Automorphic representations and associated L-functions

Corso di Laurea in Matematica

Tesi di Laurea Magistrale

Automorphic representations

and associated L-functions

22 settembre 2017

Candidato

Giordano Pennetta

Relatore

Prof. Andrea Maffei

Introduction

The aim of this thesis is to introduce the theory of automorphic repre-sentations of the group of invertible 2 × 2 matrices with coefficients in the adele ring A of a global field k. Actually we will treat in detail the case when k is a function field, and we will briefly review the theory in the number field case in the last chapter.

This theory is particularly interesting because it relates number theory with representation theory: the first step in this direction was Tate’s Ph.D. thesis (Princeton, 1950), which now can be reinterpreted as the theory of automorphic forms on GL(1, A).

An automorphic form is a smooth function φ on GL(2, k)\GL(2, A) de-pending on a character ω of A×/k×such that φ is K-finite and of moderate growth: here K denotes the compact subgroup of GL(2, A) given by the productQ

vGL(2, Ov), as v runs through the set of the valuations on k and

Ov is the ring of integers of the completion of k with respect to v, kv; we

will define these terms in section 2.2.

If k is a function field, the group GL(2, A) acts on the space of auto-morphic forms, denoted by A(GL(2, k)\GL(2, A), ω), by right translation, namely g · φ(h) = φ(hg), g, h ∈ GL(2, A), φ ∈ A(GL(2, k)\GL(2, A), ω). An automorphic representation is an irreducible admissible representation (π, V ) of GL(2, A) such that V is a subspace of the space of automorphic forms. We will study in particular the properties of automorphic cuspidal represen-tations, which are automorphic representations on the subspace of cuspidal automorphic forms: φ ∈ A(GL(2, k)\GL(2, A), ω) is said to be cuspidal if

Z k\A φ 1 x 1  g  dx = 0.

In the thesis we will also study the irreducible admissible representa-tions of GL(2, kv): we will see that it is possible to factorize an irreducible

admissible representation π of GL(2, A) as product of representations πv of

the groups GL(2, kv), so many properties of GL(2, A)-representations will

follow from local properties.

One of the most important theorems we will prove in the thesis is the Multiplicity one Theorem: this asserts that if (π, V ) and (π0, V0) are two

automorphic cuspidal representations of GL(2, A) such that almost all local factors πvand π0v are isomorphic as GL(2, kv)-representations, then V = V0.

We can reformulate this theorem: every irreducible component of any admissible representation of GL(2, A) on the space of cuspidal automorphic forms occurs with multiplicity one.

A modular form of weight k and level N is a holomorphic function f on the Poincar´e upper half plane H such that

faz + b cz + d  = χN(d)(cz + d)kf (z), z ∈ H, a b c d  ∈ Γ₀(N ), where Γ0(N ) is the subgroup of SL(2, Z) consisting of matrices (a b_{c d}) such

that c ≡ 0 (N ) and χN is a Dirichlet character modulo N (we will define

these notions in the last section of chapter 3). It can be associated an function to every modular form, and under some assumption, these L-functions satisfy a functional equation.

In the thesis we will associate to a cuspidal automorphic representation of GL(2, A) an L-function which, in a way, generalizes the L-functions as-sociated to modular forms: these L-functions will be defined as product of local L-functions. The definition of the local factors requires the study of particular representations of GL(2, kv), the spherical representations, which

are representations containing a nonzero GL(2, Ov)-fixed vector. In

partic-ular we will prove that the L-function associated to a cuspidal automorphic representation satisfies a functional equation.

In our dissertation the notion of Whittaker model is fundamental: the Whittaker model of an irreducible admissible representation (π, V ) of GL(2, A) is a representation of GL(2, A) isomorphic to V on the space of smooth K-finite functions W on GL(2, A) such that W is of moderate growth and there exists a nontrivial character ψ of A/k such that

W 1 x 1  g  = ψ(x)W (g) for any x ∈ A, g ∈ GL(2, A).

We will prove that the Whittaker model, when it exists, it is unique up to isomorphism and that automorphic cuspidal representations have a Whittaker model: the existence and the uniqueness of this model will be crucial in the proof of Multiplicity one Theorem and in the theory of the L-functions.

In the first chapter we will introduce the notion of idempotented algebra and we will study the properties of the relative modules: we will see how if {Hi | i ∈ I} is a family of idempotented algebras and H is their restricted

tensor product, then every simple admissible H-module is the restricted tensor product of simple admissible Hv-modules.

INTRODUCTION iii This is because we will associate to the groups GL(2, kv) and GL(2, A)

an associative idempotented algebra, the Hecke algebra, and we will see that the category of smooth modules over these algebras is equivalent to the category of smooth representations of GL(2, kv) and GL(2, A).

In the second chapter we will define the Whittaker model of a repre-sentation (π, V ) of GL(2, A). Even in this case, we will define firstly the Whittaker model of a representation of GL(2, kv), and then we will use

uniqueness of the local model to prove the uniqueness in the global case. Furthermore, here we will define automorphic forms and automorphic representations (section 2.2), and we will prove the Multiplicity one Theorem and the functional equation for the associated L-function: the methods used to prove the functional equation are very similar to those used in Tate’s thesis to prove a functional equation for an L-function associated to a character of k×\A×.

As we said before, in the third chapter we will briefly describe the dif-ferences between the function field case and the number field case: we will be able to generalize all the results proved for a function field k, but we need to change the definition of representation of GL(2, A). In fact, when k is a number field, a representation of GL(2, A) is not a vector space V on which GL(2, A) acts, but it is a vector space with a structure of GL(2, Af

)-representation and a structure of (g∞, K∞)-module; we will define these

terms in section 3.1.

Finally, in section 3.3, we will illustrate shortly the connection between modular forms and cuspidal automorphic representations. If k = Q, there is an isomorphism between the space of cuspidal modular forms Sk(Γ0(N ), χN)

Introduction i

1 Tensor product theorem 2

1.1 Modules over idempotented algebras . . . 2 1.2 Representation of GL(2) over a p-adic field . . . 12

2 Whittaker model and L-functions 18

2.1 Existence and uniqueness of Whittaker models . . . 18 2.2 Automorphic forms and Multiplicity one theorem . . . 41 2.3 L-functions associated to automorphic representations . . . . 45

3 Number field case 55

3.1 Tensor product Theorem . . . 57 3.2 Whittaker model and Automorphic Forms . . . 58 3.3 Modular forms associated to automorphic representations . . 62

Bibliography 70

Chapter 1

Tensor product theorem

In this chapter we will define the notions of smooth and admissible representation for the groups GL(2, F ) and GL(2, A), where F is a non-archimedean local field and A is the adele ring of a function field k. Our aim is to describe the irreducible admissible representations of GL(2, A) in terms of local irreducible admissible representations: we will accomplish this with the Tensor product Theorem.

To do this, in the first part we will approach the problem from a more abstract point o view by introducing the idempotented algebras. In par-ticular we will see that if H is the restricted tensor product of a family of idempotented algebras {Hv | v ∈ Σ}, then the product of simple admissible

Hv-modules is a simple admissible H-module and conversely every simple

admissible H-module is of this form.

Then we will relate these concepts: we will prove that there is a 1-1 corre-spondence between smooth GL(2, F )-representations and smooth modules over an idempotented algebra, the Hecke algebra associated to GL(2, F ). This allows us to express the Tensor product Theorem for irreducible ad-missible representations of GL(2, A) in terms of product of simple adad-missible modules over idempotented algebras.

1.1

Modules over idempotented algebras

Let Ω be an algebraically closed field. An idempotented algebra is a pair (H,E ) where H is an Ω-algebra and E is a subset of idempotent elements such that:

• ∀ e1, e2 ∈ E there exists e0 in E such that e0ei = eie0 = ei;

• ∀ φ ∈ H, ∃ e ∈ E such that eφ = φe = φ.

If e is an idempotent element, we call H[e] := eHe (it is a ring with unit e since e · (ehe) = ehe = (ehe) · e because e2 = e), and given M an H-module

we write M [e] for the H[e]-module eM . A module M is said to be smooth if M =S

e∈EM [e], admissible if it is smooth and M [e] is finite dimensional

over Ω for all e ∈ E .

If we write e ≥ f when ef = f e = f , E is a directed set with respect the partial order ≥. A subset E◦ ⊂ E is said to be cof inal if for every e ∈ E there exists f ∈ E◦ such that f ≥ e.

Next propositions point out the relevance of idempotent elements: to study important properties of a module M over an idempotented algebra H like simplicity or H-modules isomorphism class it will be sufficient to focus our attention to the H[e]-module M [e].

Proposition 1.1.1. Let M be a nonzero smooth H-module. M is simple if and only if M [e] is simple or zero as H[e]-module ∀ e in a cofinal subset E◦. Proof. Let assume that M [e] is simple or zero as H[e]-module for every e ∈ E . If N ⊂ M is a non trivial smooth H-submodule, then since M and N are smooth there exist e1, e2 ∈ E such that M [e1] 6= 0 and N [e2] 6= 0. If

we take e ∈ E such that e > e1 and e > e2, then we find N [e] ⊂ M [e] a

proper H[e]-submodule, contradicting our hypothesis.

Assume now that M is simple, and let N ⊂ M [e] a non trivial H[e]-submodule. Let consider the H-module H · N ∩ M [e]. It is contained in N and now we show that the converse is true also: so H · N ∩ M [e] is an H-submodule of M (H · N ∩ M [e] ⊂ M [e] ⊂ M ), and this contradicts our assumption. We takePn

i=1hiwi an arbitrary element of H · N ∩ M [e].

We observe that if x ∈ M [e], then e · x = x because there exists a suitable element m in M such that x = e·m and e2= e. Now, since wi∈ M [e], ewi=

wi and e ·Pni=1hiwi =Pni=1hiwi⇒Pi=1n hiwi = ePni=1hiwi=Pni=1ehiwi

=Pn

i=1ehiewi∈ N because N is an H[e]-module and ehie ∈ H[e].

Theorem 1.1.2. (Burnside’s theorem)

Let R be an Ω-algebra with unit, M a simple R-module and a finite dimensional vector space on Ω. The homomorphism φ : R → EndΩ(M )

which maps r ∈ R 7→ φ(r)(m) = r · m is surjective. Moreover EndR(M )

is one dimensional and consists of the scalar endomorphisms m 7→ λm, λ ∈ Ω.

Proposition 1.1.3. Let M and N be two H-modules simple and admissible. Then M ∼= N ⇔ M [e] ∼= N [e] for all e in a cofinal subset E◦.

Proof. ⇒ is obvious.

For the converse, if e0 ∈ E then by our hypothesis there exists an

iso-morphism of H[e0]-modules σ0: M [e0] → N [e0].

If e1 ≥ e0 then e0he0 = e1e0he0e1 and e0 · m = e1e0· m, so H[e0] ⊂

CHAPTER 1. TENSOR PRODUCT THEOREM 4 H[e1]-modules M [e1] → N [e1] whose restriction to M [e0] is σ0; in fact by

hypothesis there exists a H[e1]-modules isomorphism σ1. Then

σ1(M [e0]) = σ1(e0M [e1]) = e0σ1(M [e1]) = e0N [e1] = N [e0]

where the first and the last equalities come from the fact that M [e0] = e0M

= e0e1M = e0M [e1] and similarly for N [e0]. So σ1 induces an isomorphim

M [e0] → N [e0] which is a constant multiple of σ0by the Burnside’s theorem,

and we can normalize it to have this constant equal to 1. M is a smooth module, so M =S

e∈EM [e] and for what we have just seen the isomorphism

σ(m) = σ0(m) if m ∈ M [e0] is well defined. It is an H-modules isomorphism:

if h · m ∈ M [e0] we take e1 ≥ h and e ≥ e0, e1 so

σ(hm) = σe(e1he1m) = e1he1σe(m) = hσ(m)

This concludes the proof of the Proposition.

Proposition 1.1.4. Let R be a ring, e, f be idempotent elements such that ef = f e = e and M an R-module. Then there exists an idempotent element e0 such that f = e + e0 and M [f ] = M [e] ⊕ M [e0]. If R is an Ω-algebra, M [e] is finite dimensional over Ω and simple as R[e]-module, then HomR[e](M [e],M [f ]) is one dimensional.

Proof. Let e0 = f − e; e02 = f2 − ef − f e + e2 _{= f − e − e + e = f − e}

= e0, thus e0 is idempotent. We observe that ee0 = ef − e2 = e − e = 0. If m ∈ M [f ] ⇒ m = f m = em + e0m ⇒ M [f ] = M [e] + M [e0]. To show that the sum is direct, let m be an element in M [e] ∩ M [e0]. Then m = em and m = e0m, so m = ee0m = 0.

Now suppose that M [e] is simple as R[e]-module. Since ee0 = 0, R[e] acts as zero on M [e0] therefore HomR[e](M [e], M [e0]) = 0 ⇒ HomR[e](M [e],M [f ])

∼

= HomR[e](M [e],M [e]) ∼= Ω by the Burnside’s theorem.

Now we study the tensor product of algebras an their modules: if A and B are algebras over an algebraically closed field Ω we may form the tensor product A ⊗ΩB, then we will see that simple modules over A ⊗ΩB are

tensor products M ⊗ N where M is a simple A-module and N is a simple B-module. We will apply this result to the tensor product of idempotented algebras.

Proposition 1.1.5. Let A and B be algebras with unit over the field Ω (not necessarily algebraically closed). Let R := A ⊗ B and let P be a simple R-module that is finite dimensional over Ω. Then there exists a simple A-module M and a simple B-A-module N such that P is isomorphic to a quotient of M ⊗ N , and moreover M and N are uniquely determined by P .

Proof. Since P is finite dimensional over Ω, every submodule of minimal dimension is a simple A-module because we can identify A and B with their images under the homomorphism a 7→ a ⊗ 1 and b 7→ 1 ⊗ b respectively, so P contains a simple A-module M . Let N1 := HomA(M , P ), and we make

N1 a B-module with the action b · · · n1(m) = b · · · n1(m).

The map φ : M ⊗ N1 → P , φ(m ⊗ n1) = n1(m) is an R-module

homo-morphism:

φ((a⊗b)(m⊗n1)) = φ(am⊗bn1) = bn1(am) = ba · · · n1(m) = (a⊗b)φ(m⊗n).

Since the inclusion map M → P is contained in N1, N1 is nonzero so it

contains a simple B-module N . It follows that P is isomorphic to a quotient of M ⊗ N because φ is nonzero on M ⊗ N and P is simple.

As an A-module, P is isomorphic to a finite number of copies of M , so the A-isomorphism class of M is uniquely determined by P and similarly the B-isomorphism class of N is uniquely determined by P .

Proposition 1.1.6. Let A and B be algebras with unit over the alge-braically closed field Ω and let R = A ⊗ B. If M and N are simple modules over A and B respectively and finite dimensional vector space over Ω, then M ⊗ N is simple as R-module and every simple R-module finite dimensional over Ω has this form for uniquely determined M and N (up to isomorphism). Proof. The homomorphisms φM : A → EndΩ(M ) and φN : B → EndΩ(N )

defined as in Burnside’s theorem are surjective, thus to show that M ⊗ N is simple as R-module it is sufficient to show that it is simple as EndΩ(M ) ⊗

EndΩ(N )-modules. The map EndΩ(M ) ⊗ EndΩ(N ) → EndΩ(M ⊗ N ) which

maps λ1 ⊗ λ2 7→ (λ1⊗ λ2)(m ⊗ n) = λ1(m) ⊗ λ2(n) is surjective, so it is

sufficient to show that M ⊗ N is simple as an EndΩ(M ⊗ N )-module, and

this is obvious.

To conclude we may use the Proposition 1.1.5.

Let (H1, E1), (H2, E2) be two idempotented algebras. Their tensor

prod-uct is the idempotented algebra (H, E ), where H = H1⊗ΩH2 and

E = {e1⊗ e2 | ei ∈ Ei}.

Theorem 1.1.7. Let (H1, E1) and (H2, E2) be idempotented algebras over

Ω, and let (H, E ) be their tensor product. If M1 and M2 are simple

admis-sible H1 and H2 modules respectively, then M1⊗ M2 is a simple admissible

H-module and every simple admissible H-module has this form, with M1

and M2 uniquely determined up to isomorphism.

This theorem clearly is true for the tensor product of a finite number of idempotented algebras.

CHAPTER 1. TENSOR PRODUCT THEOREM 6 Proof. Let e1 ∈ E1 and e2 ∈ E2. By Proposition 1.1.1 M1[e1] if nonzero is a

simple H1[e1]-module and M2[e2] if nonzero is a simple H2[e2]-module. By

Proposition 1.1.6, M1[e1] ⊗ M2[e2] is a simple H1[e1] ⊗ H2[e2]- module and

observing that M1[e1] ⊗ M2[e2] = (M1⊗ M2)[e1⊗ e2] and H1[e1] ⊗ H2[e2] =

(H1⊗H2)[e1⊗e2], then M1⊗M2is simple as H-module again by Proposition

1.1.1. It is admissible because for every idempotent e1⊗ e2, (M1⊗ M2)[e1⊗

e2] ∼= M1[e1] ⊗ M2[e2] and so it is a product of two finite-dimensional vector

spaces by admissibility of M1 and M2.

Let M be an admissible simple module over H = H1⊗ H2. There exist

e0₁ ∈ E1 and e02 ∈ E2 which we fix such that M [e01⊗ e02] 6= 0 by admissibility

of M . Let E_i0 = {ei ∈ Ei|ei ≥ e0i} for i = 1, 2. As in the proof of Proposition

1.1.3, (e0₁⊗ e0

2)M [e1⊗ e2] = M [e01⊗ e02] 6= 0 for every e1 ∈ E10, e2 ∈ E20, thus

by Proposition 1.1.1 M [e1⊗ e2] is a simple H[e1 ⊗ e2]-module and it is a

finite dimensional vector space on Ω by admissibility of M . Therefore there exist simple Hi[ei]-modules Mi(e1, e2) (i = 1, 2) such that

M [e1⊗ e2] ∼= M1(e1, e2) ⊗ M2(e1, e2).

We now proof that the H2[e2]-isomorphism class of M2(e1, e2) depends only

on e2 showing that if e1, f1 ∈ E1◦ then M2(e1, e2) ∼= M2(f1, e2). We can

suppose f1 ≥ e1 because we can take g1 ≥ f1, e1 and prove that M2(e1, e2)

and M (f1, e2) are both isomorphic to M2(g1, e2). By Proposition 1.1.4, f1

= e1 + e01 and M [f1⊗ e2] ∼= M [e1⊗ e2] ⊕ M [e01⊗ e2]. As H2[e2]-modules,

M [e1⊗ e2] and M [f1⊗ e2] are isomorphic to a finite direct sum of M2(e1, e2),

M2(f1, e2) respectively, and since the simple summand in the decomposition

of a module are uniquely determined up to module isomorphism, we have an H2[e2]-isomorphism M2(e1, e2) → M2(f1, e2). Clearly the same is true

for M1(e1, e2), therefore

M [e1⊗ e2] ∼= M1(e1) ⊗ M2(e2).

We now prove that if fi≥ ei ∈ Ei, i = 1, 2, then

dimΩHomHi[ei](M (ei), M (fi)) = 1.

The modules M [e1⊗ e2] and M [f1⊗ e2] are isomorphic to a finite number

of copies of M1(e1) and M1(f1) respectively as H1[e1]- and H1[f1]-modules;

moreover by Proposition 1.1.4, dimΩHomH[e1⊗e2](M [e1 ⊗ e2], M [f1 ⊗ e2])

= 1 hence in particular there exist at least one nonzero H1[e1]-modules

homomorphism M1(e1) → M1(f1) ⇒ dim HomH1[e1](M1(e1), M1(f1)) ≥ 1.

The map

HomH1[e1](M1(e1), M1(f1)) ⊗ HomH2[e2](M2(e2), M2(f2)) →

Hom_H[e1⊗e2](M1(e1) ⊗ M2(e2), M1(f1) ⊗ M2(f2)),

is an injective homomorphism; thus

dim HomH1[e1](M1(e1), M1(f1)) · dim HomH2[e2](M2(e2), M2(f2)) ≤ 1

and therefore

dim Hom_H1[e1](M1(e1), M1(f1)) = dim HomH2[e2](M2(e2), M2(f2)) = 1

Let e1 ≤ f1 ≤ g1 ∈ E1◦. We fix λ(e1◦, e1) ∈ HomH1[e1](M1(e

◦

1), M1(e1)) and

λ(e◦₁, f1) ∈ HomH1[e1](M1(e

◦

1), M1(f1)) and define λ(e1, f1) to be the unique

homomorphism in Hom(M1(e1), M1(f1)) such that

λ(e◦₁, f1) = λ(e1, f1) ◦ λ(e◦1, e1).

It is easy to see that

λ(e1, g1) = λ(f1, g1) ◦ λ(e1, f1);

in fact

λ(e◦₁, g1) = λ(f1, g1) ◦ λ(e◦1, f1) = λ(f1, g1) ◦ λ(e1, f1) ◦ λ(e◦1, e1).

Thus (M (e1), λ(e1, f1) | e1∈ E1◦) is a directed family of abelian groups and

homomorphisms, so we can take the direct limit M1 = lim_→ M (e1)

By definition of direct limit, there exist for each e1 ∈ E1◦ an homomorphism

λ(e1) : M1(e1) → M1 such that λ(e1) = λ(f1) ◦ λ(e1, f1) for all f1 ≥ e1.

We define on M1 a structure of H1-module as follows: let h ∈ H1, m1 ∈

M1. There exists an idempotent e such that e ≥ h, and by definition of

idempotented algebra there exist e ∈ E such that e ≥ e and e ≥ e◦₁. Thus h ∈ H1[e], because h = eh = he ⇒ h = ehe, so we can define h · m1 =

(h · m1(e1)|e1 ∈ E1◦), because h · m1(e1) is defined on the component relative

to M (e) and hence in all components relative to f1 ≥ e, and the maps

λ(e, f1) are H1[e]-homomorphisms.

The direct limit of abelian groups is isomorphic to the quotientL M (e_i)/N where N is the subgroup generated by all elements

xe1f1 = (0, . . . , x, 0, . . . , −λ(e1, f1)(x), 0, . . . ))

hence since λ(e1, f1) are all injective the maps λ(e1) are injective too, so we

can replace M1(e1) with the H1[e1]-module M1[e1] because these modules

are simple.

We can repeat this construction to find a simple H2-module M2 such

that M2[e2] ∼= M2(e2) for all e2 ∈ E2◦, therefore

CHAPTER 1. TENSOR PRODUCT THEOREM 8 and by Proposition 1.1.3 it follows that M ∼= M⊗M2 (it follows from the

proof of that Proposition that M ∼= N ⇔ M [e] ∼= N [e] for any e in a cofinal subset of E ) Finally, let e1 ∈ E1◦. Then M [e1⊗ e◦2 ∼= M1[e1] ⊗ M2[e◦2], so by

Proposition 1.1.5 M1[e1] is uniquely determined (up to isomorphism), thus

M1 is uniquely determined by Proposition 1.1.3.

In order to extend this theorem to the product of infinitely many idempo-tented algebras, we have to introduce a new class of elements: the spherical elements. In particular we will see that the H-modules isomorphism class of an H-module M is determined by the H[e]-modules isomorphism class of M [e] where e is a spherical element, while with Proposition 1.1.3 we have to check the isomorphism class of M [e] for every e in a cofinal subset of E .

So let (H, E ) be an idempotented algebra on the algebraically closed field Ω. An element e◦∈ E is said to be spherical if there exists an antiinvolution ι : H → H such that ι_{x = x ∀x ∈ H[e}◦_{]; with antiinvolution we mean a}

linear map of order 2 such that ι(xy) = ιyιx. If x, y ∈ H[e◦], then xy = ι(xy) = ιyιx = yx, therefore H[e◦] is a commutative ring. Given M a simple admissible H-module, M [e◦] is a simple H[e◦]-module and thus is one dimensional over Ω: if R is any commutative ring with unit and V is a simple R-module, then for every r ∈ R the map λr(v) = r · v is a R-module

homomorphism because

λr(sv) = rs · v = sr · v = sλr(v);

hence by Schur’s Lemma it is a constant multiple of identity and so every subspace W ⊂ V is an R-submodule, and since V is simple it must be one dimensional.

We introduce the notion of smooth functional: let e◦ a spherical idem-potent algebra (H, E ), let ι be the corresponding antiinvolution and M a smooth H-module; a linear functional λ : M → Ω is smooth if there exists e ∈ E such that λ(m) = λ(em) ∀m ∈ M . We denote with cM the space of these functional. We can give to cM the structure of an H-module by

φλ(m) = λ(ιφm) φ ∈ H, m ∈ M, λ ∈ cM This define a structure of H-module: in fact

φ1φ2λ(m) = λ(ι(φ1φ2)m) = λ(ιφι2φ1m)

φ1(φ2λ)(m) = φ2λ(ιφ1m) = λ(ιφι2φ1m)

while

(φ1+ φ2)λ(m) = λ(ι(φ1+ φ2)m) = λ(ιφ1m +ιφ2m) =

by linearity of ι and λ for every φ1, φ2 ∈ H, m ∈ M . To show that it is a

smooth module, we first observe that if e ∈ E then (ιe)2 = ιeιe = ι(e2) =

ι_{e ⇒} ι_{e ∈ E .}

Since λ ∈ cM is smooth, there exists e ∈ E such that λ(em) = λ(m) ⇒

ι_{eλ(m) = λ(}ι₍ι_{e)m) = λ(em) = λ(m) ∀m ∈ M : we have proved that for}

every λ ∈ cM there exists an idempotent element such that λ is fixed by the action of this element, so cM = S

e∈EM [e].c

If M is admissible, cM is admissible too; in fact if {m1, . . . , mn} is an

Ω-basis of M [e], then the functional λi(mj) = δji is a basis for cM [e]: it is

clear that this forms a basis and since ∀m ∈ M [e] we have em = m, it follows that λi(em) = λi(m) so λi is smooth.

Let M be a simple module. If e is spherical then M [e] is one dimensional; if M [e] is nonzero, then cM [e] is nonzero: if m◦ ∈ M [e◦] we call ˆm◦ the

functional in cM [e◦] given by

em = ˆm◦(m)m◦ (1.1)

for all m ∈ M .

Proposition 1.1.8. Let R be an algebra over the field k, and let M1, M2 be

two simple R-modules which are finite dimensional as k-vector spaces. By a matrix coef f icient of a module M we mean a function c : R → k such that c(r) = L(r · m), where L: M → k is a linear functional. If M1 and M1 have

a matrix coefficient in common, then they are isomorphic as R-modules. Proof. By our hypothesis there exist Li: Mi → k and xi ∈ Mi such that

c(r) = Li(r · xi) where c is the common matrix coefficient. Let u ∈ R such

that c(u) 6= 0.

If M1 and M2 are not isomorphic, considering the R-module M = M1⊕

M2 we can find an element e ∈ R which acts as the identity on M1 and

as zero on M2: in fact the projections f1 : M → M1 commute with every

R-module endomorphism of M so we can apply the density Theorem, which ensures the existence of e ∈ R such that e · x = f1(x) for all x ∈ M . Then

we have an absurd, because

c(ue) = L1(ue · x1) = L1(u · x1) = c(u) 6= 0

= L2(ue · x2) = L2(0) = 0,

therefore M1 and M2 must be isomorphic.

We often will write hm, λi instead of λ(m).

Theorem 1.1.9. Let (H, E ) be an idempotented algebra over the field Ω, and let e◦ be a spherical element. Let M and N be simple admissible H-modules such that M [e◦] 6= 0 and N [e◦] 6= 0. If M [e◦] ∼= N [e◦] as H[e◦ ]-modules, then M ∼= N as H-modules.

CHAPTER 1. TENSOR PRODUCT THEOREM 10 Proof. Let ˆm ∈ cM [e◦] and ˆn ∈ bN [e◦] defined as in eq. (1.1). It is sufficient to show that

hφm◦, ˆmi = hφn◦, ˆni ∀φ ∈ H.

In fact if this happens then c(φ) = ˆm(φm) defines a common matrix coeffi-cient between M and N and so we conclude by Proposition 1.1.8.

Let φ0 = e◦φe◦. Since e◦m◦ = m◦ and e◦m = ˆˆ m,

hφm◦, ˆmi = hφe◦m◦, e◦mi = hφˆ 0m◦, ˆmi

where the last equality follows from the fact thatι_e◦ _{= e}◦_.

Now being M [e◦] ∼= N [e◦] as H[e◦]-modules, by definitions of ˆm and ˆn we have the thesis.

We now give the definition of restricted tensor product.

Let Σ be a set of indices and for every v ∈ Σ, let Vv be a vector space.

We fix for almost all v ∈ V a nonzero vector x◦_v ∈ Vv. Let O be the set

consisting of the finite subsets S ⊂ Σ such that if v /∈ S, x◦_v is defined. Then O is a directed set with respect the inclusion. We put ∀ S ⊂ S0 _{∈ O}

λS,S0 : O v∈S Vv → O v0_∈S0 Vv0 x 7→ x ⊗v∈S0_−Sx◦_v.

The restricted tensor product of the vector spaces Vvwith respect the vectors

x◦_v is the direct limit O v Vv := lim_→( O v∈S Vv, λS,S0|S ∈ O)

In practice the elements ofN

vVv are finite linear combination of vectors of

the form ⊗xv where xv = x◦v for almost all v.

If (Hv,Ev) (v ∈ Σ) is a family of idempotented algebras and we specify

an element e◦_v ∈ Ev for almost all v ∈ Σ, then the restricted tensor product

H of the Hv with respect to the ev is itself an idempotented algebra with

set of idempotents E consisting of tensors ⊗vev with ev = e◦v for almost all

v. Let Mv be a simple admissible Hv-module, and let assume that Hv[e◦v]

is commutative for almost all v. Then M [e◦_v] is one dimensional over Ω if it is not zero. Thus if we assume that M [e◦_v] is one dimensional for almost all v, we fix elements m◦_v and we may form the restricted tensor product M = L Mv with respect the m◦v: this is clearly a module over H with

componentwise multiplication.

Proposition 1.1.10. Let Rv a family of rings with unit ev indexed by a set

γ : R → Ω be a ring of homomorphism. Then there exists for each v ∈ Σ a ring homomorphism γv : Rv → Ω such that

γ(⊗vrv) =

Y

v

γv(rv).

We assume that a ring homomorphism between two rings maps the unit of the first one in the unit of the second one, so almost all factors in the product are one and thus it is well defined.

Proof. We have a ring homomorphism iv : Rv → R which maps xv 7→

xv⊗w6=vew. Then it is sufficient to put γv = γ ◦ iv and to observe that r =

⊗_vrv =

Q

viv(rv): clearly γ(r) =

Q

vγv(rv).

Theorem 1.1.11. Let (Hv,Ev) (v ∈ Σ) be an indexed family of

idempo-tented algebras over Ω, and for almost all v let e◦_v ∈ E_v be a spherical idempotent. Let (H,E ) be the restricted tensor product of the Hv with

re-spect to the e◦_v. Let Mv be a simple admissible Hv-module for each v ∈ Σ

and for almost all v let m◦_v be a specified nonzero element in Mv[e◦v]. Let

M be the restricted tensor product of the Mv with respect to the m◦v. Then

M is a simple admissible H-module. Conversely every simple admissible H-module is of this form, with uniquely determined modules Mv.

Proof. We observe that if e = ⊗ev ∈ E is a given idempotent, then there

exists a finite subset S ⊂ Σ such that ev = e◦v if v ∈ Σ − S and hence such

that H[ev] is commutative, so M [ev] is one dimensional for v /∈ S. Since

N

v /∈SMv[ev] is spanned by ⊗v /∈Se◦v, the tensor product with this vector gives

us an isomorphism O v∈S Mv[ev] → O v∈Σ Mv[ev] = M [e]

and since the finite product (if nonzero) is simple by Proposition 1.1.1 and Theorem 1.1.7, we obtain that M is simple again by Proposition 1.1.1.

It is obvious that M is admissible because M [e] is isomorphic to a finite tensor product of finite dimensional spaces.

Now let M be a simple admissible H-module. We make further assump-tions: we assume that e◦_v is spherical idempotent ∀ v and that if e := ⊗ve◦v,

then M [e] 6= 0. Then since e is spherical M [e] is one dimensional; let m be a generator and consider the ring homomorfism γ : H[e] → Ω such that h · m = γ(h)m, h ∈ H[e] (M [e] is one dimensional and an H[e]-module so if h ∈ H[e] then h m is a multiple of m). Since H[e] = N

vHv[e ◦

v], by

Propo-sition 1.1.10 we can factor γ as γ(⊗vhv) = Qvγv(hv) where γv : H[e◦v] → Ω

is a ring homomorphism.

We may decompose H = Hv ⊗ Hv0 where Hv0 =

N

w6=vHw. Thus by

CHAPTER 1. TENSOR PRODUCT THEOREM 12 and H_v0 respectively such that M [e] ∼= Mv[e◦v] ⊗ Mv0[e0v], where e0v = ⊗w6=ee◦w.

As in the proof of Proposition 1.1.10 we may see h ∈ H[e] as Q

viv(hv)

and therefore (N

vMv)[e] (tensor product restricted to nonzero elements in

Mv[e◦v]) =

N

vMv[ev] ∼= M [e] as H[e]-modules because hvmv= γv(hv)mv. It

follows by Theorem 1.1.8 thatN

vMv ∼= M as H-modules. Now we conclude

by composing this special case with Theorem 1.1.7: it is enough to write H as the finite tensor product N

v∈SHv⊗ H0, where S ⊂ Σ is a finite subset

of indices with the property that if v /∈ S then e◦_v is spherical idempotent, and H0 =N

v /∈SHv.

1.2

Representation of GL(2) over a p-adic field

Let F be a non-archimedean local field, that is the completion of a global field with respect to a non-archimedean absolute value | |v.

Let O := {a ∈ F ||a|v ≤ 1} the ring of integers of F : it is a discrete

valu-ation ring (DVR) with maximal ideal p = (ωv), and every ideal is generated

by a power of ωv. We denote by ordv: F → Z∪{∞} the valuation defined by

ordv(0) = ∞ and ordv(ωrvu) = r if u ∈ O×, the group of invertible elements

in the ring of integers; further, we normilize the absolute value | |v such that

|x|_v = q−ordv(x)

v , where qv denotes the cardinality of the residue field

O/p.

Theorem 1.2.1. Let F be a non-archimedean local field, O its ring of integers. Then O is an open subgroup of the additive group F , and it is a maximal compact subring of F . Furthermore F is a locally compact totally disconnected non-discrete field.

Proof. The topology on F is induced by the metric | |, so the open balls B(x, r) = {y ∈ F ||x − y|v < r} are a basis of open sets and a subset U ⊂ F

is a open neighborhood of x if it contains an open ball for a small enough r. F in non-discrete: it is enough to show that every neighborhood of 0 intersects F \{0}, and this is clear since {a ∈ F ||a|v ≤ r} always contains a

sufficiently large power of ω.

Next, we observe that p is an open subgroup in F and it has finite index in O, so O can be recovered by a finite number of open cosets and hence it is open.

Thus O is closed and so it is compact in F , because F is complete and O is limited with respect to | |v (so O is relatively compact).

If a subring K ⊂ F properly contains O, then there exists a ∈ K such that |a|v > 1 and thus, since it contains every power of a, it cannot be

compact, so O is maximal as compact subring.

Finally F is locally compact because O is a compact neighborhood of zero, so by translation every a ∈ F has a compact neighborhood.

Let H be a group. Then H acts on itself by left and right translations, namely

λx(a) = xa

ρx(a) = ax−1

A lef t (resp. right) Haar measure µ on H is a nonzero Borel measure invariant by left (resp. right) translation by elements of H, i.e. such that µ(λaM ) = µ(M ) (resp. µ(ρaM ) = µ(M )) for all borelian subsets M ⊂ H.

If µ is a left Haar measure on H, then cµ with c ∈ R≥0 is again a left Haar

measure, so Haar measures cannot be unique; furthermore if H is abelian then µ(M ) = µ(aM ) = µ(M a), hence µ is also a right invariant measure, so left and right Haar mesures coincide if the group is abelian. Finally, if H admits a left Haar measure µ, then it admits a right Haar measure: if we define ˆµ(M ) = µ(M−1), then ˆµ(M x−1) = µ(xM−1) = µ(M−1) = ˆµ(M ), hence ˆµ is a right Haar measure. It can be shown that a locally compact group admits a left (and so also a right) Haar measure.

By Theorem 1.2.1. the local non-archimedean local field F is a locally compact additive group so it admits a left Haar measure, which is a right Haar measure too because F is abelian. We denote with dx this measure, and we normalize it in order to give O volume one.

Let G := GL(2, F ). We consider on the space of 2×2 matrix with entries in the field F , Mat2(F ), the product topology, and since G is an open subset

we give it the subspace topology. Thus G is a locally compact group, and K := GL(2, O) is a compact subgroup. If dg =Q

i,jdgi,j denotes the product

measure on F4, then |det(g)|−2dg is both left and right invariant measure on G (we denote again with dg the restriction to G of the product measure on Mat2(F )), so G is unimodular.

A representation of G is a pair (π, V ) where V is a complex vector space and π : G → End_C(V ) is an homomorphism. The representation π is said to be smooth if for any v ∈ V the stabilizer {g ∈ G|π(g)v = v} is an open subgroup of G; if furthermore for any open subgroup U ⊂ G the space of U -fixed vectors {v ∈ v|π(u)v = v ∀ u ∈ U } is finite dimensional. We call a nonzero vector in VK spherical vector, an the representation itself is called spherical if it contains a nonzero spherical vector.

Let HGbe the space of the smooth function on G with compact support:

here smooth simply means locally constant. We make HGan algebra without

unit with the convolution

(φ1∗ φ2)(g) =

Z

G

φ1(gh−1)φ2(h)dh (1.2)

where dh is the normalized Haar measure on G such that K has volume one. This algebra is called the Hecke algebra of G. The group G acts on HGby

CHAPTER 1. TENSOR PRODUCT THEOREM 14 left and right translations, respectively

(λ(h)f )(g) = f (h−1g) (ρ(h)f )(g) = f (gh)

If K0 is any open compact subgroup, let HK0 the subspace of the K0

-biinvariant functions: clearly this space is invariant under convolution, and it has an identity element

εK0(g) = ( vol(K0)−1 g ∈ K0 0 otherwise (1.3) In fact φ ∗ εK0(g) = Z G φ(gh−1)εK0(h)dh = vol(K0) −1Z K0 φ(g)dh = φ(g) since φ is K0-invariant. When K0 = K, we call HK the spherical Hecke

algebra.

Theorem 1.2.2. (Elementary divisor theorem)

Let R be a principal ideal domain, let Λ1 be a free R-module of rank

n and let Λ2 be a free R-submodule of rank n. Then there exist a basis

{ξ₁, . . . , ξn} of Λ1 and nonzero elements D1, . . . , Dn in R such that Di+1|Di

and such that {D1ξ1, . . . , Dnξn} is a basis for Λ2.

Proposition 1.2.3. (p-adic Cartan decomposition)

A complete set of double coset representatives for K\G/K consists of diagonal matrices

ωn1

ωn2



where n1 ≥ n2 are integers.

Proof. Let g ∈ G. We take N ∈ Z such that ω−Ng ∈ K. We apply the elementary divisor theorem to the principal ideal domain O, with Λ1 = O2

and Λ2 the sublattice of Λ1 spanned by rows of ω−Ng.

By the elementary divisor theorem there exists a basis ξ1, ξ2 of Λ1 and

D1, D2 ∈ O such that D2|D1 and such that D1ξ1, D2ξ2 is a basis for Λ2. We

can assume that ωND1= ωn1 and ωND2 = ωn2 because (as all the elements

in O) they are powers of ω multiplied by a unit, and multiplying D1 and D2

by a unit does not change the properties which define them. The matrix ξ with rows ξ1 and ξ2 is then an element of K and the rows of diag(D1, D2)

span the same lattice as ω−Ng, so there exists k ∈ K such that kω−Ng = diag(D1, D2)ξ: it follows that ξ and diag(ωn1, ωn2) as required.

To show that distinct diagonal matrices diag(ωn1_{, ω}n2_{) generate disjoint}

double cosets, we observe that g lies in the double coset K\diag(ωn1_{, ω}n2_)/K

if and only if the fractional ideal generated by the coefficients of g is (ωn2

and the fractional ideal generated by det(g) is (ωn1+n2_{), so the integers n}

1

Theorem 1.2.4. The spherical Hecke algebra HK is commutative.

Proof. Matrix transposition is an antiinvolution on G which induces an an-tiinvolution on HG, namely ι_{φ(g) = φ(}T_{g) (1.4)} ι is an antiinvolution: ι_φ 2∗ιφ1 (g) = Z G ι_φ 2(gh−1)ιφ1(h)dh = Z G φ2((Th)−1 Tg)φ1(Th)dh = = Z G φ1(Tgh−1)φ2(h)dh = φ1∗ φ2(Tg) = =ι (φ1∗ φ2) (g)

where the third equality follows from the change of variables (Th)−1 Tg 7→ h since transposition map preserves the measure.

Since HK has a basis (over C) consisting of the characteristic functions

of the double cosets of K, by Cartan decomposition ι is the identity map on H_K, so HK is commutative.

Let (π,V ) a smooth representation of G. Then we can define an action of HG on V , denoted again with π, by

π(φ)v = Z

G

φ(g)π(g)vdg. (1.5)

This integral is a finite sum: in fact since π is smooth, v is fixed by an open subgroup H and the compact support of φ is a finite union of left cosets giH.

Thus we can choose H small enough so that φ is constant on this cosets (φ is locally constant) and replace the integral with the finite sum

vol(H)−1X

i

φ(gi)π(gi)v

Proposition 1.2.5. Let (π,V ) be a smooth representation of G. The fol-lowing conditions are equivalent:

1. π is irreducible;

2. V is simple as HG module;

3. VK0 _{if nonzero is simple as H}

K0-module for all K0 ⊂ G open compact

subgroup;

Proof. Clearly 2⇒1 because if V has a G-invariant subspace, then this sub-space is HG-invariant because of the equation (1.5).

Vice versa, let W ⊂ V an invariant subspace for the action of HG. If

CHAPTER 1. TENSOR PRODUCT THEOREM 16 such that π(g)w /∈ W . Since w is fixed by an open subgroup N , if φ is the characteristic function of gN divided by the volume of gN then

π(φ)w = Z G φ(h)π(h)w dh = 1 vol(gN ) Z gN π(h)w dh = = 1 vol(gN ) Z gN π(g)wdh = π(g)w /∈ W, and this is absurd because W is HG-invariant. Thus 1⇒2.

If W is an HG-submodule, then we can find K0 small enough such that

WK0 _{is a proper subspace of V}K0 _{and this is not possible by hypothesis,}

thus 3⇒2.

Finally, let W0be a proper HK0-submodule of V

K0_{. Let}Pn

i=1π(φi)wibe

a nonzero element in π(HG)W0∩ VK0. K0stabilizes every element of W0, so

π(εK0)(wi)=wi and π(εK0)(

Pn

i=1π(φi)wi)=

Pn

i=1π(φi)wi, so it follows that

Pn

i=1π(φi)wi=P π(εK0∗ φi∗ εK0)wi ∈ W0 since wi ∈ W0, εK0∗ φi∗ εK0 ∈

HK0 and W0 is an HK0-module. We proved that π(HG)W0∩ V

K0 _{= W}

0 is

a proper HG-submodule of V .

We observe that HG is an idempotented algebra: if K0 ⊂ G is an open

compact subgroup, then εK0 defined as in equation (1.3) is an idempotent

element: εK0 ∗ εK0(g) = Z G εK0(gh −1_)ε K0(h)dh = 1 vol(K0) Z K0 εK0(gh −1_{)dh =} = 1 vol(K0) Z K0h εK0(g)dh = εK0(g)

since dh is a right invariant measure. With similar calculation one can show that if K0 ⊂ K1are open compact subgroups, then εK0∗εK1 = εK1, and since

the open and compact subgroups K(ωn) = { 1+ω_ωnn_1+ωωnn
} form a basis of

neighborhoods of identity, it follows that HGis an idempotented algebra (it

is enough to choose n such that K(ωn) is contained in the support of φ ∈ HG

to have the second condition in the definition of idempotented algebra). Proposition 1.2.6. Let V be a smooth HG-module. Then there exists a

smooth representation π of G such that φ · x = π(φ)x (defined in equation (1.5)), for all φ ∈ HG, x ∈ V .

Proof. Let x ∈ V . Since V is smooth, there exists an open compact subgroup K0 such that x ∈ V [εK0], where εK0 is defined as in eq. (1.3). Let g ∈ G.

We define

π(g)x = εgK0 · x.

This definition does not depend on K0: let K1 be another open compact

subgroup such that x ∈ V [εK1], and let K2 = K0∩ K1. Then εgK2 ∗ εKi =

εgki for i = 0, 1, so we have

Now we have to show that it is a representation of G: let g, h ∈ G and K, H subgroups which stabilize x and π(h)x respectively. If we choose H so small that h−1Hh ⊂ K, then εgH∗ εhK = εghK, so it results that

π(g)π(h)x = εgH(εhK)x = εgH∗ εhKx = εghK = π(gh)x,

so π is a G-representation. Smoothness follows directly from the fact that V is smooth.

Let k be a function field and for every place v let kv be the completion

of k with respect to the absolute value | |v induced by v and Ov the ring

of integers (remember that every place in k is non archimedean). The adele ring of k, wich we denote with A, is the restricted direct product of the kv

with respect to Ov: an adele is an element (av) ∈Q_vkv such that av ∈ Ov

for almost all v (with almost all we mean all but finitely many).

A base of open neighborhood in A is given by the subsets of the form Q

vXv where Xv is open in kv and Xv = Ov for almost all v. With the

normalizations on the local Haar measures as above we can define a product measure on A, namely dx =Q

vdxv.

The group GA:= GL(2, A) of the invertible matrices with coefficients in

A does not have the subspace topology: it can be thought as the restricted direct product of the groups Gv := GL(2, kv) with respect the maximal

compact subgroups Kv := GL(2, Ov). It follows from the unimodularity of

GL(2, kv) that GL(2, A) is unimodular.

We call KA=QvKv: by the Tychanoff theorem it is a compact subgroup

of GA.

A representation (π,V ) of GAis admissible if for every irreducible

finite-dimensional representation ρ of K the ρ-isotypic part V (ρ) of V is finite dimensional and moreover every vector in V is K-finite: this means that the vector space spanned by {π(k)v | k ∈ K} is finite dimensional for all v ∈ V . Theorem 1.2.7. (Tensor Product Theorem)

Let (π, V ) be an irreducible admissible representation of GA. Then there

exists for each place v an irreducible admissible representation (πv, Vv) of

Gv such that for all v, Vv is spherical and π is the restricted tensor product

of the πv.

Proof. Let Hv := HGL(2.kv), and let e

◦

v be the characteristic function of Kv;

we normalize the Haar measures so that Kv has volume one, then e◦v is

idempotent; e◦_v is spherical: in fact the transpose map of Gv induces an

antiinvolution on Hv which fixes Hv[e◦v] by Proposition 1.2.3.

We define the global Hecke algebra HGA to be the restricted tensor

prod-uct of the local Hecke algebras Hvwith respect to Hv[e◦v]. Since there is a 1-1

correspondence between smooth Hv-modules and smooth Gv-representation,

we can see an irreducible admissible representation of GAas a simple

Chapter 2

Whittaker model and

L-functions

In this chapter we will introduce the main object of the thesis, the auto-morphic representations. These are some admissible representations of the group of 2 × 2 matrices with coefficients in the adele ring A of a function field.

In the first part of the chapter we will introduce the Whittaker model of a GL(2, A)-representation (π, V ), that is a space of functions on GL(2, A) such that the right translation by elements in GL(2, A) gives a representation isomorphic to π.

We will prove that Whittaker model is unique up to isomorphism, and we will see how the uniqueness will lead us to the Multiplicity one Theorem: this theorem asserts that two automorphic cuspidal representations of GL(2, A) such that almost all local factors given by Tensor Product Theorem 1.2.7 are isomorphic actually are equals.

In the last part, we will use the existence of a Whittaker model for an automorphic cuspidal representation to determinate explicitly almost all fac-tors in the decomposition of these representations. Once we have done this, we will associate to automorphic cuspidal representations an L-function, and we will see that it satisfies a functional equation.

2.1

Existence and uniqueness of Whittaker models

Before starting with the definition of Whittaker functional and the rela-tive properties, we need some result concerning the representations of GL(2) over a p-adic field.

Let F be a non-archimedean local field, and let O, p = (ω) and q as in section 1.2.

If Γ is any compact totally disconnected group, we denote by bΓ the set of equivalence classes of finite-dimensional irreducible representations of Γ

whose kernel is open and hence of finite index in Γ. In particular, if Γ is finite, we know by the Maschke’s theorem that if (π, V ) is a representation on G then

V =M

ρ∈bΓ

V (ρ),

where V (ρ) is the ρ-isotypic part of V .

Proposition 2.1.1. Let (π, V ) be a smooth representation of GL(2, F ), and let K = GL(2, O). Then

V = M

ρ∈ bK

V (ρ).

The representation π is admissible if and only if V (ρ) is finite-dimensional for every ρ ∈ bK.

Proof. Let v ∈ V ; since π is smooth, there exists an open compact subgroup K0 of K such that π(g)v = v for every g ∈ K0. Then if Γ is the finite group

K/K0 we have v ∈ VK0 ₌M ρ∈bΓ V (ρ) ⊂ X ρ∈ bK V (ρ),

so V is the sum of the spaces V (ρ).

The sum is direct: if it is not, there is a relation P

ρ∈Scρvρ = 0, where

S is a finite subset of bK, vρ ∈ V (ρ) and the constants cρ are not all zero.

If K0 is the intersectionT_ρ∈Sker(ρ), then we contradict Maschke’s theorem

with Γ = K/K0, so the sum must be direct.

Finally, if π is admissible then V (ρ) is finite dimensional because V (ρ) ⊂ Vker(ρ) and this space is finite dimensional because ker(ρ) is open.

Vice versa, if π is not admissible, then there exists an open normal subgroup K0 of K such that VK0 is infinite dimensional. But VK0 is the

direct sum of the spaces V (ρ) with ρ ∈ \K/K0, and since this set is finite,

there must exist ρ such that V (ρ) is infinite dimensional.

An intertwining operator between two representations (π, V ), (σ, W ) of a group G is a linear map T : V → W such that T ◦ π(g) = σ(g) ◦ T for all g ∈ G.

Proposition 2.1.2. (Schur’s Lemma)

Let (π, V ) be an irreducible admissible representation of GL(2, F ). Let T : V → V be an intertwining operator for π. Then there exists a complex number λ such that T (v) = λv for all v ∈ V .

Proof. Let K0 be an open compact subgroup such that VK0 is nonzero.

CHAPTER 2. WHITTAKER MODEL AND L-FUNCTIONS 20 because it commutes with π. Then T an eigenvalue λ on VK0_{, and the}

kernel of T − λI is a nonzero GL(2, F )-invariant subspace, so it must be all V because π is irreducible.

It follows that the center Z(F ) of GL(2, F ) acts by scalars on (π, V ), so there exists a quasicharacter (a non unitary character) ω of F× such that

πz z



v = ω(z)v, z ∈ F×, v ∈ V. (2.1)

Let (π, V ) be an admissible representation of GL(2, F ). By distribution we mean a linear functional on the Hecke algebra HGL(2,F ). Let denote

by D(GL(2, F )) the space of distributions. Since φ ∈ HGL(2,F ) is locally

constant and compactly supported, there exists an open compact subgroup K0 such that φ ∈ HK0. By eq. (1.5) the image of π(φ) is contained in the

K0-fixed vectors, and this is a finite dimensional vector space so π(φ) has

finite rank: we define the trace of π(φ) to be the trace of the endomorphism π(φ)|U of U , where U is any finite-dimensional subspace of V containing the

image of φ. We call character of π the distribution χ : HGL(2,F ) → C,

φ 7→ tr(π(φ)).

The character of an irreducible admissible representation π determines com-pletely the representation:

Theorem 2.1.3. Let (π1, V1) and (π2, V2) be two irreducible admissible

representations of GL(2, F ). If the character of π1 and π2 agree, then π1 ∼=

π2.

Proof. By Proposition 1.1.3 it is sufficient to prove that VK1

1 ∼= V K1

2 as HK1

-modules for every open compact subgroup K1 of GL(2, F ). Both V1K1 and

VK1

2 are finite dimensional vector spaces because π1 and π2 are admissible.

We suppose that they are not isomorphic. As we observed in the proof of Proposition 1.1.8, there exist φ ∈ HK1 acting as identity on V

K1

1 and as zero

on VK1

2 . Then, if χ1 and χ2 are the characters of π1 and π2 respectively, it

results that dim(VK1

1 ) = χ1(φ) = χ2(φ) = 0 and by interchanging the roles

of VK1

1 and V

K1

2 we find that dim(V K1 2 ) = 0. So if nonzero, V K1 1 and V K1 2

are isomorphic as HK1-modules.

We now introduce an important and useful representation of GL(2, F ), the contragredient representation: the contragredient representation of the admissible representation (π, V ) is the representation (ˆπ, ˆV ), where ˆV is the space of smooth linear f unctional on V and, if Λ ∈ ˆV , then ˆπ(g)Λ(v) = Λ(π(g−1)v); a linear functional Λ : V → C is said to be smooth if there

exists some open subgroup K0 such that Λ(π(g)v) = Λ(v) for all g ∈ K0

and v ∈ V .

The contragredient representation is smooth: if Λ ∈ bV , then there exist an open subgroup K0 such that Λ(π(g)v) = Λ(v) for all g ∈ K0 and v ∈ V .

If g ∈ K0, then ˆπ(g)Λ(v) = Λ(π(g−1)v) = Λ(v), so every smooth linear

functional is stabilized by an open subgroup K0.

Furthermore ˆπ is admissible if π is admissible: by Proposition 2.1.1, V

=L

ρV (ρ) and since every Λ in bV is smooth, Λ is nonzero only on a finite

number of summands, so we have b

V =M

ρ

V (ρ)∗,

where V (ρ)∗ is the dual space of V (ρ). By admissibility of π, V (ρ) is finite-dimensional and therefore V (ρ)∗ is finite-dimensional, so ˆπ is admissible.

With Theorem 2.1.3 we can make explicit the contragredient represen-tation:

Theorem 2.1.4. Let (π, V ) be an irreducible admissible representation of GL(2, F ). We define the following representation on the same vector space V : π1(g):= π(Tg−1). Then ˆπ ∼= π1.

To prove the theorem we need the following result, which we give without proof: if a distribution on GL(2, F ) is invariant under conjugation, the is also invariant under transpose. This result may not be surprising since in GL(2, F ) every element is conjugate to its transpose, but the proof is not simple.

Proof. Since the character χ of the representation is invariant under con-jugation, it is invariant under transpose. Let φ0(g) = φ(g−1) and φ00(g) = φ(Tg−1). Then π1(φ) = π(φ00) so the character of π1 is φ 7→ χ(φ00), which

equals χ(φ0). We may conclude by observing that π(φ) and ˆπ(φ0) have the same trace because they are adjoints of each others, so by Theorem 2.1.3, ˆ

π ∼= π1.

By a quasicharacter of a locally compact group G we mean a continuous homomorphism χ : G → C×; a character is an unitary quasicharacter, that is a quasicharacter χ such that |χ(g)| = 1 for all g ∈ G.

Let ψ be a nontrivial additive character of F and (π, V ) be a smooth representation of GL(2, F ). There exists a unique integer m such that ψ is trivial on p−m but not on p−m−1. We call p−m the conductor of ψ.

A linear functional Λ : V → C is called W hittaker f unctional if there exists a nontrivial additive character ψ such that

Λ  π1 x 1  v  = ψ(x)Λ(v),

CHAPTER 2. WHITTAKER MODEL AND L-FUNCTIONS 22 for all x ∈ F and v ∈ V .

Our aim is to prove that a representation admits at most one nonzero Whittaker functional, up to scalar multiplies. In the proof of this result, we will call ρ and λ the actions of GL(2, F ) on itself, on HGL(2,F ) and on

D(GL(2, F )) given by:

ρ(g)(x) = xg−1, λ(g)(x) = gx, g, x ∈ GL(2, F ); (2.2)

(ρ(g)f )(x) = f (xg), (λ(g)f )(x) = f (g−1x), f ∈ H_{GL(2,F )}; (2.3) (ρ(g)T )(f ) = T (ρ(g−1)f ), (λ(g)T )(f ) = T (λ(g−1)f ), T ∈ D(GL(2, F )).

(2.4) Furthermore we define an involution ι : GL(2, F ) → GL(2, F ) given by

ι(g) = ω0 Tgω0, ω0 =  1 1  (2.5) It is an involution: if g ∈ GL(2, F ), then ι(ι(g)) = ω0 Tι(g)ω0 = ω0ω0gω0ω0 = g.

This induces an involution on HGL(2,F ) and on D(GL(2, F )), given

re-spectively by ι(φ)(g) = φ(ι(g)) and ι(∆)(φ) = ∆(ι(φ)). If N (F ) is the subgroup {(1 u₁) | u ∈ F }, then ι is the identity on N (F ): let (1 u₁) ∈ N (F ). Then ι1 u 1  =  1 1  1 u 1   1 1  =u 1 1   1 1  =1 u 1 

Proposition 2.1.5. Let ∆ be a distribution which satisfies the following λ(u)∆ = ψN(u)−1∆, ρ(u)∆ = ψN(u)∆, (2.6)

for every u ∈ N (F ), where ψN is the character of N (F ) such that (1 x1) 7→

ψ(x). Then ∆ is stable under ι.

Theorem 2.1.6. Let (π, V ) be an irreducible admissible representation of GL(2, F ). Then the space of Whittaker functionals on V if nonzero is one-dimensional.

Proof. Let π0 be the representation of GL(2, F ) on the vector space V de-fined by π0(g) = π(ι(g−1)). This defines a representation:

π0(gh) = π(ι((gh)−1)) = π(ω0 T(gh)−1ω0) = π(ω0(T(gh))−1ω0) = = π(ω0(ThTg)−1ω0) = π(ω0 Tg−1ω0ω0 Th−1ω0) =

The endomorphism π(ω0) is a GL(2, F )-representations morphism between (π0, V ) and (π1, V ) given in Theorem 2.1.4, in fact

π(ω0) ◦ π0(g) = π(ω0) ◦ π(ι(g−1)) = π(ω0ω0 Tg−1ω0) = = π(Tg−1ω0) = π1(g) ◦ π(ω0).

Then π(ω0) is an isomorphism because of irreducibility of the representa-tions, so π0 is isomorphic to the contragradient representation of π by Theo-rem 2.1.4, and we can find a pairing h , i : V × V → C such that hπ(g)v, wi = hv, π(ι(g))wi.

Thus if Λ is any smooth linear functional on V , we denote by [Λ] the vector in V such that Λ(v) = hv, [Λ]i.

We use eq. (1.5) to define L ∗ φ(v) = L(π(φ)v) =

Z

GL(2,F )

L(π(g)v)φ(g)dg,

where L is a linear functional on V and φ ∈ HGL(2,F ). Then L ∗ φ is a

smooth linear functional on V and L ∗ (φ1∗ φ2) = (L ∗ φ1) ∗ φ2:

(L ∗ φ1) ∗ φ2(v) = Z GL(2,F ) (L ∗ φ1)(π(h)v)φ2(h)dh = = Z GL(2,F ) Z GL(2,F ) L(π(g)π(h)v)φ1(g)dg  φ2(h)dh = = Z Z L(π(gh)v)φ1(g)φ2(h)dhdg g7→gh−1 = = Z Z L(π(g)v)φ1(gh−1)φ2(g)dhdg = L ∗ (φ1∗ φ2)(v).

Let v ∈ V , G ∈ GL(2, F ), φ ∈ H_{GL(2,F )} and L be a linear functional on V . Then hv,π(g)[L ∗ φ]i = hπ(ι(g)v, [L ∗ φ]i = L ∗ φ(π(ι(g))v) = = Z GL(2,F ) L(π(h)π(ι(g))v)φ(h)dhh7→hι(g −1₎ = Z GL(2,F ) L(π(h)v)φ(hι(g−1))dh = = hv, [L ∗ ρ(ι(g−1))φ]i, so π(g)[L ∗ φ] = [L ∗ ρ(ι(g−1))φ]. (2.7) If Λ is smooth, then hv, [Λ ∗ φ]i = Λ ∗ φ(v) = Λ(π(φ)v) = hπ(φ)v, [Λ]i = = hv, π(ι(φ))[Λ]i,

CHAPTER 2. WHITTAKER MODEL AND L-FUNCTIONS 24 therefore we have

[Λ ∗ φ] = π(ι(φ))[L]. (2.8)

Let Λ be a Whittaker functional, v ∈ V , y ∈ F and φ ∈ HGL(2,F ). Then

hv,[Λ ∗ λ1 y 1  φ]i = (Λ ∗ λ1 y 1  φ)(v) = = Z GL(2,F ) Λ(π(g)v)φ1 y 1 −1 gdg = = Z GL(2,F ) Λπ1 y 1  π(g)vφ(g)dg = = ψ(y)(Λ ∗ φ)(v), so it results [Λ ∗ λ1 y 1  φ] = ψ(y)[Λ ∗ φ]. (2.9)

If we fix Λ1 and Λ2 two nonzero Whittaker functionals, then we define

∆ ∈ D(GL(2, F )) by

∆(φ) = Λ2([Λ1∗ φ]).

It follows from eq. (2.9) and eq. (2.7) that ∆ satisfies eq. (2.6), so it is invariant under ι.

If L is any linear functional on V , then V = {[L ∗ σ] | σ ∈ H_{GL(2,F )}}. In fact this is a GL(2, F )-module by (2.7) and it is contained in V . Since V is irreducible, if contains a nonzero element then it will be the whole V . If L is nonzero, then there exists v ∈ V such that L(v) 6= 0. Since π is smooth, the stabilizer of v is open so if φ ∈ HGL(2,F ) has compact support K0 contained

in this stabilizer and such thatR_{GL(2,F )}φ(g)dg = 1, then L ∗ φ(g) = Z GL(2,F ) L(π(g)v)φ(g)dg = Z K0 L(π(g)v)φ(g)dg = = Z K0 L(v)φ(g)dg = L(v) 6= 0.

Next we show that if Λ1∗ φ = 0, then Λ2∗ φ = 0: it follows from eq. (2.7)

that [Λ1∗ ρ(g)φ] = 0, so Λ1∗ ρ(g)φ = 0. Furthermore,

0 = Λ2([Λ1∗ ρ(g)φ]) = ∆(ρ(g)φ) = ∆(ι(ρ(g)φ)) =

= Λ2([Λ1∗ λ(ι(g−1))ι(φ)]),

so if we exchange g with ι(g−1) we obtain

Let us observe that if σ ∈ HGL(2,F ) then by eq. (1.2) and (2.2) (σ∗ι(φ))(g) = Z GL(2,F ) σ(gh−1)(ιφ)(h)dhh7→h −1_g = Z GL(2,F ) σ(h)(λ(h)ι(φ)(g))dh, so by eq. (2.10) and (2.8) we obtain Λ2(π(φ)[Λ1∗ σ]) = Λ2([Λ1∗ σ ∗ ι(φ)])

= 0 for all σ ∈ H_{GL(2,F )}

We have just seen that V = {[Λ1∗ σ | σ ∈ HGL(2,F )}, so we obtain Λ2∗ φ

= 0.

Let T : V → V , defined by T ([Λ1∗ φ]) = [Λ2∗ φ]; it is defined on all of

V and it is well defined for what we have just proved. Furthermore, T is an intertwining operator, because

T (π(g)[Λ1∗ φ]) = T ([Λ1∗ ρ(ι(g−1))φ]) = [Λ2∗ ρ(ι(g−1))φ] =

= π(g)[Λ2∗ φ] = π(g)T ([Λ1∗ φ]).

Thus we can apply the Schur’s Lemma to find a constant c such that T (v) = cv. It remains to prove that Λ2= cΛ1: we take v ∈ V and φ ∈ HGL(2,F )such

that the support of φ is contained in the stabilizer of v and R

GL(2,F )φ(g)dg

= 1. Then

Λ2(v) = (Λ2∗ φ)(v) = hv, [Λ2∗ φ]i = chv, [Λ1∗ φ]i = cΛ1(v),

and the proof is complete.

We now define the Whittaker functionals in the global case, and we will see that Theorem 2.1.6 will lead us to a uniqueness result in the global case too.

Let k be a function field, A its adele ring. We can embed k ,→ A with the diagonal map x 7→ (. . . , x, . . . ). The adeles of this type are called principal. The field k regarded as image in A through the diagonal map is a discrete subset: it is sufficient to show that 0 has a neighborhood which does not contain other principal adeles, and U := {α ∈ A||αv|v < 1 ∀v} is what we

are looking for because by the product formulaQ

v|α|v = 1 for all α ∈ k.

Moreover A/k is compact with the quotient topology.

Let (π, V ) be an irreducible admissible representation of GL(2, A). By a W hittaker f unctional we mean a linear functional Λ : V → C such that

Λ  π 1 x 1   v  = ψ(x)Λ(v)

for all x ∈ A and v ∈ V , where ψ is a nontrivial character of A/k.

The uniqueness in the local case imply the uniqueness in the global case. Theorem 2.1.7. Let Λ be a Whittaker functional for the irreducible ad-missible representation (π, V ) of GL(2, A). By Theorem 1.2.7 we have π =

CHAPTER 2. WHITTAKER MODEL AND L-FUNCTIONS 26 ⊗vπv where (πv, Vv) (v ∈ Σ) is an irreducible admissible representation of

GL(2, kv), and the tensor product is restricted to a family of vectors ξv◦∈ Vv

such that ξ_v◦ is spherical (GL(2.Ov) invariant). Then for each v there exists

a Whittaker functional Λv of Vv such that Λv(ξ◦v) = 1 for almost al v and

Λ(⊗vξv) =

Y

v

Λv(ξv). (2.11)

Moreover the space of Whittaker functionals on V if nonzero is one dimen-sional.

Proof. If Λ 6= 0, then there exists a pure tensor ξ◦ such that Λ(ξ◦) 6= 0, so we can suppose ξ◦ = ⊗vξv◦ and Λ(ξ◦) = 1. For any place w, we define the

injecion

iw : Vw→ V

ξw7→ ξw⊗v6=wξv◦.

We put Λw := Λ ◦ iw: Λw is a Whittaker functional for Vw:

Λw(πw(1 x1)ξw) = Λ((πw(1 x1)ξw) ⊗v6=wξ◦v) = Λ(π(1 x1)ξw⊗v6=wξv◦) =

= ψ(x)Λ(ξw⊗v6=wξ◦v) = ψw(x)Λ(ξw⊗ ξv◦) =

= ψwΛw(ξw)

where x is the adele whose w-th component is xw and the others are 0 and

ψw is the composition of ψ with xw 7→ (0, . . . , xw, 0, . . . ). Furthermore

Λw(ξw◦) = Λ(ξ ◦ w⊗ ξ ◦ v) = Λ(ξ ◦ ) = 1.

Lastly we have to prove the equation (2.11); it is well defined because almost all factors on the right are 1. We prove it by induction on the cardinality of a finite set S = {v ∈ Σ|ξv = ξv◦if v /∈ S}.

Thus equation (2.11) can be rewritten Λ(⊗vξv) =

Y

v∈S

Λv(ξv)

because if v /∈ S then ξ_v = ξ_v◦ and Λv(ξv◦) = 1.

If S is empty then we have already proved that both sides of (2.11) are equal to 1.

Now we suppose ξv = ξv◦ ∀v /∈ S ∪ {w}. The linear functional xw 7→

Λ(xw⊗v6=wξv) is a Whittaker functional fo Vv, so by Theorem 2.1.6 there

exists a constant c such that

Λ(xw⊗v6=wξv) = cΛw(xw)

To evaluate c we take xw= ξcircw , hence by induction hypothesis c =

Q

v∈SΛv(ξv),

so we obtain equation (2.11) when ξv = ξv◦ for v /∈ S ∪ {w}.

The existence of a Whittaker functional for the representation (π, V ) of GL(2, F ) is equivalent to the existence of a space of functions which allows us to realize the representation π on a concrete space: we will call this space W hittaker model for the representation (π, V ) with respect to ψ.

Theorem 2.1.8. Let F be a archimedean local filed, let ψ be a non-trivial character of F and (π, V ) a representation of GL(2, F ). Then there exists a space W of function on GL(2, F ) such that it is isomorphic to V and, for W ∈ W, it results

W 1 x 1  g  = ψ(x)W (g), x ∈ F, g ∈ GL(2, F ), if and only if π admits one nonzero Whittaker functional.

Proof. Let Λ a nonzero Whittaker functional. We put, for any ξ ∈ V , Wξ(g)

= Λ(π(g)ξ). Then {Wξ| ξ ∈ V } is a Whittaker model for (π, V ):

− W_ξ1 x 1  g  = Λ  π 1 x 1   π(g)ξ  = ψ(x)Λ(π(g)ξ) = ψ(x)Wξ(g) − ρ(g)W_ξ(h) = Wξ(hg) = Λ(π(h)π(g)ξ) = Wπ(g)ξ(h) − Wλ1ξ1+λ2ξ2(g) = Λ(π(g)(λ1ξ1+ λ2ξ2) = Λ(π(g)λ1ξ1+ π(g)λ2ξ2) = = Λ(π(g)λ1ξ1) + Λ(π(g)λ2ξ2) = λ1Wξ1(g) + λ2Wξ2(g)

It follows that ξ 7→ Wξ is a GL(2, F )-representations isomorphism.

Conversely, if there exists a Whittaker model for (π, V ) with respect to ψ, then Λ(ξ) = Wξ(1) is a Whittaker functional for V :

Λπ1 x 1   ξ= W_{π( 1 x} 1)ξ (1) = Wξ  1 ·1 x 1   = = ψ(x)Wξ(1) = ψ(x)Λ(ξ)

We now want to determinate the spherical representations of GL(2, F ); let F be a non-archimedean local field, let O be its ring of integers and let K = GL(2.O). Let B(F ) be the subgroup of GL(2, F ) consisting of upper triangular matrices.

Proposition 2.1.9. (Iwasawa decomposition) GL(2, F ) = B(F )K

This result is true also for n > 2 and it can be shown by induction on n, where the base case is the following proof.

CHAPTER 2. WHITTAKER MODEL AND L-FUNCTIONS 28 Proof. Let g = (a b_{c d}) ∈ GL(2, F ). First consider the case d = 0. Then

a b c 0   1 1  =b a c  ∈ B(F ), so being ( 1 1 ) ∈ K, we obtain g = (g(11))(11)−1 ∈ B(F )K.

Next we consider the case d 6= 0. If cd−1 ∈ O, then a b c d  =a − bcd −1 _b d   1 cd−1 1  ∈ B(F )K.

If cd−1∈ O, we can multiply g on the right by an opportune h ∈ K so that/ gk verify the previous assumption. Then we find b ∈ B(F ) and h0 ∈ K such that gh = bh0, thus g = bhh0−1∈ B(F )K.

Let χ1 and χ2 be quasicharacters of F×; we assume that χ1 and χ2 are

nonramif ied, that is they are trivial on O×. Let us define a character χ on B(F ) by

χy1 ∗ y2



= χ1(y1)χ2(y2).

Let B(χ1, χ2) := IndGL(2,F )_{B(F )} (χ), that is the representation giver by the

right translation of GL(2, F ) on the space of smooth functions f : G → C such that f y 1 ∗ y2  g  = δy1 ∗ y2 1/2 χy1 ∗ y2  f (g),

where δ is a character of B(F ), the so called modular character, which we now explicit. Let dlx be a left Haar measure on the group G. Then, for

a fixed y ∈ G, dlxy is another left Haar measure, so by uniqueness of the

Haar measure there exists a character δ of G, the modular character, such that dlxy = δ(y)dlx. We have already observed in section 1.2 that drx =

δ(x)−1dlx is a right Haar measure.

In the case G = B(F ), if b =1 x 1  y1 y2  ∈ B(F ),

then drb = dxd×y1d×y2 is the right Haar measure on B(F ) and dlb =

|y₁|−1|y₂| is the left Haar measure, where d×y = |y|−1dy is the Haar mea-sure on F×. Thus the modular character of B(F ) is δ(b) = |y1||y2|−1, so the

functions in B(χ1, χ2) verify f y 1 ∗ y2  g  =     y1 y2     1/2 χ1(y1)χ2(y2)f (g). (2.12)

Proposition 2.1.10. The representation B(χ1, χ2) is admissible.

Proof. Let f (g) ∈ B(χ1, χ2). From Iwasawa decomposition, f is determined

by what values it assume on K. Further f is locally constant, so for every h ∈ K let Uh be a compact neighborhood of 1 such that f (h · Uh) = f (h). Since

K is compact, we may cover it with a finite number of such neighborhoods and if U denotes their intersection, then f (h · U ) = f (h), so B(χ1, χ2) is

smooth.

Next let K0 e a compact open subgroup in K. If f ∈ B(χ1, χ2)K

0

, then f (hh0) = f (h) for all h ∈ K, h0 ∈ K0. Since K0 has finite index in K and f is determined by the values it assume on the cosets K\K0, it follows that B(χ1, χ2)K

0

) is admissible.

It can be shown that the representation B(χ1, χ2) is irreducible except

when χ1χ−12 (y) = |y| or χ1χ−12 (y) = |y|−1. When irreducible we indicate it

with π(χ1, χ2) and call it principal series representation. It is a spherical

representation: Let

φK(bh) = δ

1

2χ(b), (2.13)

where b ∈ B(F ) and h ∈ K; it is defined on all GL(2, F ) by Iwasawa decomposition. We now show that it is well defined, so suppose that bh = b0h0. Then b = b0h0h−1 and h0h−1 = b0−1b ∈ K ∩ B(F ), and since χ1 and

χ2 are trivial on O× it results that δ

1

2χ(h0h−1) = 1 ⇒ φ_K(bh) = φ(b0h0)

⇒ φ_K is well defined.

Next we have to prove that φK satisfies equation (2.12):

φK y 1 ∗ y2  bh= δ12χ y 1 ∗ y2  b= δ12y1 ∗ y2  χy1 ∗ y2  δ12(b)χ(b) =     y1 y2     1 2 χ1(y1)χ2(y2)φ(bk), so φK∈ π(χ1, χ2).

Finally it is clear that φK is spherical, because it depends only on b ∈

B(F ).

Let (π, V ) be an irreducible admissible representation of GL(2, F ). Then VK is at most one dimensional because it is a module for the commutative algebra HK, the spherical Hecke algebra introduced in section 1.2. Thus

there exists a character ξ of HK such that

π(φ)vK = ξ(φ)vK,

for any φ ∈ HK and vK ∈ VK; we call ξ the character of HK associated

with the spherical representation (π, V ). We will see that the character of an irreducible admissible spherical representation determines the representation up to isomorphism.