Cohomology of GL(2)

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Cohomology of GL (2)

A Thesis

submitted to

Indian Institute of Science Education and Research Pune in partial fulfillment of the requirements for the

BS-MS Dual Degree Programme by

Shiva Chidambaram

Indian Institute of Science Education and Research Pune Dr. Homi Bhabha Road,

Pashan, Pune 411008, INDIA.

April, 2015

Supervisor: Prof. A. Raghuram

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Acknowledgments

I would like to express my sincere gratitude to my thesis supervisor Prof. A. Raghuram, for his constant motivation and help throughout the course of the project. He helped clarify my innumerable doubts and guided me in the right direction to tackle the problem. His passion for Mathematics and overall exuberance is contagious. It has kept me determined throughout the year and has made me look forward to a career in academia with much gusto.

I would also like to thank Dr. Debargha Banerjee, Dr. Steven Spallone and Dr. Amit Hogadi for helping me assess my progress regularly.

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Abstract

LetF be a number field. LetG=GL(2) overF. LetAandAf denote the ring of adeles and finite adeles ofF respectively. LetK_∞denote the maximal compact subgroup of G_∞=Y

ν G(F_ν) thickened by the center, where the product runs over all archimedean placesνofF andF_νdenotes the completion ofF atν. For a fixed open compact subgroup K_f ⊂G(Af), letS^G_K

f =G(F)\G(A)/K_fK_∞be a locally symmetric space attached toG. Let r[d] be an irreducible representation ofG of highest weightd, and letFd denote the corresponding sheaf onS^G_K

f. The goal of this project is to understand the cohomology H^•(S^G_K

f,Fd) via its relation to the theory of automorphic forms onG. This relation arises due to the isomorphismH^•(S^G_K

f,Fd)'H^•(g_∞,K_∞;C^∞(GF\G_A/K_f)⊗R[d]^∨). A

specific problem is to understand the inner cohomology denotedH_!^•(S^G_K

f,Fd), which by definition is the image of compactly supported cohomology in full cohomology:

H_!^•(S^G_K_f,Fd) :=Im

³

H_c^•(S^G_K_f,Fd)−−−−→H^•(S^G_K_f,Fd)

´ .

It is known that inner cohomology contains cuspidal cohomology, which is genered by cusp forms onG. The problem is to classify inner cohomology classes which are not cuspidal. In this thesis, we deal withF =QandF =Q(p

−n) wheren is a square free positive integer. We give a description of the inner cohomology mentioned above in these two cases.

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Chapter 1 Introduction

We start by mentioning a theorem in Eberhard Freitag’s Hilbert Modular Forms ([8], Chapter III, Theorem 6.3). It gives a description of the Betti numbersb^qfor an arbitrary congruence subgroupΓofSL(2,R)ⁿ. These are the numbers that give the dimensions of the sheaf cohomology groupsH^q(Γ)=H^q(Hⁿ/Γ,C) corresponding to the constant sheaf given byC, considered as vector spaces overC.

b^q=dim_CH^q(Γ). (1.0.1)

If we denote the set of archimedean places ofF byS, the completion ofF at a placeν byF_ν, the adele ring ofF byA, the finite adele ring byAf, the direct productY

ν∈S

GL(2,F_ν) byGL(2)_∞, and its maximal compact subgroup thickened by the center ofGL(2)_∞byK_∞, the objectsH^q(Γ) are exactly isomorphic to certain sheaf cohomology groupsH^q(S^G_K

f,Fd) for certain open compact subgroupsK_f ofGL(2,Af) depending onΓ. To be precise, for an open compact subgroupK_f ⊆GL(2,Af), the spaceS^G_K

f mentioned above, is actually the locally symmetric spaceGL(2,F)\GL(2,A)/K_fK_∞attached toG=GL(2), andFd denotes the sheaf onS^G_K

f derived from a highest weight irreducible representation ofGL(2)_∞, denoted (r[d],R[d]), with highest weightd=(d_ν)_ν∈S.

The theorem also gives the dimensions of various different subspaces of H^q(Γ), namely the Eisenstien cohomology, square integrable cohomology, inner cohomology, cuspidal cohomology and universal cohomology. We know from Chapter III, Theorem 6.3 of [8], that for all degrees other than zero, the full cohomology breaks up as a direct sum of Eisenstein cohomology and square integrable cohomology; and the square integrable cohomology is the same as inner cohomology, which by definition is the image of com-

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2 CHAPTER 1. INTRODUCTION pactly supported cohomology in full cohomology, for all these degrees. Moreover, inner cohomology at any degreeq is the direct sum of universal and cuspidal cohomologies.

We are interested particularly in the inner cohomology classes that are not cuspidal, i.e., the universal cohomology. In this thesis, we have tried to understand this universal part of the cohomology for some special cases of number fieldsF, namely,F =Qand F =Q(p

−n),n≥0. We take the approach of Waldspurger [12] using automorphic forms onGL(2), making use of the following isomorphism of the sheaf cohomology group to a certain relative lie algebra cohomology, as explained in [5].

H^•(S^G_K

f,Fd)'H^•(g_∞,K_∞;C^∞(G_F\G_A)^K^f ⊗R[d]^∨). (1.0.2) Let (π,E) be an irreducible automorphic representation ofGL(2) overF. We first find out whenE has non-trivial Lie algebra cohomology. Propositon I.4 of [12] gives us a classification of irreducible cohomological representations forGL(2,R) andGL(2,C) and describes the cohomology groups as well. We also use Theorem I.5.1 of [12] that captures all the possible automorphic representations ofGL(2) overF, that could contribute to inner cohomology. This set includes only cuspidal and one-dimensional representations with infinity type isomorphic to one of a finite set of representations depending on the weightd. Further, cuspidals contribute to cuspidal cohomology, and hence we try to find the one-dimensional representations that contribute to inner cohomology. These exactly correspond to universal cohomology classes.

We start by stating the classification of irreducible admissible representations of GL(2,R) and GL(2,C), and then define Lie Algebra cohomology, also called (g,K) cohomology, and talk about irreducible admissible cohomological representations and their cohomology groups in Chapter 2. This is basically a summary of certain results following [7], [11], [4]. In Chapter 3, we set up the basic notations to talk about the sheaf cohomology groupsH^•(S^G_K

f,Fd) and state their relation to the relative lie algebra cohomology groups given in 1.0.2. We also introduce the inner cohomology groups, define the spectrum and describe the set of irreducible automorphic representations that are candidates to contributing to inner cohomology, following [5], [10] and [12]. We discuss Theorem I.5.1 of [12] and also give an outline of the proof of it given there. In Chapter 4, we give a summary of results given in [8] that we mentioned in the beginning. Then, we go on to calculate the inner cohomology using our approach involving the language of automorphic forms for the case ofF =Q. In Chapter 5, we do the same for the case of an imaginary quadratic field.

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Chapter 2 Cohomological Representations of Archimedean GL (2)

2.1 Classification of irreducible admissible ¡

g , K ¢

modules

We follow [7] and [11] in this section for classifying irreducible admissible representations ofGL(2,R) andGL(2,C) respectively. The classification is up to infinitesimal equivalence as

³ g l¡

2,R¢ ,O(2)

´

and respectively

³ g l¡

2,C¢ ,U(2)

´

modules.

LetG=GL¡ 2,R¢

. LetK denote the maximal compact subgroupO(2) ofG,K⁰=SO(2) andgdenote the Lie algebrag l¡

2,R¢

ofG. Let

Z =



 1 0 0 1



, H= −i





0 1

−1 0



, L=1 2





1 −i

−i −1



, R=1 2



 1 i i −1



. (2.1.1) be elements of the complexificationg_Cof the Lie algebrag. These form a basis ofg_Cas a complex vector space. LetU¡

g_C^¢denote the universal enveloping algebra ofg_C. The elements∆=⁻₄¹

³

H²+2RL+2LR´

andZ lie in the center ofU¡

g_C^¢. Since the irreducible unitary representations ofK⁰are one dimensional and are parametrized by the integers, withk∈Zdenoting the characterσk

¡k_θ¢

=e^{i k}^θ, wherek_θ=





cosθ sinθ

−sinθ cosθ



, we may

denote theσk-isotypic componentV¡ σk

¢of a representation¡ π,V¢

simply byV¡ k¢

. The set of all integersksuch thatV¡

k¢

6=0 is called the set ofK⁰-types ofV. With notations as in [12], the irreducible admissible¡

g,K¢

modules forGare then classified based on the K⁰-types and the action ofZ and∆.

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4 CHAPTER 2. COHOMOLOGICAL REPRESENTATIONS OF ARCHIMEDEANGL(2) 1. The finite dimensional representations π, obtained by twisting the symmetric

powers of the standard representation ofGL¡ 2,R¢

onC², by the characters of the formχ◦det, whereχis a character ofR^×.

π=Sym^h−2³ C²´

⊗ µ

det¹⁺^a−h² sgn(det)^²

¶

, wherea,h∈Z,h≥2, a≡h (mod 2) and

²∈© 0, 1ª

. A representation of this type will be denoted (r[d],R[d]), withd=(h,a,²) as above. Ford as above, we letd⁰=(h,a,²⁰) where²⁰=²−1 and ˇd=(h,−a,²). The representationr[ ˇd] is in fact isomorphic to the dualr[d]^∨of the representationr[d].

The representationr[d] is ofK⁰-typeΣ⁰¡ h¢

=©

l∈Z|l≡h (mod 2),−h<l<hª . Z acts by the scalara, while the Casimir element∆acts by the scalar ^h₂³

1−^h₂´ . 2. Letχ1,χ2be characters ofR^×defined asχi

¡y¢

=sgn¡ y¢_²_i¯

¯y¯

¯

si

fori =1, 2. Letχbe the character of the Borel subgroupB(R) defined as

χ



 y1 x

0 y2



=χ1

¡y₁¢ χ2

¡y₂¢

. (2.1.2)

Ifχ1χ⁻₂¹6=sgn^²|·|^k⁻¹where²∈© 0, 1ª

andk is an integer of the same parity as², then theK-finite vectors of the induced representation Ind^G_B(_R₎¡

χ1,χ2

¢ofGform an irreducible admissible¡

g,K¢

-module denotedπ¡ χ1,χ2

¢. If²=²1+²2 (mod 2),µ= s₁+s₂ andλ= ¹₄¡

s₁−s₂+1¢ ¡

s₂−s₁+1¢

, then the set ofK⁰-types of π¡ χ1,χ2

¢ is

©k∈Z|k≡² (mod 2)ª

, Z acts byµand∆byλ. These are called principal series representations and are denotedP_µ¡

λ,²¢

. Ifµ=0, they are denoted simply as P¡

λ,²¢ .

3. Letχ1,χ2be characters ofR^×defined as χ1(y)=y^a⁺²^h⁻¹¯

¯y¯

¯

1

2sgn(y)^², χ2(y)=y^a⁻²^h⁺¹¯

¯y¯

¯

−1

2 sgn(y)^².

(2.1.3)

forh ≥1,a ≡h (mod 2) and ²∈© 0, 1ª

. Then, χ1χ⁻¹₂ =sgn^²⁰|·|^h−1 where ²⁰ ≡h (mod 2). Letχbe defined as in Equation 2.1.2. LetHdenote the¡

g,K¢

-module of K-finite vectors in Ind^G_B(R)¡

χ1,χ2

¢, thenHhas an irreducible invariant subspace H₀with set ofK⁰-types equal toΣ^±¡

h¢

=©

l∈Z|l≡h (mod 2),l≥horl≤ −hª . We see thatZ acts on this space byaand∆byλ=^h₂³

1−^h₂´

. These are called discrete series representations ifh≥2 and limits of discrete series ifh=1 and are denoted

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2.1. CLASSIFICATION OF IRREDUCIBLE ADMISSIBLE¡ G,K¢

MODULES 5

π[d] whered=(h,a,²). It is to be noted thatπ[d]'π[d⁰]. Further, the quotient H/H₀is an irreducible admissible¡

g,K¢

-module isomorphic tor[d], i.e., we have an exact sequence of (g,K) modules:

0−−−−→H₀_{−−−−→}H_{−−−−→}r[d]−−−−→0. (2.1.4)

Now, letG=GL¡ 2,C¢

. Then,Gis the product of its center denotedZ¡ C¢

'C^×and SL₂¡

C¢

. This product is not direct and the two subgroups intersect in {±I}, which is the center ofSL₂¡

C¢ .

Supposeπis an irreducible admissible representation ofSL₂¡ C¢

, and letεbe the character through which {±I} acts. Letχbe an extension of²to a quasi-character of Z¡

C¢

. Then there is a unique extension ofπto a representation ofGL₂¡ C¢

, on which Z¡

C¢

acts via the characterχ. Conversely, given an irreducible admissible representation ofGL¡

2,C¢

with central characterχ, we can consider the restriction toSL¡ 2,C¢

, which is also irreducible. Further, the original representation ofGL¡

2,C¢

is got back from this representation ofSL¡

2,C¢

, by such a construction using the characterχ. Hence, it is enough to classify irreducible admissible representations ofSL¡

2,C¢ . LetK denote the maximal compact subgroupU¡

2,C¢

of unitary matrices inGandg denote the real Lie algebrag l¡

2,C¢

ofG. Let us denote the complexification ofgbyg_C. We may identifyg_Cwithg^Lgby the mapping

X ⊗1+Y⊗i 7−→ (X+i Y)⊕( ¯X+iY¯).

This also identifies the universal enveloping algebraU(g_C) withU(g)N

U(g). Let∆andZ be elements in the center ofU(g l(2,C)) defined above. Let∆1=∆⊗1,∆2=1⊗∆,Z₁=Z⊗1 andZ₂=1⊗Z be elements ofU(g)N

U(g)'U(g_C). Then, they are in fact elements in the center ofU(g_C). A representation ofG is said to be admissible if its restriction to the groupSU(2,C) of unitary matrices of determinant 1, breaks up into a direct sum of irreducible representations ofSU(2,C) each occuring with finite multiplicity.

Letn≥0. LetVnbe the vector space of complex homogeneous polynomials in two variables of degreen. ConsiderSU(2,C) acting on this space as follows.

ρ

 a b

 

 z₁

 

 a b

₋₁ z₁



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6 CHAPTER 2. COHOMOLOGICAL REPRESENTATIONS OF ARCHIMEDEANGL(2)

where



 a b c d



∈SU(2,C). Then, (ρn,Vn) is an irreducible representation ofSU(2,C) of dimensionn+1. In fact, any irreducible representation ofSU(2,C) is equivalent toρnfor somen≥0.

We classify all irreducible admissible representations of GL(2,C) using the above classification of irreducible representations ofSU(2,C). Let

µi(z)=z^sⁱ⁺¹²^(aⁱ⁻^bⁱ⁾z¯^sⁱ⁻¹²^(aⁱ⁻^bⁱ⁾. (2.1.6) be quasi-characters ofC^×, wheresi ∈C, ai andbi are non-negative integers witha₁≥ a2, b1≥b2and aibi =0 fori =1, 2. Letµ(z)=µ1µ⁻₂¹(z)=z^s⁺¹²^(a⁻^b)z¯^s⁻¹²^(a⁻^b) withs = s₁−s₂,a=a₁−a₂,b=b₁−b₂such thatab=0. This implies that eithera₁=a₂=0 or b₁=b₂=0. The quasi-charactersµ1andµ2together define a quasi-character of the Borel subgroupB(C) of upper triangular matrices.



 α β

0 γ



 7−→ µ1(α)µ2(γ).

Let us denote the induced representation ofGobtained from the above character by B(µ1,µ2). The action of elements∆1,∆2,Z₁andZ₂ofZ(U(g_C)) onB(µ1,µ2) is given by scalars as described below.

ρ(∆1)≡1 2



 Ã

s+a−b 2

!2

−1



; ρ(Z1)≡s₁+s₂+1

2(a₁−b₁+a₂−b₂).

ρ(∆2)≡1 2



 Ã

s+b−a 2

!2

−1



; ρ(Z2)≡s₁+s₂+1

2(b₁−a₁+b₂−a₂).

(2.1.7)

Further,B(µ1,µ2) is admissible and contains the representationsρnofSU(2,C) if and only ifn≥a+bandn≡a+b (mod 2). Moreover,ρnoccurs with multiplicity exactly 1 in B(µ1,µ2) in this case. We will denote the unique subspace ofB(µ1,µ2) that is isomorphic to the representationρnofSU(2,C) byB(µ1,µ2,ρn).

1. Ifµ(z) is not of the formz^pz¯^qwithp,q∈Zandp,q≥1 orp,q≤1, thenB(µ1,µ2) is irreducible as a representation ofG.

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2.2. RELATIVE LIE ALGEBRA COHOMOLOGY OR(G,K)-COHOMOLOGY 7 2. Ifµ(z)=z^pz¯^qwithp,q∈Z,p,q≥1, there is a unique proper invariant subspace

V = X

n≥p+q n≡p+q (mod 2)

B(µ1,µ2,ρn).

We will denote the representation ofGonV byσ(µ1,µ2) and the representation ofGon the quotient spaceB(µ1,µ2)/V byπ(µ1,µ2).σ(µ1,µ2) is in fact equivalent to the representationB(ν1,ν2) of 1, for charactersν1,ν2 such thatν1ν2=µ1µ2, ν1ν⁻₂¹(z)=z^pz¯⁻^q. On the other hand,π(µ1,µ2) is an irreducible admissible finite dimensional representation ofG.

3. Ifµ(z)=z^pz¯^qwithp,q∈Z,p,q≤ −1, there is a unique proper invariant subspace

V = X

|p−q|≤n<p+q n≡p+q (mod 2)

B(µ1,µ2,ρn).

We will denote the representation ofG onV byπ(µ1,µ2) and the representation ofGon the quotient spaceB(µ1,µ2)/V byσ(µ1,µ2). The representationsπ(µ1,µ2) andσ(µ1,µ2) are in fact isomorphic to the representationsπ(µ2,µ1) andσ(µ2,µ1) of 2.

Any irreducible admissible representation ofGL(2,C) is isomorphic to a representa- tionπ(µ1,µ2) for some charactersµ1,µ2ofC^×as in Equation 2.1.6. Further, the finite dimensional irreducible admissible representations given above may also be realised as

Sym^h¹⁻¹(C²)Sym^h²⁻¹(C²)⊗(det¹⁺

a1−h1 2 det¹⁺

a2−h2

2 ) (2.1.8)

for integersh_i,a_i such thath_i≥2,a_i≡h_i (mod 2). We will denote this representation by r[d] and its space byR[d], whered=(h₁,h₂,a₁,a₂).

2.2 Relative Lie algebra cohomology or ( g , K )-cohomology

Let us consider the reductive Lie groupG=GL(2,R) orGL(2,C). Letg₌g l(2,R) org l(2,C) denote its corresponding Lie algebra of all 2x2 matrices with real and respectively complex entries. LetK⁰=O(2,R) orU(2,C) be a maximal compact subgroup ofG, depending on

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8 CHAPTER 2. COHOMOLOGICAL REPRESENTATIONS OF ARCHIMEDEANGL(2) ofK. As a real vector space,kis two dimensional ifG=GL(2,R) and five dimensional if G=GL(2,C).

We know that the adjoint action ofK onggiven by

Ad(k)X =k X k⁻¹ ∀k∈K,X ∈g (2.2.1) leaves the sub-algebrakinvariant. Thus, it induces an action on the quotientg/k. We complexify this space to get a representation ofK on¡

g/k^¢_C₌g/k_⊗_C. For convenience, we will omit theCaltogether and just denote the space asg/kin the future. Note that, g/kis now a two or three dimensional complex Lie algebra, according to whether we are working withGL(2,R) orGL(2,C). We will work with the following basis of the complex vector spaceg/kforGL(2,R) andGL(2,C).

ForG=GL(2,R), we will work with the basis

z₁=i⊗X−1⊗Y, z₂=i⊗X+1⊗Y. (2.2.2)

where X =



 1

−1



andY =



 1 1



. The action ofSO(2,R) on g/k decomposes into a direct sum of charactersΘ2⊕Θ₋2with respect to this basis, where Θn denotes the characterk_θ7→e⁻^{i nθ}. In other words,k_θ·z₁=Ad(k_θ)z₁=e⁻^2iθz₁=Θ2(k_θ)z₁andk_θ·z₂= Ad(k_θ)z₂=e^2iθz₂=Θ₋2(k_θ)z₂. Further, if we let w =



 1

−1



∈O(2,R), thenw·z₁= Ad(w)z₁=z₂andw·z₂=Ad(w)z₂=z₁. Moreover, it is obvious that the adjoint action by elements ofZ(R) is trivial. Thus, we know explicitly the action ofK=O(2,R)Z(R) ong/k.

ForG=GL(2,C), we will work with the following basis ofg/k:

w₁=



 1

−1



, w₂=



 1 1



, w₃=



 i

−i



. (2.2.3)

In both cases, the action ofK ong/kinduces an action ofK onVqg/k. Let (π,V) be a (g,K) module. LetC^q(g,K;V)=HomK(Vqg/k,V), whereK acts onVqg/kas described

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2.2. RELATIVE LIE ALGEBRA COHOMOLOGY OR(G,K)-COHOMOLOGY 9 above. Let us define cochain mapsd:C^q(g,K;V)−→C^q+1(g,K;V) as

dη³

X0∧X1∧ · · · ∧Xq

´

=

q

X

i=0

(−1)ⁱXi·η³

X0∧ · · ·Xˆi· · · ∧Xq

´

+ X

0≤i<j≤q

(−1)ⁱ^+jη³

[X_i,X_j]∧X₀∧ · · ·Xˆ_i· · ·Xˆ_j· · · ∧X_q´ .

(2.2.4)

forη∈C^q(g,K;V), where ˆ over an argument means it should be excluded. It can be verified that the mapsdare well defined and further thatd²=0, i.e., the composite map C^q(g,K;V)−→C^q⁺¹(g,K;V)−→C^q⁺²(g,K;V) is zero for everyq. So, this indeed defines a cochain complex which we will denote byC^•(g,K;V).

Now, theq^{t h}relative Lie algebra cohomology or (g,K) cohomology groupH^q(g,K;V) is defined to be theq^{t h} cohomology group of this cochain complex,

H^q(g,K;V)=H^q¡

C^•(g,K;V)¢ .

Let us try to compute H⁰(g,K;V). By definition, it is the kernel of the map d : C⁰(g,K;V)−→C¹(g,K;V). So, letη∈C⁰(g,K;V) be such thatdη=0. So, for allX ∈g/k, dη(X)=0 i.e., X ·η(1)=0. So, η is in fact a (g,K) module homomorphism. Thus, H⁰(g,K;V)=Hom(g,K)(C,V). Forq≥0, we know thatH^q(g,K;V)=Ext^q₍_g_,K₎(C,V), where Cdenotes the trivial (g,K) module withgacting as zero andK acting as identity. This is shown in Chapter I, Section 2 of [4] by taking a special projective resolution ofC, which we sketch here.

LetV be aK module. Then we get an action ofkonV by differentiating. LetL=g_⊗_kV forq≥0 thinking ofgas akmodule by right translation. ThenLaffords a representation ofgby left translation on the first componentgofL. The action ofkonL obtained by restriction gives the usual tensor product representationg_⊗_kV withkacting ongby the adjoint action. This is true because

Y ·(X⊗v)=[Y,X]⊗v+X⊗Y ·v=Y.X⊗v−X.Y ⊗v+X⊗Y ·v=Y.X ⊗v forY ∈k,X ∈gandv∈V.

LetL_q=g_⊗_k^V^q(g/k). Then, by the above construction, we get thatL_qis a representation ofg, and hence a (g,K) module. It is actually known ([4], Chapter I, Section 2.4,

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10 CHAPTER 2. COHOMOLOGICAL REPRESENTATIONS OF ARCHIMEDEANGL(2)

∂q:L_q−→L_q₋₁by

∂q(X⊗X₁∧ · · · ∧Xq)=

q

X

i=1

(−1)ⁱ⁻¹Xi·X⊗X₁∧ · · ·Xˆi· · · ∧Xq

+ X

1≤i<j≤q

(−1)ⁱ^+jX⊗[X_i,X_j]∧X₁∧ · · ·Xˆ_i· · ·Xˆ_j· · · ∧X_q.

(2.2.5)

Moreover, let²:L₀=g_−→_Cbe the augmentation map. Then the chain complex

· · · −−−−→Lq

∂q

−−−−→Lq−1

∂q−1

−−−−→ · · ·−−−−→^∂¹ L₀−−−−→^² C−−−−→0

is exact and hence is the desired projective resolution ofC. Now, applying the functor Homg(_,V), we get the complex

Homg(C,V)−−−−→^² Homg(L₀,V)−−−−→ · · ·Hom^∂¹ g(Lq−1,V)−−−−→^∂^q Homg(Lq,V)−−−−→ · · · Now, observing that Homg(L_q,V)'Hom_k(Vq(g/k),V)'Hom_K(Vq(g/k),V) tells us that the last complex is exactly the complexC^•(g,K;V) together with the mapsd, that was used in defining the relative lie algebra cohomology. Thus, we get that

H^q(g,K;V)=Ext^q₍_g_,K₎(C,V). (2.2.6)

2.3 Cohomological Representations of GL(2, R ) and GL (2, C )

An irreducible admissible (g,K) moduleV ofGL(2,R) orGL(2,C) is said to be of cohomological type or a cohomological representation ifH^•(g,K;V⊗F)6=0 for some finite dimensional irreducible representationF ofGL(2,R) or respectivelyGL(2,C).

We would like to have an explicit classification of all irreducible cohomological representations ofGL(2,R) andGL(2,C). We will need a Lemma to help prove this classification result, which we state below. This is due to D. Wigner and is popularly known as Wigner’s Lemma.

Lemma 1.

Let U,V be two(g,K)modules. Let the center Z¡ U(g)¢

of the universal enveloping alge- bra U(g)act on U and V by the infinitesimal charactersχU andχV. IfχU 6=χV, then Ext^q₍_g_,K₎(U,V)=0for all q≥0.

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2.3. COHOMOLOGICAL REPRESENTATIONS OFGL(2,R)ANDGL(2,C) 11 Proof. The proof we give here is from Chapter 1, Section 4 of [4]. It is trivial to see that Ext⁰₍_g_,K₎(U,V)=Hom₍_g_,K₎(U,V)=0 if the infinitesimal charactersχU andχV are different.

We know that forq≥1, the group Ext₍^q_g_,K₎(U,V) is canonically isomorphic to the group of allqlength exact sequences with first termV and last termU upto an equivalence under the Baer sum. Since the infinitesimal characters are not equal, we may find an element z∈Z¡

U(g)¢

such thatχU(z)=0 andχV(z)=1. Now, for anyS∈Ext₍^q_g_,K₎(U,V) given by an exact sequence:

S≡ 0−→V −→X_q−→X_q−1−→ · · · −→X₁−→U −→0,

the elementzdefines an endomorphism ofSgiven by the action ofzon each of theX_i’s, U andV. So, the exact sequenceχV(z)·S, which is the pushout ofSby the mapV −→^z V is equivalent to the pullbackS·χU(z) ofSby the mapU−→^z U. Sincezwas chosen such thatχU(z)=0,χV(z)=1, we get that 1.S'S.0, which implies thatS'0. Thus, we have shown that Ext^q₍_g_,K₎(U,V)=0.

Letd=(h,a,²) ord=(h₁,a₁,h₂,a₂) and let (r[d],R[d]) denote the finite dimensional irreducible representation ofGL(2,R) orGL(2,C) as defined in Section 2.1. The following theorem determines all the irreducible admissible representations ofGL(2,R) and respectivelyGL(2,C) that are of cohomological type.

Theorem 1.

Let(π,V)be an irreducible(g,K)module for GL(2,R)or respectively GL(2,C)such that H^•(g,K;V⊗R[d]^∨)6=0. Then,π'r[d],r[d⁰]orπ[d]for GL(2,R)andπ'r[d]orπ[d]for GL(2,C).

Proof. We already know from Section 2.2 thatH^q(g,K;V⊗R[d]^∨)=Ext^q₍_g_,K₎(C,V⊗R[d]^∨).

SinceR[d] is finite dimensional,V⊗R[d]^∨'Hom¡

R[d],V¢

. Further, since Hom(X, _) is right adjoint to _⊗X, we get that

H^q(g,K;V⊗R[d]^∨)'Ext^q₍_g_,K₎¡

C, Hom(R[d],V)¢

'Ext^q₍_g_,K₎¡

C⊗R[d],V¢ 'Ext^q₍_g_,K₎¡

R[d],V¢

. (2.3.1)

Thus, by Wigner’s Lemma, we get that for this cohomology to be non-zero, we want the infinitesimal charactersχR[d]andχV to be equal, which tells us thatZ and the Casimir element∆act by the scalarsaand ^h₂³

1−^h₂´

respectively. This proves the theorem since

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12 CHAPTER 2. COHOMOLOGICAL REPRESENTATIONS OF ARCHIMEDEANGL(2) (g,K) modules forGL(2,R) and respectivelyGL(2,C) with the desired action ofZ¡

U(g)¢ .

It may be noted that the only infinite dimensional cohomological representations for GL(2,R) are the discrete series representations.

Next, we state a well known proposition (Proposition I.4, [12]) that describes the cohomology groupsH^q(g,K;V⊗R[d]^∨) for different (g,K) modules (π,V). We note by Theorem 1 thatH^q(g,K;V⊗R[d]^∨)=0 forπ6'r[d],r[d⁰] orπ[d] forGL(2,R) andπ6'r[d] orπ[d] forGL(2,C).

Let (π,V) be a (g,K) module forGL(2,R) orGL(2,C). In the following proposition, we will denoteCⁿ(g,K;V⊗R[d]^∨) simply byCⁿandHⁿ(g,K;V⊗R[d]^∨) byHⁿforn≥0, exactly as in [12].

Proposition 1.

1. Let G=GL(2,R).

(a) Ifπis not isomorphic to r[d], r[d⁰]orπ[d], then Hⁿ={0}for all n.

(b) Ifπ'r[d], H⁰'Cand Hⁿ={0}for all n6=0.

(c) Ifπ'r[d⁰], H²'Cand Hⁿ={0}for all n6=2.

(d) Ifπ'π[d], C¹=H¹'Cand Cⁿ=Hⁿ={0}for all n6=1.

2. Let G=GL(2,C).

(a) Ifπis not isomorphic to r[d]orπ[d], then Hⁿ={0}for all n.

(b) Ifπ'r[d], H⁰'H³'Cand Hⁿ={0}for all n6=0, 3.

(c) Ifπ'π[d], C¹=H¹'C, C²=H²'Cand Cⁿ=Hⁿ={0}for all n6=1, 2.

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Chapter 3 Cohomology of GL (2)

3.1 Basics

LetF be a number field. LetAdenote its ring of adeles,Af the ring of finite adeles,A^× the group of ideles. Further, for all placesνofF, letF_νdenote the completion ofF atν. LetS₁,S₂denote the set of all real and respectively complex places ofF. LetS=S₁∪S₂ denote the set of all archimedean places ofF and ¯Sdenote the set of all embeddings ofF inC. Let us fix a section fromSto ¯Sso thatνrepresents an embedding corresponding to the placeν. Let ¯νrepresent the conjugate of the embeddingνforν∈S2.

For any algebraic groupHdefined overF, letH_F =H(F),H_A=H(A),H_f =H(Af),H_ν= H(F_ν) for all placesνofF andH_∞=Y

ν∈S

H_ν. LetG=GL(2)/F,Z its center andB the subgroup of upper triangular matrices. Letg_ν denote the lie algebra ofG_ν for everyν∈S andg_∞₌^L_ν∈_Sg_ν. LetK_ν =O(2,R)Z(R) for a real placeνand K_ν =U(2,C)Z(C) for a complex placeν. LetK_∞=Y

ν∈S

K_ν. Letk_νdenote the Lie algebra ofK_νandk_∞₌^L_ν∈_Sk_ν. Let us denote the Hecke algebra ofGbyH, and letHF,HA,Hf,HνandH∞denote the respective Hecke algebras ofG_F,G_A,G_f,G_νandG_∞. Now, given any open compact subgroupK_f ⊆G_f, we attach a locally symmetric space

S^G_K

f =G(F)\G(A)/K_∞K_f. (3.1.1)

Letd=¡

(h_ν)_ν∈S¯, (a_ν)_ν∈S¯, (²ν)_ν∈S₁¢

0, 1ª

for allν∈S₁. Letd_ν=(h_ν,a_ν,²_ν) ifν∈S₁and (h_ν,a_ν,hν¯,aν¯) ifν∈S₂. Then for eachν∈S,r[d_ν] denotes an irreducible finite dimensional representation as in

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14 CHAPTER 3. COHOMOLOGY OFGL(2) Section 2.1. Consider the following representation ofG_∞, denoted byr[d].

r[d]=O

ν∈S

r[d_ν]. (3.1.2)

It is an irreducible finite dimensional representation ofG_∞since each of ther[d_ν] are irreducible and finite dimensional. We will denote the space of this representation byR[d]. The quantitydis said to be a dominant integral weight corresponding to the representation (r[d],R[d]). The contragredient (r[d]^∨,R[d]^∨) is isomorphic toN

ν∈Sr[ ˇd_ν].

We will work with the contragredient representation for convenience.

The representationr[d]^∨ofG_∞defines a sheafFd of vector spaces on the spaceS^G_K

f

by

Fd(U)= n

s:p⁻¹(U)−→R[d]^∨ ¯

¯ s(g x)=r[d]^∨(g)s(x) ∀g∈G_F,x∈G_Ao

, (3.1.3) wherep:G_A/K_∞K_f −−−−→S^G_K

f is the projection map. The objects of interest to us are the sheaf cohomology groupsH^•(S^G_K

f,Fd). The sheaf cohomology groupsH^•(S^G_K

f, _) are by definition, the right derived functors of the left exact functorH⁰(S^G_K

f, _). This is called the global sections functor and is defined as

H⁰(X, _)(G)=H⁰(X,G)=G(X)

for any topological spaceX and a sheafGonX. The cohomology ofS^G_K

f with coefficients in the sheafFd, denotedH^•(S^G_K

f,Fd) are calculated by taking any injective resolution, say 0−−−−→Fd ²

−−−−→I⁰−−−−→^d⁰ I¹−−−−→ · · ·^d¹ of the sheafFd, applying the global sections functor, dropping the first term of the resulting cochain complex and computing cohomology. We get the following cochain complex after applying the global sections functor to the injective resolution 0−−−−→Fd ²

−−−−→I^•mentioned above.

0−−−−→H⁰(S^G_K

f,Fd)−−−−→^² H⁰(S^G_K

f,I⁰)−−−−→ · · ·^d⁰ −−−−→^d^q−1 H⁰(S^G_K

f,I^q)−−−−→^d^q H⁰(S^G_K

f,I^q⁺¹)· · · After dropping the first term, we get

H⁰(S^G_K_f,I⁰)−−−−→^d⁰ H⁰(S^G_K_f,I¹)−−−−→ · · ·^d¹ −−−−→^d^q⁻¹ H⁰(S^G_K_f,I^q)−−−−→^d^q H⁰(S^G_K_f,I^q⁺¹)· · ·

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3.2. RELATION TO LIE ALGEBRA COHOMOLOGY 15 So, theq^{t h}cohomology group ofS^G_K

f with coefficients inFd is given by H^q(S^G_K

f,Fd)=H^q(H⁰(S^G_K

f,I^•))=kerd_q/ Imd_q−1.

Though the definition a priori involves a choice of an injective resolution ofFd, it can be shownH^q(S^G_K

f,Fd) is independent of this choice ([13], Chapter III, Theorem 1.1A ).

3.2 Relation to Lie algebra cohomology

It is a basic proposition in the theory of Sheaf Cohomology that we may use any acyclic resolution ofFd to compute the cohomology groups, instead of the injective resolution 0−−−−→Fd ²

−−−−→I^• used in the definition in previous section. In particular, we may use the de-Rham complex. It is the following resolution [9] of the constant sheafRM on a manifoldM, using the sheavesΩ^k_M of smoothk-forms, that assign to any open setU of M, the setΩ^k_M(U) of smoothk-forms onU.

0−−−−→RM

−−−−→i Ω⁰_M−−−−→^d⁰ Ω¹_M−−−−→ · · ·^d¹ (3.2.1) Tensoring the above resolution by a sheafGgives an acyclic resolution ofG. If we use such a resolution of the sheafFd defined in Equation 3.1.3, the resulting cochain complex, obtained after applying the global sections functor and discarding the first term, can be reinterpreted in terms of the complex computing relative Lie algebra cohomologies. This yields us the following isomorphism

H^•(S^G_K

f,Fd)'H^•(g_∞,K_∞;C^∞(G_F\G_A)^K^f ⊗R[d]^∨), (3.2.2) whereC^∞(G_F\G_A) represents the space of all smooth functions onG_Athat are left invariant underG_F andC^∞(G_F\G_A)^K^f represents the subspace of functions that are right invariant underK_f. The spaceC^∞(G_F\G_A) contains the spaceC_cusp^∞ (G_F\G_A) of all cusp forms onG_A. The inclusionC_cusp^∞ (G_F\G_A),−→C^∞(G_F\G_A) induces an injection in coho-

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16 CHAPTER 3. COHOMOLOGY OFGL(2) commutative diagram

H^•(S^G_K

f,Fd) ^' ^//H^•(g_∞,K_∞;C^∞(G_F\G_A)^K^f ⊗R[d]^∨)

H_cusp^• (S^G_K

f,Fd)

?OO

' //H^•(g_∞,K_∞;C_cusp^∞ (G^? _F\G_A)^K^f ⊗R[d]^∨)

OO

We know by multiplicity one theorem ([7], Chapter III, Theorem 3.3.6) for the action ofG_Aon the space of cusp formsC_cusp^∞ (G_F\G_A), that any cuspidal automorphic representation ofG_Aappears in the direct sum decomposition ofC_cusp^∞ (GF\G_A) with multiplicity one. Using this direct sum decomposition, we get that

H_cusp^• (S^G_K

f,Fd) ' M

π=N₀

νπ_ν

H^•(g_∞,K_∞;π∞⊗R[d]^∨)⊗πf, (3.2.3)

where the direct sum is over all irreducible cuspidal automorphic representationsπ, with π_∞andπf denoting the respective spaces of the representationsπ_∞=N

ν∈Sπ_νofG_∞ andπf =N₀

ν6∈Sπ_νofG_f. We say that an irreducible cuspidal automorphic representation πcontributes to the cuspidal cohomologyH^•(S^G_K

f,Fd) ifH^•(g_∞,K_∞;π_∞⊗R[d]^∨)6=0.

3.3 Inner Cohomology

LetX be a locally compact topological space andFbe a sheaf of abelian groups onX. Let U be an open subset ofX andsbe a section ofF overU i.e.,s∈F(U). We denote the set of all pointsx∈U such thats|x6=0 in the stalkFx by Supp(s)= |s|. It is clear that|s|is a closed subset ofU. We may work with the functorH_c⁰(X, _) of taking global sections with compact support, which is again left exact. Now, cohomology with compact supports is defined to be the right derived functors ofH_c⁰(X, _). ClearlyH_c⁰(X,G) is a subgroup of H⁰(X,G) for every sheafG onX. Thus, we get a map fromH_c^q(X,F)−→H^q(X,F) for everyq≥0. This map is in general not injective and the image of this map is called the inner cohomology and denotedH_!^q(X,F).

H_!^•(X,F) = Im¡

H_c^•(X,F)−−−−→H^•(X,F)¢

. (3.3.1)

We are particularly interested in the inner cohomology groupsH_!^•(S^G_K

f,Fd) which by

Cohomology of GL(2)