two classifications for the group GL(2, R). 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 푊R ...... 5 2.3 퐿-function and 휖-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 GL(2, R) ...... 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 GL(2, R) 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 픤 퐵, 퐻 are Borel and Cartan subgroups of 퐺

퐾0 is the maximal compact subgroup of 퐺0 with complexiﬁcation 퐾 푈(픤) is the universal enveloping algebra of 픤 with center 푍 (픤) 2 1.3 Preliminaries

Deﬁnition 1.3.1 A connected linear algebraic group 퐿 over R is reductive if the base change 퐿C is reductive, which is equivalent to the largest smooth connected unipotent normal subgroup of 퐿C being trivial.

Deﬁnition 1.3.2 A real Lie group 퐺0 is reductive if there exists a linear algebraic group 퐿 over R whose identity component (in the Zariski topology) is reductive and

a homomorphism 휑 : 퐺0 → 퐿(R) with ker 휑 ﬁnite and im 휑 open in 퐿(R).

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 GL(푛, C), 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 (휌, 푉) of 퐺0 on a complex Hilbert 푉 휌 퐾 space is admissible if |퐾0 is unitary and each irreducible representation of 0 휌 푉 푣 푉 occurs with at most finite multiplicity in |퐾0 . If is admissible, a vector ∈ is called 퐾0-finite if it lies in a finite-dimensional space under the action of 퐾0.

Deﬁnition 1.3.7 A (픤, 퐾)-module is a vector space 푉 with actions of 픤 and 퐾 such that for any 푣 ∈ 푉, 푘 ∈ 퐾, 푋 ∈ 픤, and 푌 ∈ 픨 (the Lie algebra of 퐾), we have

푘 ·(푋 · 푣) = (Ad(푘)푋)·(푘 · 푣) 퐾푣 ⊂ 푉 spans a ﬁnite-dimensional subspace on which 퐾 acts continuously 푑 푡푌 푣 푌 푣 ( 푑푡 exp( )· )|푡=0 = ·

Given an admissible representation (휌, 푉) of 퐺0, we can consider the space 푉퐾 of

퐾0-ﬁnite vectors. Since the action of 퐾0 on 푉퐾 is locally ﬁnite, it extends to an

action of 퐾. Moreover, it is shown that 푉퐾 is dense in 푉 and inherits a 픤 action. The actions of 퐾 and 픤 are compatible in the sense that (휌, 푉퐾 ) is a (픤, 퐾)-module. [6, Chapter III]. We also have the following result [6, Chapter VIII].

Proposition 1.3.8 (휌, 푉) is irreducible if and only if (휌, 푉퐾 ) is irreducible.

Another result by [10] shows that: 3

Proposition 1.3.9 Every irreducible (픤, 퐾)-module is isomorphic to (휌, 푉퐾 ) for some irreducible admissible representation (휌, 푉) of 퐺0.

Deﬁnition 1.3.10 Two admissible representations of 퐺0 are inﬁnitesimally equivalent if their underlying (픤, 퐾)-modules are isomorphic.

We would like to classify irreducible admissible representations of 퐺0 up to inﬁnites- imal equivalence, or equivalently, irreducible (픤, 퐾)-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 (휌, 푉) be an irreducible admissible representation of 퐺0 and 0 푣, 푣 be 퐾0-finite vectors. Then a 퐾0-finite matrix coefficient of 휌 is a function 0 2+휖 푥 ↦→ h휌(푥)푣, 푣 i. If all 퐾0-finite matrix coefficients of 휌 are in 퐿 (퐺) for ∀휖 > 0, we say 휌 is a tempered representation.

Proposition 2.1.3 (Iwasawa) Let 픤0 be the Lie algebra of 퐺0 and 픤0 = 픨0 + 픭0 be the Cartan decomposition. Then there is moreover a decomposition 픤0 = 픨0 +픞0 +픫0 such that 픞0 is abelian, 픫0 is nilpotent, and [픞0 ⊕ 픫0, 픞0 ⊕ 픫0] = 픫0.

Deﬁnition 2.1.4 Let 퐴픭, 푁픭 be the analytic subgroups of 퐺0 with Lie algebras 픞0 and 픫0 respectively. Let 푀픭 be the centralizer of 픞0 in 퐾0. The subgroup 푀픭 퐴픭푁픭 is call a minimal parabolic subgroup of 퐺0. Any closed subgroup 푄0 containing

푀픭 퐴픭푁픭 is called a standard parabolic subgroup.

Proposition 2.1.5 Any standard parabolic subgroup 푄0 has a Langlands decomposition 푄0 = 푀0 퐴0푁0, where 푀0 is reductive, 퐴0 is abelian, and 푁0 is nilpotent.

Example 2.1.6 For GL(2, R), the only proper parabolic is the subgroup of upper triangular matrices with ( !) ( ! ) ( !) ±1 0 푥 0 1 푧 푀0 = 퐴0 = : 푥, 푦 > 0 푁0 = 0 ±1 0 푦 0 1

Theorem 2.1.7 (Langlands) [8] The equivalence classes (up to inﬁnitesimal equivalence) of irreducible admissible representations of 퐺0 are in bijection with all triples (푀0 퐴0푁0, [휎0], 휈0) satisfying

a) 푀0 퐴0푁0 is a parabolic subgroup of 퐺0 containing 푀픭 퐴픭푁픭

b) 휎0 is an irreducible tempered (unitary) representation of 푀0 and [휎0] its equivalence class 5 휈 0 휈 , 훼 > c) 0 is a member of 픞0 such that Reh 0 i 0 for every positive restricted root not vanishing on 픞0

In particular, the correspondence is that (푀0 퐴0푁0, [휎0], 휈0) corresponds to the 퐺0 class of the unique irreducible quotient of Ind (휎0 ⊗ 휈0 ⊗ 1). 푀0 퐴0푁0

퐺0 Speciﬁcally, the induced representation Ind (휎0 ⊗ 휈0 ⊗ 1) is given by ﬁrst 푀0 퐴0푁0 considering all 퐹 ∈ 퐶(퐺0, 푉) with

−(휈0+휌) log 푎 −1 퐹(푥푚푎푛) = 푒 휎0(푚) 퐹(푥)

퐺0 −1 and deﬁning Ind (휎0 ⊗ 휈0 ⊗ 1)(푔)퐹(푥) = 퐹(푔 푥). 푀0 퐴0푁0

Deﬁnition 2.1.8 The parameters (푀0 퐴0푁0, [휎0], 휈0) are called the Langlands parameters of the given irreducible admissible representation.

2.2 Representations of the Weil Group 푊R

× Deﬁnition 2.2.1 The Weil group of R, denoted 푊R, is the nonsplit extension of C × × 2 −1 × by Z/2Z given by 푊푅 = C ∪ 푗C , where 푗 = −1 and 푗푐 푗 = 푐 for ∀푐 ∈ C .

For reasons that will become clear, we are interested in classifying 푛-dimensional

complex representations of 푊R whose images consist of semisimple elements (i.e. semisimple matrices in GL(푛, C)). We refer to them as semisimple representations.

Lemma 2.2.2 One-dimensional representations of C× are of the form 푧 ↦→ 푧휇푧휈 with 휇, 휈 ∈ C and 휇 − 휈 ∈ Z. (Note that C× 푆1 × R+, and representations of the components are known.)

Proposition 2.2.3 One-dimensional representations 휑 of 푊R are of the form

, 푡 휑 푧 푧 푡 휑 푗 h+ i : ( ) = | |R and ( ) = 1 , 푡 휑 푧 푧 푡 휑 푗 h− i : ( ) = | |R and ( ) = −1 where 푡 can be any complex number.

Proof : Suppose 휑( 푗) = 푤 ∈ C× and ∀푧 ∈ C×, 휑(푧) = 푧휇푧휈. Then since 휑(푧) = 푤휑(푧)푤−1 = 휑( 푗푧 푗−1) = 휑(푧), it follows that 휇 = 휈, so 휑(푟푒푖휃) = 푟2휇. Since 푤2 = 휑( 푗2) = 휑(−1) = 1, 푤 = ±1. Let 푡 = 2휇. It is easy to check that the formulas above indeed give valid representations. 6 Proposition 2.2.4 Equivalent classes of irreducible two-dimensional semisimple

representations 휑 of 푊R are of the form

h푙, 푡i : 휑(푟푒푖휃)훼 = 푟2푡 푒푖푙휃훼 and 휑( 푗)훼 = 훽 휑(푟푒푖휃)훽 = 푟2푡 푒−푖푙휃 훽 and 휑( 푗)훽 = (−1)푙훼 where 훼, 훽 form a basis of the two dimensional vector space and 푙 ∈ Z+, 푡 ∈ C.

Proof : Given such a representation 휑, pick a basis 푎, 푏 such that 휑(C×) consist of diagonal matrices. By the lemma, we can suppose 휑(푧)푎 = 푧휇1 푧휈1 푎 and 휑(푧)푏 = 휇 휈2 0 푧 2 푧 푏. Since 휑 is irreducible, either 휇1 ≠ 휇2 or 휈1 ≠ 휈2. Let 푎 := 휑( 푗)푎. Then

휑(푧)푎0 = 휑( 푗푧 푗−1)푎0 = 휑( 푗푧)푎 = 푧휈1 푧휇1 푎0

0 If 휇1 = 휈1, then 푎 ∈ C푎 and C푎 is an invariant subspace. Contradiction! Thus 0 푎 ∈ C푏 by the equation above, which means 휈 := 휈1 = 휇2 and 휇 := 휇1 = 휈2. Since 휑( 푗)−1 = 휑( 푗)휑(−1) = (−1)휇−휈 휑( 푗), we have

휑(푧)푎 = 푧휇푧휈푎 and 휑( 푗)푎 = 푎0 휑(푧)푎0 = 푧휈푧휇푎0 and 휑( 푗)푎0 = (−1)휇−휈푎

Considering instead the basis 푎0, (−1)휇−휈푎 and in view of the symmetry, we can suppose without the loss of generality that the integer 푙 := 휇 − 휈 is positive. Let 푡 1 휇 휈 훼 푎 훽 푎0 = 2 ( + ), = , and = . Then the result follows immediately.

Proposition 2.2.5 Every ﬁnite-dimensional semisimple representation 휑 of 푊R is fully reducible, and each irreducible representation has dimension one or two.

Proof : Given a representation 휑 acting on 푉, since 휑(C×) consists of commuting

diagonalizable matrices, 푉 is the direct sum of spaces 푉휇,휈 on which 휑(푧) acts by 휇 휈 푧 푧 . Meanwhile, 휑( 푗)푉휇,휈 = 푉휈,휇. If 휇 = 휈, we can choose a basis of eigenvectors

for 휑( 푗) in 푉휇,휈, so the span of each vector is an one-dimensional invariant subspace

under 휑(푊R). If 휇 ≠ 휈, pick a basis 푢1, 푢2, ··· 푢푟 of 푉휇,휈. Then C푢푖 ⊕ C휑( 푗)푢푖 is a 푉 푉 É푟 푢 휑 푗 푢 two-dimensional invariant subspace. Moreover, 휇,휈 ⊕ 휈,휇 = 푖=1 C 푖 ⊕ C ( ) 푖.

2.3 퐿-function and 휖-factor

Given any irreducible admissible representations 휌 of 퐺0, we can attach two invari- ants, called the 퐿-function and 휖-factor. For simplicity, we deﬁne them here for the 2휋푖푥 special case of 퐺0 = GL(푛, R). Fix an additive character 휓 of R given by 푥 ↦→ 푒 . 7

Let 푀푛 (R) be the collection of 푛 × 푛 matrices and I0 be the space of functions 푀 퐾 푃 푥 휋 Í 푥2 푃 푛 ( ) → R of the form ( 푖 푗 ) exp(− 푖 푗 ) for any polynomial . For any 퐾0-finite matrix coefficient 푐 of 휌, ∀ 푓 ∈ I0, and 푠 ∈ C, we define ∫ 휁 ( 푓 , 푐, 푠) = 푓 (푥)푐(푥)| det 푥|푠푑×푥 푀푛 (R) × −푛 Here 푑 푥 = | det 푥| 푑푥, where 푑푥 is a fixed invariant measure for 푀푛 (R).

Proposition 2.3.1 If 휌 is irreducible, then all 휁 ( 푓 , 푐, 푠) converge in the right half plane and extend to a meromorphic function (in terms of 푠) on C. Moreover, there

is a finite collection {(푐푖, 푓푖)} such that ∑︁ 퐿(푠, 휌) = 휁 ( 푓푖, 푐푖, 푠) 푖 푐, 푓 , 휁 푓 , 푐, 푠 1 푛 푃 푓 , 푐, 푠 퐿 푠, 휌 푃 푠 satisfies ∀( ) ( + 2 ( − 1)) = ( ) ( ) with a polynomial in . In particular 퐿(푠, 휌) is uniquely defined this way (up to a scalar).

Now for ∀ 푓 ∈ I0, we deﬁne ∫ 푓ˆ = 푓 (푦)휓(Tr(푥푦))푑푦 푀푛 (R) where 푑푦 is the self-dual Haar measure on 푀푛 (R). Then 푓ˆ ∈ I0.

Proposition 2.3.2 If 휌 is irreducible, there exists a meromorphic function 훾(푠, 휌, 휓) 푓 푐 휁 푓ˆ , 푐, 푠 1 푛 훾 푠, 휌, 휓 휁 푓 , 푐, 푠 1 푛 independent of and with ( ˇ 1− + 2 ( −1)) = ( ) ( + 2 ( −1)) −1 for ∀ 푓 ∈ I0 and any 퐾0 ﬁnite matrix coeﬃcient 푐 of 휌. Here 푐ˇ(푥) = 푐(푥 ).

Deﬁnition 2.3.3 We refer to 퐿(휌, 푠) as the 퐿-function of 휌. The 휖-factor is deﬁned as 훾(푠, 휌, 휓)퐿(푠, 휌) 휖 (푠, 휌, 휓) = 퐿(1 − 푠, 휌˜) where 휌˜ is the admissible dual of 휌.

We can also attach an 퐿-function and an 휖-factor to a representation of 푊R as follows.

Deﬁnition 2.3.4 Given an irreducible ﬁnite-dimensional semisimple representation

휑 of 푊R, we can attach an 퐿-function

푠+푡 +  휋− 2 푠 푡 휑 , 푡  Γ( 2 ) for given by h+ i   − 푠+푡+1 푠+푡+1 퐿(푠, 휑) = 휋 2 휑 , 푡 Γ( 2 ) for given by h− i   −(푠+푡+ 푙 ) 푙  ( 휋) 2 (푠 + 푡 + ) 휑 h푙, 푡i  2 2 Γ 2 for given by  8 and an 휖-factor:

 1 for 휑 given by h+, 푡i   휖 (푠, 휑, 휓) = 푖 for 휑 given by h−, 푡i   푖푙+1 휑 푙, 푡  for given by h i  Now to state the local Langlands correspondence, we need one last ingredient.

Roughly speaking, the 퐿-group of a connected reductive group 퐺0 over R is given by the semi-direct product of a complex reductive group with dual root datum to 퐺0 퐿 and the Galois group Gal(C/R). We denote it by 퐺0. Concretely,

Deﬁnition 2.3.5 The 퐿-groups of GL(푛, R), SL(푛, R), SO(2푛+1, R), and SO(2푛, R) are GL(푛, C), PGL(푛, C), Sp(푛, C), and SO(2푛, C) respectively.

2.4 Main Results

Theorem 2.4.1 (Local Langlands Correspondence) There is a bijection between 퐿 semisimple representations of 푊R into 퐺0 and 퐿-packets of irreducible admissible representations of 퐺0.

Remark 2.4.2 This theorem means we have a partition of all the irreducible admissible representations of 퐺0, indexed by semisimple representations of 푊R into 퐿 퐺0. An 퐿-packet is a collection of representations of 퐺0 that correspond to the same representation of 푊R, which is called the 퐿-parameter of that packet.

Proposition 2.4.3 The 퐿-parameter and representations in its corresponding 퐿- packet have the same 퐿-function and 휖-factor.

A priori, there can be multiple representations in a single 퐿-packet. However, when

퐺0 = GL(푛, R), each 퐿-packet has size one. In particular,

Theorem 2.4.4 There is a bijection between the set of equivalence classes of 푛- dimensional semisimple complex representations of 푊R and the set of equivalence classes of irreducible admissible representations of GL(푛, R).

To make the correspondence explicit, we ﬁrst deﬁne some classes of representations of GL(1, R) and GL(2, R). Note that for 푗 = 1, 2, any matrix 푋 ∈ GL( 푗, R) admits a decomposition 푋 = 푌 푍 where 푌 ∈ SL±( 푗, R) and 푍 is positive scalar. Denote the ± , 푙 + 퐷+ two characters of SL (1 R) as 1 and sgn. For ∈ Z , let 푙 be the representation 9 of SL(2, R) on the space of analytic functions 푓 on the upper half plane satisfying k 푓 k2 = ∫ ∫ | 푓 (푧)|2푦푙−1푑푥푑푦 < ∞ given by ! 푎 푏 푎푧 + 푐 퐷+ 푓 (푧) = (푏푧 + 푑)−푙−1 푓 ( ) 푙 푐 푑 푏푧 + 푑

± 퐷 = SL (2,R) (퐷+) ( , ) ∀푡 ∈ Consider 푙 indSL(2,R) 푙 . For GL 1 R , C, we have representations

1 ⊗ | · |푡 : 푋 ↦→ |푍|푡 sgn ⊗| · |푡 : 푋 ↦→ sgn(푌)|푍|푡

Similarly for GL(2, R), we have

푡 푡 퐷푙 ⊗ | det(·)| : 푋 ↦→ 퐷푙 (푌)| det(푍)|

휑 푊 휑 É 푗 휑 휑 푛 Now given a representation of of R, suppose = 푖=1 푗 , where 푗 is a 푗 - dimensional representation with 푛 푗 = 1 or 2. We associate to 휑 푗 to a representation

휓 푗 of GL(푛 푗 , R) in the following way

h+, 푡i → 1 ⊗ | · |푡 h−, 푡i → sgn ⊗| · |푡 푡 h푙, 푡i → 퐷푙 ⊗ | det(·)|

푡 휑 푛−1 푡 Let 푗 be the complex number corresponding to 푗 . Suppose wlog that 1 Re 1 ≥ 푛−1 푡 ≥ · · · ≥ 푛−1 푡 퐷 Î 푗 (푛 , ) 푈 2 Re 2 푗 Re 푗 . Let = 푖=1 GL 푗 R and the block strictly upper (푛, ) É 푗 휓 휓0 triangular subgroup of GL R . We can extend 푖=1 푗 to a representation on 퐷푈 by making it trivial on 푈.

GL(푛,R) 푛, Proposition 2.4.5 The induced representation ind퐷푈 of GL( R) has a unique irreducible quotient 휓, which is admissible.

Remark 2.4.6 The bijection in Theorem 2.4.4 is given by 휑 ↦→ 휓 described above. 10 C h a p t e r 3

BEILINSON–BERNSTEIN LOCALIZATION

3.1 Representation of Lie Algebras If we choose a positive root system Δ+ for 픤 then the set of roots of 픤 is given by Δ+ ∪ {0} ∪ Δ− and we have a decomposition Ê Ê 픤 = 픤훼 ⊕ 픥 ⊕ 픤훼 훼∈Δ+ 훼∈Δ− É − É where 픥 is the Cartan subalgebra. Let 픫 = 훼∈Δ+ 픤훼 and 픫 = 훼∈Δ− 픤훼. Then 픟 := 픥 ⊕ 픫 is a Borel subalgebra. Let Π = {훼 푗 }1≤ 푗≤푛 be a collection of simple roots.

훼 훼∨ 훼∨ 훼 Proposition 3.1.1 For each 푖 ∈ Π, there exists 푖 ∈ 픥 such that 푖 ( 푖) = 2 and ∗ ∗ ∨ 푠 ∨ (Δ) = Δ, where the map 푠 ∨ : ℎ → ℎ is given by 푣 ↦→ 푣 − 훼 (푣)훼. 훼푖 ,훼푖 훼푖 ,훼푖

∗ Deﬁnition 3.1.2 The subgroup of GL(픥 ) generated by {푠 ∨ } ≤ ≤ is called the 훼푖 ,훼푖 1 푖 푛 Weyl group, denoted 푊. We also deﬁne the following subsets of 픥∗:

The fundamental weights 휋푖 are deﬁned by 훼푖 (휋 푗 ) = 훿푖 푗 휌 Í푛 휋 1 Í푛 훼 = 푖=1 푖 = 2 푖=1 푖 푄+ 훼 É푛 훼 is the nonnegative integral span of { 푖}, i.e. 푖=1 Z≥0 푖 푃+ É푛 휋 = 푖=1 Z≥0 푖, with elements referred to as dominant weights Elements in −푃+ are referred to as anti-dominant weights 휆 ∗ 푎∨ 휆 < 푃+ Elements in { ∈ 픥 : 푖 ( ) 0} ⊂ − are called regular weights

É ∗ Given a representation 푉 of 픤, we can write 푉 = 푉휆, where all 휆 ∈ 픥 appearing in the decomposition are called weights of 푉.

Theorem 3.1.3 Any irreducible representation 푉 of 픤 has a unique maximal weight 휆 and a unique minimal weight 휇 with respect to the partial ordering given by 푄+, with 휆 dominant and 휇 anti-dominant. Moreover, for ∀휆 ∈ 푃+ there is a unique representation of 픤, denoted 퐿+(휆) with maximal weight 휆.

Deﬁnition 3.1.4 It is known that the center 푍 (픤) ⊂ 푈(픤) acts on 퐿+(휆) by scalars, which deﬁnes a map 휒휆 : 푍 (픤) → C, called the central character associated to 휆.

Consider the map 푓 : 푈(픥) → 푈(픥) induced by the linear map 픥 → 푈(픥) with ℎ ↦→ ℎ + 휌(ℎ)1. Since 푈(픤) = 푈(픥) ⊕ (픫푈(픤) +푈(픤)픫−), ∀푧 ∈ 푍 (픤) can be written 11 as 푧 = ℎ + 푛 for some ℎ ∈ 푈(픥) and 푛 ∈ (픫푈(픤) + 푈(픤)픫−). The Harish-Chandra homomorphism is the map 훾 : 푍 (픤) → 푈(픥) given by 훾(푧) = 푓 (ℎ). Moreover, for the central character, we have 휒휆 (푧) = 휆(훾(푧)).

Theorem 3.1.5 (Harish-Chandra) The Harish-Chandra homomorphism is injec- tive and an isomorphism onto 푈(픥)푊·, with the dotted action of 푊 on 픥∗ given by 푤 · 휆 = 푤(휆 + 휌) − 휌.

3.2 Algebraic D-modules In this subsection, 푋 denotes a smooth algebraic variety over C, with structure sheaf

O푋 and tangent sheaf T푋 . For simplicity, we only discuss the aﬃne case here, but the deﬁnitions can be extended to general cases and the propositions hold as well.

Definition 3.2.1 A linear map 퐷 : O푋 → O푋 is a differential operator of order 0 if it is multiplication by a function. For 푘 ∈ Z+, 퐷 is a differential operator of order 푘 if for any function 푓 ∈ O푋 , the commutator [퐷, 푓 ] is a differential operator f order 푘 − 1. Let 퐷 푋,푘 be the collection of differential operators of order 푘 and 퐷 Ð∞ 퐷 푋 = 푘=0 푋,푘

Concretely, we have the following description of 퐷 푋 .

Proposition 3.2.2 The graded algebra (퐷 ) is isomorphic to (T ). gr 푋 SymO푋 푋

Deﬁnition 3.2.3 A (left) D-module is an O푋 -quasicoherent sheaf of (left) modules over 퐷 푋

Now consider any line bundle L on 푋 with dual sheaf L∨.

Definition 3.2.4 A map 퐷 : L → L is a differential operator of order 0 on L if it is multiplication by a function. For 푘 ∈ Z+, 퐷 is a differential operator of order 푘 on L if for any function 푓 ∈ O푋 , the commutator [퐷, 푓 ] is a differential operator of order 푘 − 1. Denote the collection of all such differential operators by 퐷L.

퐷 퐷 ∨ Proposition 3.2.5 Concretely, we have L = L ⊗O푋 푋 ⊗O푋 L .

3.3 Main Results

Deﬁnition 3.3.1 The quotient 퐺/퐵 is called the ﬂag variety of 퐺.

Proposition 3.3.2 The 퐺-equivariant vector bundles on 퐺/퐵 are in bijection with representations of 퐵. 12 Proposition 3.3.3 The one-dimensional representations of 퐵 are in bijection with characters of 퐻 (i.e. integral characters of 픥).

Hence given an integral weight 휆 ∈ 픥∗, we can attach an one dimensional representation C휆 of 퐵 and a 퐺-equivariant line bundle L(휆)

∗ Deﬁnition 3.3.4 For an integral weight 휆 ∈ 픥 , we deﬁne 퐷휆 as the sheaf 퐷L(휆+휌).

We denote by 퐷휆 (퐺/퐵) the category of 퐷휆-modules that are quasi-coherent as

O퐺/퐵-modules, and by 픤 -mod휆 the category of 푈(픤)-modules on which 푍 (픤) acts

by the character 휒휆. Then we have the following crucial result:

Theorem 3.3.5 (Beilinson–Bernstein Localization) If 휆 is regular, then the global

section functor Γ : 퐷휆 (퐺/퐵) → 픤 -mod휆 is an equivalence of categories with quasi-

inverse given by 퐷휆 ⊗푈(픤) (·)

This is still not exactly what’s needed for our purpose, as we are intersted in classifying (픤, 퐾)-modules. As it turns out, the appropriate notion corresponding to

(픤, 퐾)-modules with central character 휒휆 on the geometric side is the category 퐾 퐷휆 (퐺/퐵) of “퐾-equivariant” 퐷휆-modules on 퐺/퐵. In other words, we have an 퐾 equivalence of categories 퐷휆 (퐺/퐵) (픤, 퐾) -mod휆. Thus to classify irreducible 퐾 (픤, 퐾)-modules, it suﬃces to classify simple objects in 퐷휆 (퐺/퐵) , for which the following two results are necessary.

퐾 Proposition 3.3.6 For every simple M ∈ 퐷휆 (퐺/퐵) , there is a 퐾-orbit 푄 of 퐺/퐵 퐾 and a simple N ∈ 퐷휆 (푄) such that M 푗!∗(N). (For a detailed discussion of

the functor 푗!∗, see [3, Section 5].)

Proposition 3.3.7 Fix a point 푥 in a 퐾-orbit 푄. Let 퐾푥 be the stabilizer of 푥 in 퐾. 퐷 푄 퐾 휆 퐾0 Then 휆 ( ) = 0 unless integrates to a character of the identity component 푥 퐾 퐷 푄 퐾 퐾 퐾0 of 푥, in which case 휆 ( ) Rep( 푥/ 푥 ), the category of representations.

Remark 3.3.8 The integrality condition means we can ﬁrst conjugate 퐾푥 so that it 퐻 퐾0 is contained in and then ﬁnd a character of 푥 whose derivative coincides with the restriction of 휆.

3.4 Induction in the Geometric Context Parallel to the procedure of parabolic inductions, there is also a notion of induction

for (픤, 퐾)-modules. In particular, let 퐾퐻 = 퐾 ∩ 퐵. 13 푟퐺 , 퐾 , 퐾 Deﬁnition 3.4.1 The Jacquet functor 퐻 is given by (픤 ) -mod → (픥 퐻) -mod with 푀 ↦→ 푈(픥) ⊗푈(픟) 푀, in other words, taking 픫 coinvariants. Notably, it admits 푖퐺 a right adjoint, denoted 퐵 .

Remark 3.4.2 On the level of 퐾-modules, 푖퐺 is equivalent to the functor ind퐾 [5]. 퐵 퐾퐻 14 C h a p t e r 4

AN EXAMPLE OF GL(2, R)

In this section, we investigate the connection between the classiﬁcations of irreducible admissible representations of GL(2, R) through the “Langlands” approach and the “geometric” approach.

4.1 Notation

퐺0 = GL(2, R) 퐺 = GL(2, C) 퐾 = O(2, C) 퐵 is the subgroup of upper triangular matrices 퐻 is the subgroup of diagonal matrices

4.2 Langlands Approach By Remark 2.4.6, all irreducible admissible representations of GL(2, R) are param- + eterized in the following way, with 푙 ∈ Z , 푡, 푡1, 푡2 ∈ C satisfying Re 푡1 ≥ Re 푡2:

푡 푡 h1, 푡1, 1, 푡2i : (1 ⊗ | · | 1 )⊗(1 ⊗ | · | 2 ) 푡 푡 h1, 푡1, sgn, 푡2i : (1 ⊗ | · | 1 )⊗(sgn ⊗| · | 2 ) 푡 푡 hsgn, 푡1, 1, 푡2i : (sgn ⊗| · | 1 )⊗(1 ⊗ | · | 2 ) 푡 푡 hsgn, 푡1, sgn, 푡2i : (sgn ⊗| · | 1 )⊗(sgn ⊗| · | 2 ) 푡 h푙, 푡i : 퐷푙 ⊗ | det(·)| Here ⊗ refers to the procedure of taking induced representations and irreducible quotient as described in Remark 2.4.6.

Proposition 4.2.1 The center 푍 (픤픩(2, C)) is generated by 퐼 and the Casimir Δ = 1 퐻2 퐻 퐸퐹 − 4 ( − 2 + 4 ), where ! ! ! ! 1 0 0 −푖 − 푖 1 푖 1 퐼 = 퐻 = 퐸 = 2 2 퐹 = 2 2 푖 1 푖 1 푖 0 1 0 2 2 2 − 2

Proposition 4.2.2 [2] For the representations (sgn휖1 ⊗| · |푡1 ) ⊗ (sgn휖2 ⊗| · |푡2 ) of

GL(2, R) with 휖1, 휖2 ∈ {1, 2}, the central character 휒휆 is given by 1 퐼 ↦→ 푡 + 푡 , Δ ↦→ [1 − (푡 − 푡 )2] 1 2 4 1 2 15 푡 1 푡 푡 휖 휖 휖 Let = 2 ( 1 − 2 + 1) and = 1 + 2. Then moreover If 2푡 . 휖 (mod 2), the representation above is irreducible. Otherwise, it has two irreducible factors: a ﬁnite representation (trivial if and 푡 1 only if = 2 ) and a discrete series representation (a limit of discrete series if 푡 1 and only if = 2 ).

Remark 4.2.3 As all irreducible representations come from parabolically induced ones, we will focus on ﬁnding a geometric interpretation of parabolic induction.

4.3 Geometric Approach By explicit computation, there are two 퐾-orbits on 퐺/퐵, represented by ( !) ( !) 1 0 1 0 푝 := and 푞 := 0 1 푖 1 respectively. Moreover, the stabilizer groups are given by

( 푥 ! ) 0 2 2 퐾푝 = : 푥 = 푦 = 1 0 푦 ( 푥 푦! ) 2 2 퐾푞 = : 푥 + 푦 = 1 −푦 푥

퐾 퐾 퐾0 Note that 푝 is discrete with size four, so Rep( 푝/ 푝) has four simple objects. 휆 퐾0 Moreover, the condition from Proposition 3.3.7 that integrates to a character of 푝 is always satisﬁed. These D-modules correspond to the irreducible representations 휖 휖 hsgn 1 , 푡1, sgn 2 , 푡2i, where h푡1, 푡2i parametrize the weight 휆 and h휖1, 휖2i the simple 퐾 퐾0 object in Rep( 푝/ 푝). 퐾 퐾 퐾0 On the other hand, 푞 is connected, so Rep( 푞/ 푞) has exactly one simple object. Note that characters of 퐾푞 are parametrized by integers 푛 ∈ Z.

푥 푦! ↦→ (푥 + 푦푖)푛 −푦 푥

푡 These D-modules correspond to the discrete series representations 퐷푙 ⊗ | det(·)| with 푙 ∈ Z+ and 푡 ∈ C that we described in Theorem 2.4.4.

Hence as expected, the Langlands approach and the geometric approach both yield the same classiﬁcation for GL(2, R). 16 4.4 Parabolic Induction 퐾 For a dominant regular weight 휆, let Δ휆 : (픤, 퐾) -mod휆 → 퐷휆 (퐺/퐵) be the localization functor from Theorem 3.3.5. Let 푄 be the orbit of 푝 in 퐺/퐵 and 푗 the

inclusion 푄 ↩→ 퐺/퐵. Finally, let 푇푝 (−) be the functor of taking ﬁber at 푝.

Proposition 4.4.1 We have a commutative diagram:

Δ휆 휆 퐾 (픤, 퐾)-mod휆 퐷 (퐺/퐵)

푗∗ (픤, 퐾)-mod 퐷휆 (푄)퐾 = 퐷(푄)퐾

퐺 푇 (−) 푟퐻 푝 (픥, 퐾 )-mod Rep(퐾 /퐾0) 퐻 ⊗ (−) 푝 푝 C휆 푉휃

Here 휃 = 푊 · 휆 and 푉휃 = 푈(픥)/퐽휃푈(픥) where 퐽휃 is the kernel of 푈(픥) → C determined by 휆 + 휌. Notably, on the 퐾-orbit 푄, any 휆-twisting is trivialized.

Proof : By [11, Theorem 2.5], we have an isomorphism between the left derived functors 퐿푇 ◦ 퐿Δ and C ⊗퐿 (푈(픥) ⊗퐿 −). As 푗∗ doesn’t change the ﬁber and 푝 휆 휆 푉휃 푈(픟) 푟퐺 푈 퐻 is exactly (픥) ⊗푈(픟) −, the diagram follows.

Remark 4.4.2 This gives two interpretations of the localization of principal series representations. Now take right adjoints.

Γ휆 (−) 휆 퐾 (픤, 퐾)-mod휆 퐷 (퐺/퐵)

푗∗ (픤, 퐾)-mod 퐷휆 (푄)퐾 = 퐷(푄)퐾

퐺 푖퐵 , 퐾 퐾 퐾0 (픥 퐻)-mod Rep( 푝/ 푝)

We obtain two parametrizations of the principal series representations by the four 퐾 퐾0 simple objects in Rep( 푝/ 푝). On the right hand side, this is exactly the geometric 푖퐺 GL(푛,R) approach, while on the left hand side, note that 퐵 is the same as ind퐷푈 from Proposition 2.4.5, so it is essentially the Langlands approach. 17 BIBLIOGRAPHY

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