Representation Theory of Real Reductive Groups

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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

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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.

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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).

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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 isRorC, 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 certainD-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:

𝐺₀is a connected real reductive Lie group

𝐺is the complexification of𝐺₀with Lie algebra𝔤 𝐵, 𝐻are Borel and Cartan subgroups of𝐺

𝐾₀is the maximal compact subgroup of𝐺₀with complexification𝐾 𝑈(𝔤)is the universal enveloping algebra of𝔤with center 𝑍(𝔤)

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1.3 Preliminaries

Definition 1.3.1 A connected linear algebraic group 𝐿 over R is reductive if the base change 𝐿

C is reductive, which is equivalent to the largest smooth connected unipotent normal subgroup of𝐿

Cbeing trivial.

Definition 1.3.2 A real Lie group𝐺₀ is reductive if there exists a linear algebraic group𝐿overRwhose identity component (in the Zariski topology) is reductive and a homomorphism𝜑 :𝐺₀→ 𝐿(R)withker𝜑finite andim𝜑open in 𝐿(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 𝐵 of an algebraic group 𝐺 is Borel if it is maximal among all Zariski closed connected solvable subgroups. A subgroup 𝐻 ⊂ 𝐺 is Cartan if it is the centralizer of a maximal torus.

Example 1.3.5 ForGL(𝑛,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 (𝜌, 𝑉) of 𝐺₀ on a complex Hilbert space𝑉 is admissible if 𝜌|𝐾₀ is unitary and each irreducible representation of𝐾₀ occurs with at most finite multiplicity in 𝜌|𝐾₀. If𝑉 is admissible, a vector 𝑣 ∈𝑉 is called𝐾₀-finite if it lies in a finite-dimensional space under the action of𝐾₀. Definition 1.3.7 A(𝔤, 𝐾)-module is a vector space𝑉 with actions of𝔤and𝐾 such that for any𝑣 ∈𝑉 , 𝑘 ∈𝐾 , 𝑋 ∈𝔤, and𝑌 ∈𝔨(the Lie algebra of𝐾), we have

𝑘· (𝑋 ·𝑣)= (Ad(𝑘)𝑋) · (𝑘·𝑣)

𝐾 𝑣 ⊂𝑉 spans a finite-dimensional subspace on which𝐾 acts continuously (^𝑑

𝑑 𝑡exp(𝑡𝑌) ·𝑣) |_𝑡=0 =𝑌 ·𝑣

Given an admissible representation (𝜌, 𝑉) of𝐺₀, we can consider the space𝑉_𝐾 of 𝐾₀-finite vectors. Since the action of 𝐾₀ on 𝑉_𝐾 is locally finite, it extends to an action of𝐾. Moreover, it is shown that𝑉_𝐾 is dense in𝑉 and inherits a𝔤action. The actions of𝐾 and𝔤are compatible in the sense that (𝜌, 𝑉_𝐾) is a(𝔤, 𝐾)-module. [6, Chapter III]. We also have the following result [6, Chapter VIII].

Proposition 1.3.8 (𝜌, 𝑉) is irreducible if and only if(𝜌, 𝑉_𝐾)is irreducible.

Another result by [10] shows that:

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Proposition 1.3.9 Every irreducible (𝔤, 𝐾)-module is isomorphic to (𝜌, 𝑉_𝐾) for some irreducible admissible representation(𝜌, 𝑉)of𝐺₀.

Definition 1.3.10 Two admissible representations of𝐺₀are infinitesimally equiva- lent if their underlying (𝔤, 𝐾)-modules are isomorphic.

We would like to classify irreducible admissible representations of𝐺₀up to infinitesimal equivalence, or equivalently, irreducible (𝔤, 𝐾)-modules up to isomorphism.

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C h a p t e r 2

LOCAL LANGLANDS CORRESPONDENCE

2.1 Parabolic Induction

Definition 2.1.1 A unitary representation of 𝐺₀ is a continuous norm-preserving group action of𝐺₀by linear transformations on a Hilbert space.

Definition 2.1.2 Let(𝜌, 𝑉) be an irreducible admissible representation of 𝐺₀and 𝑣 , 𝑣⁰ be 𝐾₀-finite vectors. Then a 𝐾₀-finite matrix coefficient of 𝜌 is a function 𝑥 ↦→ h𝜌(𝑥)𝑣 , 𝑣⁰i. If all𝐾₀-finite matrix coefficients of𝜌are in 𝐿²⁺^𝜖(𝐺) for∀𝜖 >0, we say𝜌is a tempered representation.

Proposition 2.1.3 (Iwasawa) Let𝔤₀be the Lie algebra of 𝐺₀and𝔤₀ = 𝔨₀+𝔭₀be the Cartan decomposition. Then there is moreover a decomposition𝔤₀= 𝔨₀+𝔞₀+𝔫₀ such that𝔞₀is abelian,𝔫₀is nilpotent, and[𝔞₀⊕𝔫₀,𝔞₀⊕𝔫₀] = 𝔫₀.

Definition 2.1.4 Let 𝐴_𝔭, 𝑁_𝔭 be the analytic subgroups of 𝐺₀ with Lie algebras 𝔞₀ and𝔫₀ respectively. Let𝑀_𝔭be the centralizer of𝔞₀in𝐾₀. The subgroup 𝑀_𝔭𝐴_𝔭𝑁_𝔭 is call a minimal parabolic subgroup of 𝐺₀. Any closed subgroup𝑄₀ containing 𝑀_𝔭𝐴_𝔭𝑁_𝔭 is called a standard parabolic subgroup.

Proposition 2.1.5 Any standard parabolic subgroup𝑄₀ has a Langlands decom- position𝑄₀= 𝑀₀𝐴₀𝑁₀, where 𝑀₀is reductive, 𝐴₀is abelian, and 𝑁₀is nilpotent.

Example 2.1.6 ForGL(2,R), the only proper parabolic is the subgroup of upper triangular matrices with

𝑀₀=

( ±1 0 0 ±1

! )

𝐴₀ = (

𝑥 0 0 𝑦

!

:𝑥 , 𝑦 >0 )

𝑁₀ =

( 1 𝑧 0 1

! )

Theorem 2.1.7 (Langlands) [8] The equivalence classes (up to infinitesimal equiv- alence) of irreducible admissible representations of 𝐺₀ are in bijection with all triples(𝑀₀𝐴₀𝑁₀,[𝜎₀], 𝜈₀)satisfying

a)𝑀₀𝐴₀𝑁₀is a parabolic subgroup of𝐺₀containing𝑀_𝔭𝐴_𝔭𝑁_𝔭

b)𝜎₀is an irreducible tempered (unitary) representation of 𝑀₀and [𝜎₀] its equivalence class

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c)𝜈₀is a member of 𝔞⁰₀ such thatReh𝜈₀, 𝛼i > 0for every positive restricted root not vanishing on𝔞₀

In particular, the correspondence is that (𝑀₀𝐴₀𝑁₀,[𝜎₀], 𝜈₀) corresponds to the class of the unique irreducible quotient ofInd^𝐺_𝑀⁰

0𝐴₀𝑁₀(𝜎₀⊗𝜈₀⊗1).

Specifically, the induced representation Ind^𝐺_𝑀⁰

0𝐴₀𝑁₀(𝜎₀ ⊗ 𝜈₀ ⊗ 1) is given by first considering all𝐹 ∈𝐶(𝐺₀, 𝑉) with

𝐹(𝑥 𝑚 𝑎𝑛) =𝑒⁻⁽^𝜈⁰⁺^𝜌⁾^log^𝑎𝜎₀(𝑚)⁻¹𝐹(𝑥)

and defining Ind^𝐺_𝑀⁰

0𝐴₀𝑁₀(𝜎₀⊗ 𝜈₀⊗1) (𝑔)𝐹(𝑥) =𝐹(𝑔⁻¹𝑥).

Definition 2.1.8 The parameters(𝑀₀𝐴₀𝑁₀,[𝜎₀], 𝜈₀)are called the Langlands pa- rameters of the given irreducible admissible representation.

2.2 Representations of the Weil Group𝑊

R

Definition 2.2.1 The Weil group ofR, denoted𝑊

R, is the nonsplit extension ofC^× byZ/2Zgiven by𝑊_𝑅 =C^×∪ 𝑗C^×, where 𝑗²=−1and 𝑗 𝑐 𝑗⁻¹=𝑐for∀𝑐 ∈C^×. For reasons that will become clear, we are interested in classifying𝑛-dimensional complex representations of𝑊

Rwhose images consist of semisimple elements (i.e.

semisimple matrices in GL(𝑛,C)). We refer to them as semisimple representations.

Lemma 2.2.2 One-dimensional representations of C^× are of the form 𝑧 ↦→ 𝑧^𝜇𝑧^𝜈 with𝜇, 𝜈 ∈ Cand𝜇−𝜈 ∈Z. (Note thatC^× 𝑆¹×R⁺, and representations of the components are known.)

Proposition 2.2.3 One-dimensional representations𝜑of𝑊

Rare of the form h+, 𝑡i : 𝜑(𝑧) =|𝑧|^𝑡

R and 𝜑(𝑗) =1 h−, 𝑡i : 𝜑(𝑧) =|𝑧|^𝑡

R and 𝜑(𝑗) =−1 where𝑡can be any complex number.

Proof: Suppose 𝜑(𝑗) = 𝑤 ∈ C^× and∀𝑧 ∈ C^×, 𝜑(𝑧) = 𝑧^𝜇𝑧^𝜈. Then since 𝜑(𝑧) = 𝑤 𝜑(𝑧)𝑤⁻¹ = 𝜑(𝑗 𝑧 𝑗⁻¹) = 𝜑(𝑧), it follows that 𝜇 = 𝜈, so 𝜑(𝑟 𝑒^{𝑖 𝜃}) = 𝑟²^𝜇. Since 𝑤²= 𝜑(𝑗²) =𝜑(−1) =1, 𝑤 =±1. Let𝑡 =2𝜇. It is easy to check that the formulas above indeed give valid representations.

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Proposition 2.2.4 Equivalent classes of irreducible two-dimensional semisimple representations𝜑 of𝑊

Rare of the form

h𝑙 , 𝑡i: 𝜑(𝑟 𝑒^{𝑖 𝜃})𝛼=𝑟²^𝑡𝑒^{𝑖𝑙 𝜃}𝛼 and 𝜑(𝑗)𝛼= 𝛽 𝜑(𝑟 𝑒^{𝑖 𝜃})𝛽=𝑟²^𝑡𝑒^{−𝑖𝑙 𝜃}𝛽 and 𝜑(𝑗)𝛽= (−1)^𝑙𝛼

where𝛼, 𝛽form a basis of the two dimensional vector space and𝑙 ∈Z⁺, 𝑡 ∈C. Proof: Given such a representation𝜑, pick a basis𝑎, 𝑏such that𝜑(C^×) consist of diagonal matrices. By the lemma, we can suppose 𝜑(𝑧)𝑎 = 𝑧^𝜇¹𝑧^𝜈¹𝑎 and 𝜑(𝑧)𝑏 = 𝑧^𝜇²𝑧^𝜈²𝑏. Since𝜑is irreducible, either 𝜇₁≠ 𝜇₂or𝜈₁ ≠𝜈₂. Let𝑎⁰:=𝜑(𝑗)𝑎. Then

𝜑(𝑧)𝑎⁰=𝜑(𝑗 𝑧 𝑗⁻¹)𝑎⁰=𝜑(𝑗 𝑧)𝑎=𝑧^𝜈¹𝑧^𝜇¹𝑎⁰

If 𝜇₁ = 𝜈₁, then 𝑎⁰ ∈ C𝑎 and C𝑎 is an invariant subspace. Contradiction! Thus 𝑎⁰∈C𝑏 by the equation above, which means𝜈 :=𝜈₁= 𝜇₂and 𝜇:= 𝜇₁ =𝜈₂. Since 𝜑(𝑗)⁻¹=𝜑(𝑗)𝜑(−1) = (−1)^𝜇−𝜈𝜑(𝑗), we have

𝜑(𝑧)𝑎=𝑧^𝜇𝑧^𝜈𝑎 and 𝜑(𝑗)𝑎=𝑎⁰

𝜑(𝑧)𝑎⁰=𝑧^𝜈𝑧^𝜇𝑎⁰ and 𝜑(𝑗)𝑎⁰=(−1)^𝜇⁻^𝜈𝑎

Considering instead the basis 𝑎⁰,(−1)^𝜇−𝜈𝑎 and in view of the symmetry, we can suppose without the loss of generality that the integer 𝑙 := 𝜇− 𝜈 is positive. Let 𝑡 = ¹₂(𝜇+𝜈),𝛼 =𝑎, and𝛽 =𝑎⁰. Then the result follows immediately.

Proposition 2.2.5 Every finite-dimensional semisimple representation 𝜑 of 𝑊

R is fully reducible, and each irreducible representation has dimension one or two.

Proof: Given a representation 𝜑acting on𝑉, since 𝜑(C^×) consists of commuting diagonalizable matrices,𝑉 is the direct sum of spaces𝑉_𝜇,𝜈 on which 𝜑(𝑧) acts by 𝑧^𝜇𝑧^𝜈. Meanwhile,𝜑(𝑗)𝑉_𝜇,𝜈 =𝑉_{𝜈, 𝜇}. If𝜇 =𝜈, we can choose a basis of eigenvectors for𝜑(𝑗)in𝑉_𝜇,𝜈, so the span of each vector is an one-dimensional invariant subspace under𝜑(𝑊

R). If 𝜇≠ 𝜈, pick a basis𝑢₁, 𝑢₂,· · ·𝑢_𝑟 of𝑉_𝜇,𝜈. ThenC𝑢_𝑖⊕C𝜑(𝑗)𝑢_𝑖is a two-dimensional invariant subspace. Moreover,𝑉_𝜇,𝜈⊕𝑉_{𝜈, 𝜇}=É𝑟

𝑖=1C𝑢_𝑖⊕C𝜑(𝑗)𝑢_𝑖. 2.3 𝐿-function and𝜖-factor

Given any irreducible admissible representations𝜌of𝐺₀, we can attach two invari- ants, called the𝐿-function and𝜖-factor. For simplicity, we define them here for the special case of𝐺₀=GL(𝑛,R). Fix an additive character𝜓ofRgiven by𝑥 ↦→ 𝑒²^𝜋𝑖𝑥.

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Let 𝑀_𝑛(R) be the collection of 𝑛× 𝑛 matrices and I₀ be the space of functions 𝑀_𝑛(𝐾) → R of the form 𝑃(𝑥_{𝑖 𝑗})exp(−𝜋Í

𝑥²

𝑖 𝑗) for any polynomial 𝑃. For any 𝐾₀-finite matrix coefficient𝑐of𝜌,∀𝑓 ∈ I₀, and𝑠 ∈C, we define

𝜁(𝑓 , 𝑐, 𝑠) =

∫

𝑀_𝑛(R)

𝑓(𝑥)𝑐(𝑥) |det𝑥|^𝑠𝑑^×𝑥

Here𝑑^×𝑥 =|det𝑥|⁻^𝑛𝑑𝑥, where𝑑𝑥is a fixed invariant measure for𝑀_𝑛(R).

Proposition 2.3.1 If 𝜌 is irreducible, then all 𝜁(𝑓 , 𝑐, 𝑠) converge in the right half plane and extend to a meromorphic function (in terms of 𝑠) onC. Moreover, there is a finite collection{(𝑐_𝑖, 𝑓_𝑖)}such that

𝐿(𝑠, 𝜌) =∑︁

𝑖

𝜁(𝑓_𝑖, 𝑐_𝑖, 𝑠)

satisfies∀(𝑐, 𝑓), 𝜁(𝑓 , 𝑐, 𝑠+¹

2(𝑛−1)) =𝑃(𝑓 , 𝑐, 𝑠)𝐿(𝑠, 𝜌)with a polynomial𝑃in𝑠. In particular 𝐿(𝑠, 𝜌) is uniquely defined this way (up to a scalar).

Now for∀𝑓 ∈ I₀, we define 𝑓ˆ=

∫

𝑀𝑛(R)

𝑓(𝑦)𝜓(Tr(𝑥 𝑦))𝑑𝑦

where𝑑𝑦is the self-dual Haar measure on𝑀_𝑛(R). Then ˆ𝑓 ∈ I₀.

Proposition 2.3.2 If𝜌is irreducible, there exists a meromorphic function𝛾(𝑠, 𝜌, 𝜓) independent of 𝑓 and𝑐with𝜁(𝑓 ,ˆ 𝑐,ˇ 1−𝑠+¹

2(𝑛−1)) =𝛾(𝑠, 𝜌, 𝜓)𝜁(𝑓 , 𝑐, 𝑠+¹

2(𝑛−1)) for∀𝑓 ∈ I₀and any𝐾₀finite matrix coefficient𝑐of 𝜌. Here𝑐ˇ(𝑥) =𝑐(𝑥⁻¹).

Definition 2.3.3 We refer to 𝐿(𝜌, 𝑠) as the𝐿-function of 𝜌. The𝜖-factor is defined as

𝜖(𝑠, 𝜌, 𝜓) = 𝛾(𝑠, 𝜌, 𝜓)𝐿(𝑠, 𝜌) 𝐿(1−𝑠,𝜌˜) where 𝜌˜is the admissible dual of 𝜌.

We can also attach an𝐿-function and an𝜖-factor to a representation of𝑊

Ras follows.

Definition 2.3.4 Given an irreducible finite-dimensional semisimple representation 𝜑of𝑊

R, we can attach an𝐿-function

𝐿(𝑠, 𝜑) =











 𝜋⁻

𝑠+𝑡 2 Γ(^𝑠+𝑡

2 ) for 𝜑given byh+, 𝑡i 𝜋⁻

𝑠+𝑡+1

2 Γ(^𝑠+𝑡+¹

2 ) for𝜑given byh−, 𝑡i 2(2𝜋)⁻⁽^𝑠+𝑡+²^𝑙⁾Γ(𝑠+𝑡+ ^𝑙

2) for𝜑given byh𝑙 , 𝑡i

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and an𝜖-factor:

𝜖(𝑠, 𝜑, 𝜓) =













1 for𝜑given byh+, 𝑡i 𝑖 for 𝜑given byh−, 𝑡i 𝑖^𝑙⁺¹ for𝜑given byh𝑙 , 𝑡i

Now to state the local Langlands correspondence, we need one last ingredient.

Roughly speaking, the 𝐿-group of a connected reductive group𝐺₀overRis given by the semi-direct product of a complex reductive group with dual root datum to𝐺₀ and the Galois group Gal(C/R). We denote it by^𝐿𝐺₀. Concretely,

Definition 2.3.5 The𝐿-groups ofGL(𝑛,R),SL(𝑛,R),SO(2𝑛+1,R), andSO(2𝑛,R) areGL(𝑛,C),PGL(𝑛,C),Sp(𝑛,C), andSO(2𝑛,C)respectively.

2.4 Main Results

Theorem 2.4.1 (Local Langlands Correspondence) There is a bijection between semisimple representations of𝑊

Rinto^𝐿𝐺₀and𝐿-packets of irreducible admissible representations of𝐺₀.

Remark 2.4.2 This theorem means we have a partition of all the irreducible ad- missible representations of 𝐺₀, indexed by semisimple representations of𝑊

R into

𝐿𝐺₀. An 𝐿-packet is a collection of representations of𝐺₀ that correspond to the same representation of𝑊

R, which is called the𝐿-parameter of that packet.

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

A priori, there can be multiple representations in a single𝐿-packet. However, when 𝐺₀=GL(𝑛,R), each𝐿-packet has size one. In particular,

Theorem 2.4.4 There is a bijection between the set of equivalence classes of 𝑛- dimensional semisimple complex representations of𝑊

Rand the set of equivalence classes of irreducible admissible representations ofGL(𝑛,R).

To make the correspondence explicit, we first define some classes of representations of GL(1,R)and GL(2,R). Note that for 𝑗 =1,2, any matrix 𝑋 ∈GL(𝑗 ,R)admits a decomposition𝑋 =𝑌 𝑍 where𝑌 ∈SL^±(𝑗 ,R) and𝑍 is positive scalar. Denote the two characters of SL^±(1,R) as 1 and sgn. For𝑙 ∈ Z⁺, let 𝐷⁺

𝑙 be the representation

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of SL(2,R) on the space of analytic functions 𝑓 on the upper half plane satisfying k𝑓k²=∫ ∫

|𝑓(𝑧) |²𝑦^𝑙−1𝑑𝑥 𝑑𝑦 < ∞given by

𝐷⁺

𝑙

𝑎 𝑏 𝑐 𝑑

!

𝑓(𝑧) =(𝑏 𝑧+𝑑)⁻^𝑙⁻¹𝑓(𝑎 𝑧+𝑐 𝑏 𝑧+𝑑 )

Consider𝐷_𝑙 =ind^SL

±(2,R) SL(2,R) (𝐷⁺

𝑙). For GL(1,R),∀𝑡 ∈C, we have representations 1⊗ | · |^𝑡 : 𝑋 ↦→ |𝑍|^𝑡

sgn⊗| · |^𝑡 : 𝑋 ↦→sgn(𝑌) |𝑍|^𝑡 Similarly for GL(2,R), we have

𝐷_𝑙⊗ |det(·) |^𝑡 : 𝑋 ↦→ 𝐷_𝑙(𝑌) |det(𝑍) |^𝑡 Now given a representation of 𝜑 of𝑊

R, suppose 𝜑 = É^𝑗

𝑖=1𝜑_𝑗, where 𝜑_𝑗 is a 𝑛_𝑗- dimensional representation with𝑛_𝑗 =1 or 2. We associate to 𝜑_𝑗 to a representation 𝜓_𝑗 of GL(𝑛_𝑗,R) in the following way

h+, 𝑡i → 1⊗ | · |^𝑡 h−, 𝑡i → sgn⊗| · |^𝑡

h𝑙 , 𝑡i → 𝐷_𝑙⊗ |det(·) |^𝑡

Let𝑡_𝑗 be the complex number corresponding to 𝜑_𝑗. Suppose wlog that𝑛⁻¹

1 Re𝑡₁ ≥ 𝑛⁻¹

2 Re𝑡₂ ≥ · · · ≥ 𝑛⁻¹

𝑗 Re𝑡_𝑗. Let𝐷 =Î^𝑗

𝑖=1GL(𝑛_𝑗,R)and𝑈the block strictly upper triangular subgroup of GL(𝑛,R). We can extendÉ^𝑗

𝑖=1𝜓_𝑗 to a representation𝜓⁰on 𝐷𝑈 by making it trivial on𝑈.

Proposition 2.4.5 The induced representationind^GL(_𝐷𝑈^𝑛,^R⁾ ofGL(𝑛,R)has a unique irreducible quotient𝜓, which is admissible.

Remark 2.4.6 The bijection in Theorem 2.4.4 is given by 𝜑↦→𝜓 described above.

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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 𝔤then the set of roots of 𝔤is given by Δ⁺∪ {0} ∪Δ⁻ and we have a decomposition

𝔤 =Ê

𝛼∈Δ⁺

𝔤𝛼⊕𝔥⊕ Ê

𝛼∈Δ⁻

𝔤𝛼

where𝔥is the Cartan subalgebra. Let 𝔫 = É

𝛼∈Δ⁺𝔤𝛼 and𝔫⁻ = É

𝛼∈Δ⁻𝔤𝛼. Then 𝔟:= 𝔥⊕𝔫is a Borel subalgebra. LetΠ ={𝛼_𝑗}_1≤𝑗≤𝑛be a collection of simple roots.

Proposition 3.1.1 For each𝛼_𝑖 ∈ Π, there exists𝛼^∨

𝑖 ∈ 𝔥such that𝛼^∨

𝑖 (𝛼_𝑖) = 2and 𝑠𝛼^∨

𝑖,𝛼𝑖(Δ) = Δ, where the map𝑠

𝛼^∨

𝑖,𝛼𝑖 :ℎ^∗ → ℎ^∗is given by𝑣 ↦→𝑣−𝛼^∨(𝑣)𝛼. Definition 3.1.2 The subgroup ofGL(𝔥^∗) generated by {𝑠

𝛼^∨

𝑖,𝛼𝑖}₁_{≤𝑖≤𝑛} is called the Weyl group, denoted𝑊. We also define the following subsets of𝔥^∗:

The fundamental weights𝜋_𝑖are defined by𝛼_𝑖(𝜋_𝑗)=𝛿_{𝑖 𝑗} 𝜌 =Í^𝑛

𝑖=1𝜋_𝑖= ¹₂Í^𝑛

𝑖=1𝛼_𝑖

𝑄⁺is the nonnegative integral span of{𝛼_𝑖}, i.e. É^𝑛

𝑖=1Z^≥0𝛼_𝑖 𝑃⁺ =É^𝑛

𝑖=1Z^≥0𝜋_𝑖, with elements referred to as dominant weights Elements in−𝑃⁺ are referred to as anti-dominant weights

Elements in{𝜆∈𝔥^∗ :𝑎^∨

𝑖 (𝜆) < 0} ⊂ −𝑃⁺ are called regular weights Given a representation𝑉 of𝔤, we can write𝑉 =É

𝑉_𝜆, where all𝜆∈ 𝔥^∗ appearing in the decomposition are called weights of𝑉.

Theorem 3.1.3 Any irreducible representation𝑉 of𝔤has a unique maximal weight 𝜆and a unique minimal weight𝜇with respect to the partial ordering given by𝑄⁺, with 𝜆 dominant and 𝜇 anti-dominant. Moreover, for ∀𝜆 ∈ 𝑃⁺ there is a unique representation of𝔤, denoted 𝐿⁺(𝜆) with maximal weight𝜆.

Definition 3.1.4 It is known that the center𝑍(𝔤) ⊂𝑈(𝔤)acts on𝐿⁺(𝜆)by scalars, which defines a map 𝜒_𝜆 :𝑍(𝔤) → C, called the central character associated to𝜆. Consider the map 𝑓 : 𝑈(𝔥) → 𝑈(𝔥) induced by the linear map 𝔥 → 𝑈(𝔥) with ℎ↦→ ℎ+𝜌(ℎ)1. Since𝑈(𝔤) =𝑈(𝔥) ⊕ (𝔫𝑈(𝔤) +𝑈(𝔤)𝔫⁻),∀𝑧 ∈𝑍(𝔤)can be written

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as𝑧 = ℎ+𝑛for some ℎ ∈𝑈(𝔥) and𝑛 ∈ (𝔫𝑈(𝔤) +𝑈(𝔤)𝔫⁻). The Harish-Chandra homomorphism is the map𝛾 : 𝑍(𝔤) →𝑈(𝔥) given by𝛾(𝑧) = 𝑓(ℎ). Moreover, for the central character, we have 𝜒_𝜆(𝑧) =𝜆(𝛾(𝑧)).

Theorem 3.1.5 (Harish-Chandra) The Harish-Chandra homomorphism is injec- tive and an isomorphism onto𝑈(𝔥)^𝑊^·, with the dotted action of𝑊 on𝔥^∗ given by 𝑤·𝜆=𝑤(𝜆+𝜌) −𝜌.

3.2 Algebraic D-modules

In this subsection,𝑋denotes a smooth algebraic variety overC, with structure sheaf O𝑋 and tangent sheafT𝑋. 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 𝐷 : O𝑋 → O𝑋 is a differential operator of order 0 if it is multiplication by a function. For 𝑘 ∈ Z⁺, 𝐷 is a differential operator of order𝑘if for any function 𝑓 ∈ O𝑋, the commutator[𝐷 , 𝑓]is a differential operator f order 𝑘 −1. Let 𝐷_{𝑋 , 𝑘} be the collection of differential operators of order 𝑘 and 𝐷_𝑋 =Ð∞

𝑘=0𝐷_{𝑋 , 𝑘}

Concretely, we have the following description of𝐷_𝑋.

Proposition 3.2.2 The graded algebragr(𝐷_𝑋) is isomorphic toSym_O_𝑋(T𝑋). Definition 3.2.3 A (left) D-module is an O𝑋-quasicoherent sheaf of (left) modules over𝐷_𝑋

Now consider any line bundleL on 𝑋 with dual sheafL^∨.

Definition 3.2.4 A map𝐷 : L → L is a differential operator of order0onL if it is multiplication by a function. For𝑘 ∈Z⁺, 𝐷 is a differential operator of order 𝑘 onL if for any function 𝑓 ∈ O𝑋, the commutator [𝐷 , 𝑓] is a differential operator of order𝑘−1. Denote the collection of all such differential operators by𝐷_L. Proposition 3.2.5 Concretely, we have𝐷_L =L ⊗_O_𝑋 𝐷_𝑋 ⊗_O_𝑋 L^∨.

3.3 Main Results

Definition 3.3.1 The quotient𝐺/𝐵is called the flag variety of𝐺.

Proposition 3.3.2 The𝐺-equivariant vector bundles on𝐺/𝐵are in bijection with representations of𝐵.

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Proposition 3.3.3 The one-dimensional representations of 𝐵are in bijection with characters of𝐻(i.e. integral characters of𝔥).

Hence given an integral weight𝜆 ∈𝔥^∗, we can attach an one dimensional represen- tationC^𝜆 of𝐵and a𝐺-equivariant line bundleL (𝜆)

Definition 3.3.4 For an integral weight𝜆∈𝔥^∗, we define𝐷_𝜆 as the sheaf𝐷_{L (}_𝜆₊_𝜌₎. We denote by 𝐷_𝜆(𝐺/𝐵) the category of 𝐷_𝜆-modules that are quasi-coherent as O_𝐺/𝐵-modules, and by𝔤-mod𝜆 the category of𝑈(𝔤)-modules on which 𝑍(𝔤) acts by the character 𝜒_𝜆. Then we have the following crucial result:

Theorem 3.3.5 (Beilinson–Bernstein Localization) If𝜆is regular, then the global section functorΓ:𝐷_𝜆(𝐺/𝐵) →𝔤-mod𝜆is an equivalence of categories with quasi- inverse given by𝐷_𝜆 ⊗𝑈(𝔤) (·)

This is still not exactly what’s needed for our purpose, as we are intersted in classifying (𝔤, 𝐾)-modules. As it turns out, the appropriate notion corresponding to (𝔤, 𝐾)-modules with central character 𝜒_𝜆 on the geometric side is the category 𝐷_𝜆(𝐺/𝐵)^𝐾 of “𝐾-equivariant” 𝐷_𝜆-modules on 𝐺/𝐵. In other words, we have an equivalence of categories 𝐷_𝜆(𝐺/𝐵)^𝐾 (𝔤, 𝐾)-mod𝜆. Thus to classify irreducible (𝔤, 𝐾)-modules, it suffices to classify simple objects in 𝐷_𝜆(𝐺/𝐵)^𝐾, for which the following two results are necessary.

Proposition 3.3.6 For every simpleM ∈ 𝐷_𝜆(𝐺/𝐵)^𝐾, there is a𝐾-orbit𝑄of𝐺/𝐵 and a simpleN ∈ 𝐷_𝜆(𝑄)^𝐾 such that M 𝑗_!_∗(N ). (For a detailed discussion of the functor 𝑗_!∗, see [3, Section 5].)

Proposition 3.3.7 Fix a point𝑥 in a𝐾-orbit𝑄. Let𝐾_𝑥 be the stabilizer of𝑥 in𝐾. Then 𝐷_𝜆(𝑄)^𝐾 =0unless𝜆integrates to a character of the identity component𝐾⁰

𝑥

of𝐾_𝑥, in which case𝐷_𝜆(𝑄)^𝐾 Rep(𝐾_𝑥/𝐾⁰

𝑥), the category of representations.

Remark 3.3.8 The integrality condition means we can first conjugate 𝐾_𝑥 so that it is contained in 𝐻 and then find a character of𝐾⁰

𝑥 whose derivative coincides with the restriction of𝜆.

3.4 Induction in the Geometric Context

Parallel to the procedure of parabolic inductions, there is also a notion of induction for (𝔤, 𝐾)-modules. In particular, let 𝐾_𝐻 =𝐾∩𝐵.

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Definition 3.4.1 The Jacquet functor𝑟^𝐺

𝐻 is given by (𝔤, 𝐾)-mod → (𝔥, 𝐾_𝐻)-mod with𝑀 ↦→𝑈(𝔥) ⊗𝑈(𝔟) 𝑀, in other words, taking𝔫coinvariants. Notably, it admits a right adjoint, denoted𝑖^𝐺

𝐵.

Remark 3.4.2 On the level of𝐾-modules,𝑖^𝐺

𝐵 is equivalent to the functorind^𝐾_𝐾

𝐻 [5].

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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 𝐺₀=GL(2,R) 𝐺 =GL(2,C) 𝐾 =O(2,C)

𝐵is the subgroup of upper triangular matrices 𝐻 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, with𝑙 ∈Z⁺,𝑡 , 𝑡₁, 𝑡₂ ∈Csatisfying Re𝑡₁≥ Re𝑡₂:

h1, 𝑡₁,1, 𝑡₂i: (1⊗ | · |^𝑡¹)⊗(1⊗ | · |^𝑡²) h1, 𝑡₁,sgn, 𝑡₂i: (1⊗ | · |^𝑡¹)⊗(sgn⊗| · |^𝑡²) hsgn, 𝑡₁,1, 𝑡₂i: (sgn⊗| · |^𝑡¹)⊗(1⊗ | · |^𝑡²) hsgn, 𝑡₁,sgn, 𝑡₂i : (sgn⊗| · |^𝑡¹)⊗(sgn⊗| · |^𝑡²) h𝑙 , 𝑡i: 𝐷_𝑙 ⊗ |det(·) |^𝑡

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 𝑍(𝔤𝔩(2,C)) is generated by 𝐼 and the Casimir Δ =

−¹

4(𝐻²−2𝐻+4𝐸 𝐹), where 𝐼 = 1 0

0 1

!

𝐻 = 0 −𝑖 𝑖 0

!

𝐸 = −₂^𝑖 ¹₂

1 2

𝑖 2

!

𝐹 =

𝑖 2

1 2 1 2 −^𝑖

2

!

Proposition 4.2.2 [2] For the representations (sgn^𝜖¹⊗| · |^𝑡¹) ⊗ (sgn^𝜖²⊗| · |^𝑡²) of GL(2,R)with𝜖₁, 𝜖₂∈ {1,2}, the central character 𝜒_𝜆 is given by

𝐼 ↦→𝑡₁+𝑡₂, Δ↦→ 1

4[1− (𝑡₁−𝑡₂)²]

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Let𝑡 = ¹₂(𝑡₁−𝑡₂+1) and𝜖 =𝜖₁+𝜖₂. Then moreover

If2𝑡 . 𝜖 (mod 2), the representation above is irreducible.

Otherwise, it has two irreducible factors: a finite representation (trivial if and only if𝑡 = ¹

2) and a discrete series representation (a limit of discrete series if and only if𝑡 = ¹₂).

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 two𝐾-orbits on𝐺/𝐵, represented by (

𝑝 := 1 0 0 1

! ) and

(

𝑞 := 1 0 𝑖 1

! )

respectively. Moreover, the stabilizer groups are given by

𝐾_𝑝 = (

𝑥 0 0 𝑦

!

:𝑥² =𝑦²=1 )

𝐾_𝑞 = (

𝑥 𝑦

−𝑦 𝑥

!

:𝑥²+𝑦²=1 )

Note that 𝐾_𝑝 is discrete with size four, so Rep(𝐾_𝑝/𝐾⁰

𝑝) has four simple objects.

Moreover, the condition from Proposition 3.3.7 that𝜆integrates to a character of𝐾⁰

𝑝

is always satisfied. TheseD-modules correspond to the irreducible representations hsgn^𝜖¹, 𝑡₁,sgn^𝜖², 𝑡₂i, where h𝑡₁, 𝑡₂iparametrize the weight𝜆andh𝜖₁, 𝜖₂ithe simple object in Rep(𝐾_𝑝/𝐾⁰

𝑝).

On the other hand,𝐾_𝑞is connected, so Rep(𝐾_𝑞/𝐾⁰

𝑞)has exactly one simple object.

Note that characters of𝐾_𝑞 are parametrized by integers𝑛 ∈Z. 𝑥 𝑦

−𝑦 𝑥

!

↦→ (𝑥+𝑦𝑖)^𝑛

These D-modules correspond to the discrete series representations 𝐷_𝑙 ⊗ |det(·) |^𝑡 with𝑙 ∈Z⁺and𝑡 ∈Cthat 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).

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4.4 Parabolic Induction

For a dominant regular weight 𝜆, let Δ𝜆 : (𝔤, 𝐾)-mod𝜆 → 𝐷_𝜆(𝐺/𝐵)^𝐾 be the localization functor from Theorem 3.3.5. Let𝑄be the orbit of 𝑝in𝐺/𝐵and 𝑗 the inclusion𝑄 ↩→𝐺/𝐵. Finally, let𝑇_𝑝(−) be the functor of taking fiber at 𝑝.

Proposition 4.4.1 We have a commutative diagram:

(𝔤, 𝐾)-mod𝜆 𝐷^𝜆(𝐺/𝐵)^𝐾

(𝔤, 𝐾)-mod 𝐷^𝜆(𝑄)^𝐾 =𝐷(𝑄)^𝐾

(𝔥, 𝐾_𝐻)-mod Rep(𝐾_𝑝/𝐾⁰

𝑝)

Δ𝜆

𝑗^∗

𝑟^𝐺

𝐻

𝑇_𝑝(−)

C^𝜆⊗𝑉 𝜃(−)

Here 𝜃 = 𝑊 · 𝜆 and 𝑉_𝜃 = 𝑈(𝔥)/𝐽_𝜃𝑈(𝔥) where 𝐽_𝜃 is the kernel of 𝑈(𝔥) → C determined by𝜆+𝜌. Notably, on the𝐾-orbit𝑄, any𝜆-twisting is trivialized.

Proof: By [11, Theorem 2.5], we have an isomorphism between the left derived functors𝐿𝑇_𝑝◦𝐿Δ𝜆andC^𝜆 ⊗^𝐿

𝑉𝜃

(𝑈(𝔥) ⊗^𝐿

𝑈(𝔟) −). As 𝑗^∗doesn’t change the fiber and 𝑟^𝐺

𝐻 is exactly𝑈(𝔥) ⊗𝑈(𝔟)−, the diagram follows.

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

(𝔤, 𝐾)-mod𝜆 𝐷^𝜆(𝐺/𝐵)^𝐾

(𝔤, 𝐾)-mod 𝐷^𝜆(𝑄)^𝐾 =𝐷(𝑄)^𝐾

(𝔥, 𝐾_𝐻)-mod Rep(𝐾_𝑝/𝐾⁰

𝑝)

Γ𝜆(−)

𝑗_∗

𝑖^𝐺

𝐵

We obtain two parametrizations of the principal series representations by the four simple objects inRep(𝐾_𝑝/𝐾⁰

𝑝). On the right hand side, this is exactly the geometric approach, while on the left hand side, note that 𝑖^𝐺

𝐵 is the same as ind^GL_𝐷𝑈^(𝑛,^R⁾ from Proposition 2.4.5, so it is essentially the Langlands approach.

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