automorphic forms on gl(2) - h. jacquet, r. langlands

It can be extended to a unitary representation of SL2, F on L2K, the space of square integrable functions on K.. Using the lemma we shall decomposerwith respect toK and extendrto a repr

Herve´ Jacquet and Robert P Langlands

Formerly appeared as volume #114 in the Springer Lecture Notes in Mathematics, 1970, pp 1-548

Table of Contents

Introduction ii

Chapter I: Local Theory 1

§1 Weil representations 1

§2 Representations ofGL(2, F )in the non-archimedean case 15

§3 The principal series for non-archimedean fields 58

§4 Examples of absolutely cuspidal representations 77

§5 Representations ofGL(2,R) 96

§6 Representation ofGL(2,C) 138

§7 Characters 151

§8 Odds and ends 173

Chapter II: Global Theory 189

§9 The global Hecke algebra 189

§10 Automorphic forms 204

§11 Hecke theory 221

§12 Some extraordinary representations 251

Chapter III: Quaternion Algebras 267

§13 Zeta-functions for M (2, F ) 267

§14 Automorphic forms and quaternion algebras 294

§15 Some orthogonality relations 304

§16 An application of the Selberg trace formula 320

Two of the best known of Hecke’s achievements are his theory of L-functions with charakter, which are Dirichlet series which can be represented by Euler products, and his theory of theEuler products, associated to automorphic forms onGL(2) Since a gr¨ossencharakter is an automorphic

gr¨ossen-form onGL(1)one is tempted to ask if the Euler products associated to automorphic forms onGL(2)

play a role in the theory of numbers similar to that played by theL-functions with gr¨ossencharakter.

In particular do they bear the same relation to the ArtinL-functions associated to two-dimensional

representations of a Galois group as the HeckeL-functions bear to the Artin L-functions associated

to one-dimensional representations? Although we cannot answer the question definitively one of theprincipal purposes of these notes is to provide some evidence that the answer is affirmative

The evidence is presented in §12 It come from reexamining, along lines suggested by a recent

paper of Weil, the original work of Hecke Anything novel in our reexamination comes from our point

of view which is the theory of group representations Unfortunately the facts which we need from therepresentation theory ofGL(2)do not seem to be in the literature so we have to review, in Chapter I,the representation theory ofGL(2, F )whenF is a local field §7 is an exceptional paragraph It is not

used in the Hecke theory but in the chapter on automorphic forms and quaternion algebras

Chapter I is long and tedious but there is nothing hard in it Nonetheless it is necessary andanyone who really wants to understandL-functions should take at least the results seriously for they

are very suggestive

§9 and §10 are preparatory to the Hecke theory which is finally taken up in §11 We would like to

stress, since it may not be apparent, that our method is that of Hecke In particular the principal tool isthe Mellin transform The success of this method forGL(2)is related to the equality of the dimensions

of a Cartan subgroup and the unipotent radical of a Borel subgroup ofP GL(2) The implication is that

our methods do not generalize The results, with the exception of the converse theorem in the Hecketheory, may

The right way to establish the functional equation for the Dirichlet series associated to theautomorphic forms is probably that of Tate In§13 we verify, essentially, that this method leads to the

same local factors as that of Hecke and in§14 we use the method of Tate to prove the functional equation

for the L-functions associated to automorphic forms on the multiplicative group of a quaternion

algebra The results of§13 suggest a relation between the characters of representations of GL(2)andthe characters of representations of the multiplicative group of a quaternion algebra which is verified,using the results of§13, in §15 This relation was well-known for archimedean fields but its significance

had not been stressed Although our proof leaves something to be desired the result itself seems to us

to be one of the more striking facts brought out in these notes

Both§15 and §16 are after thoughts; we did not discover the results in them until the rest of the

notes were almost complete The arguments of§16 are only sketched and we ourselves have not verified

all the details However the theorem of§16 is important and its proof is such a beautiful illustration

of the power and ultimate simplicity of the Selberg trace formula and the theory of harmonic analysis

on semi-simple groups that we could not resist adding it Although we are very dissatisfied with themethods of the first fifteen paragraphs we see no way to improve on those of§16 They are perhaps

the methods with which to attack the question left unsettled in§12.

We hope to publish a sequel to these notes which will include, among other things, a detailedproof of the theorem of§16 as well as a discussion of its implications for number theory The theorem

has, as these things go, a fairly long history As far as we know the first forms of it were assertions aboutthe representability of automorphic forms by theta series associated to quaternary quadratic forms

As we said before nothing in these notes is really new We have, in the list of references atthe end of each chapter, tried to indicate our indebtedness to other authors We could not howeveracknowledge completely our indebtedness to R Godement since many of his ideas were communicatedorally to one of us as a student We hope that he does not object to the company they are forced to keep.The notes∗were typed by the secretaries of Leet Oliver Hall The bulk of the work was done byMiss Mary Ellen Peters and to her we would like to extend our special thanks Only time can tell if themathematics justifies her great efforts.

New Haven, Conn

∗ that appeared in the SLM volume

§1 Weil representations. Before beginning the study of automorphic forms we must review the sentation theory of the general linear group in two variables over a local field In particular we have toprove the existence of various series of representations One of the quickest methods of doing this is

repre-to make use of the representations constructed by Weil in [1] We begin by reviewing his constructionadding, at appropriate places, some remarks which will be needed later

In this paragraphF will be a local field andKwill be an algebra overFof one of the followingtypes:

(i) The direct sumF ⊕ F

(ii) A separable quadratic extension ofF

(iii) The unique quaternion algebra overF.Kis then a division algebra with centreF

(iv) The algebraM (2, F )of2× 2matrices overF

In all cases we identifyF with the subfield ofKconsisting of scalar multiples of the identity Inparticular ifK = F ⊕ F we identifyF with the set of elements of the form(x, x) We can introduce aninvolutionιofK, which will send xtox ι, with the following properties:

(i) It satisfies the identities(x + y) ι = x ι + y ιand(xy) ι = y ι x ι

(ii) Ifxbelongs toF thenx = x ι

(iii) For anyxinKbothτ (x) = x + x ιandν(x) = xx ι = x ι xbelong toF

IfK = F ⊕ F andx = (a, b)we setx ι = (b, a) IfKis a separable quadratic extension ofF theinvolutionιis the unique non-trivial automorphism ofKoverF In this caseτ (x)is the trace ofxand

ν(x)is the norm ofx If Kis a quaternion algebra a uniqueιwith the required properties is known toexist.τ andνare the reduced trace and reduced norm respectively IfKisM (2, F )we takeιto be theinvolution sending

Thenτ (x)andν(x)are the trace and determinant ofx.

Ifψ = ψ F is a given non-trivial additive character ofFthenψ K = ψ F ◦τis a non-trivial additivecharacter ofK By means of the pairing

x, y = ψ K (xy)

we can identifyKwith its Pontrjagin dual The functionνis of course a quadratic form onKwhich is

a vector space overFandf = ψ F ◦ νis a character of second order in the sense of [1] Since

LetS(K)be the space of Schwartz-Bruhat functions onK There is a unique Haar measure dx

onKsuch that ifΦbelongs toS(K)and

Lemma 1.1. There is a constant γ which depends on the ψ F and K, such that for every function Φ

K

(Φ∗ f)(y) ψ K (yx) dy = γf −1 (x ι) Φ (x)

Φ∗ fis the convolution ofΦandf The values ofγare listed in the next lemma

Lemma 1.2 (i) If K = F ⊕ F or M(2, F ) then γ = 1.

(ii) If K is the quaternion algebra over F then γ = −1.

(iii) If F = R, K = C, and

ψ F (x) = e 2πiax , then

γ = a

|a| i

(iv) If F is non-archimedean and K is a separable quadratic extension of F let ω be the quadratic character of F ∗ associated to K by local class-field theory If U F is the group of units of F ∗ let m = m(ω) be the smallest non-negative integer such that ω is trivial on

U F m={a ∈ U F | α ≡ 1 (mod p m

F)} and let n = n(ψ F ) be the largest integer such that ψ F is trivial on the ideal p−n

F If a is any generator on the ideal pm+n

by a positive factor This equals

Setk = 2ifK/F is unramified and setk = ifK/F is ramified Thenν(pK) = p−k F ∩ ν(K).

Since1 + ωis twice the characteristic function ofν(K ×)the factorγis the positive multiple of



p−k F

ψ F (x) dx +



p−k F

ψ F (x) ω(x) dx.

Forand thereforeksufficiently large the first integral is0 IfK/F is ramified well-known properties

of Gaussian sums allow us to infer that the second integral is equal to

Sinceω = ω −1we obtain the desired expression forγby dividing this integral by its absolute value If

K/F is unramified we write the second integral as

ψ F (x) dx −



p−k+j+1 F

ψ F (x) dx

p−k+j F





A little algebra shows that this equals 2ω(a)q q+1 m+1 so that γ = ω(a), which upon careful inspection is

seen to equal the expression given in the lemma

In the notation of [19] the third and fourth assertions could be formulated as an equality

γ = λ(K/F, ψ F ).

It is probably best at the moment to take this as the definition ofλ(K/F, ψ F)

IfKis not a separable quadratic extension ofF we takeωto be the trivial character

Proposition 1.3 There is a unique representation r of SL(2, F ) on S(K) such that

If S(K) is given its usual topology, r is continuous It can be extended to a unitary representation

of SL(2, F ) on L2(K), the space of square integrable functions on K If F is archimedean and Φ

belongs to S(K) then the function r(g)Φ is an indefinitely differentiable function on SL(2, F ) with

Ifa = 1the relation (c) becomes

to Weil’s paper for the proof thatr is continuous and may be extended to a unitary representation of

r be the corresponding representation The constantγ  = ω(a)γ.

Lemma 1.4 (i) The representation r  is given by

In particular if ν(b) = 1 both λ(b) and ρ(b) commute with r.

We leave the verification of this lemma to the reader TakeKto be a separable quadratic extension

ofF or a quaternion algebra of centreF In the first caseν(K ×)is of index 2 inF × In the second case

ν(K ×)isF ×ifFis non-archimedean andν(K ×)has index 2 inF ×ifF isR

LetK be the compact subgroup ofK ×consisting of allxwithν(x) = xx ι= 1and letG+be thesubgroup ofGL(2, F )consisting of allgwith determinant inν(K ×).G+has index 2 or 1 inGL(2, F ).

Using the lemma we shall decomposerwith respect toK and extendrto a representation ofG+.LetΩbe a finite-dimensional irreducible representation ofK ×in a vector spaceUoverC Takingthe tensor product ofrwith the trivial representation ofSL(2, F )onUwe obtain a representation on

S(K) ⊗CU = S(K, U)

which we still callrand which will now be the centre of attention

Proposition 1.5 (i) If S(K, Ω) is the space of functions Φ in S(K, U) satisfying

Φ(xh) = Ω −1 (h)Φ(x)

for all h in K  then S(K, Ω) is invariant under r(g) for all g in SL(2, F ).

(ii) The representation r of SL(2, F ) on S(K, Ω) can be extended to a representation rΩ of G+

(iii) If η is the quasi-character of F × such that

(v) If U is a Hilbert space and Ω is unitary let L2(K, U ) be the space of square integrable functions

from K to U with the norm

The first part of the proposition is a consequence of the previous lemma LetHbe the group of

a 0



withainν(K ×) It is clear that the formula of part (ii) defines a continuous representation ofH on

S(K, Ω) MoreoverG+is the semi-direct ofHandSL(2, F )so that to prove (ii) we have only to showthat

which is a consequence of the last part of the previous lemma

To prove (iii) observe that

and thata2= ν(a)belongs toν(K ×) The last two assertions are easily proved

We now insert some remarks whose significance will not be clear until we begin to discuss thelocal functional equations We associate to everyΦinS(K, Ω)a function

It is easily verified that

Thus we may regard rΩ as acting on the space V of functions ϕΦ, Φ ∈ S(K, Ω) The effect of a

matrix inB+ is given by (1.5.3) The matrix

a 0



corresponds to the operatorω(a) η(a)I Since

G+is generated byB+, the set of scalar matrices, andw =

Ifα K : h → |h| K is the module ofKthen

Z(µα 1/2 K ⊗ Ω, Φ)

is known to converge forRe ssufficiently large and the function itself is essentially a local zeta-function

in the sense of Tate The integral defining

Z(µ −1 α 1/2 K ⊗ Ω −1 , Φ )

converges forRe s sufficiently small, that is, large and negative Both functions can be analyticallycontinued to the whole s-plane as meromorphic functions There is a scalar C(µ)which dependsanalytically onssuch that

if the measures are suitably normalized.

We want to extend some of these results to the caseK = F ⊕ F We regard the element ofK

as defining a row vector so thatK becomes a right module forM (2, F ) IfΦbelongs toS(K)andg

for all Φ in S(K) and all g in G F

It is easy to prove part (ii) for g inSL(2, F ) In fact one has just to check it for the standard

generators and for these it is a consequence of the definitions of Proposition 1.3 The formula of part (ii)therefore defines an extension ofrtoGL(2, F )which is easily seen to satisfy the condition of part (i).LetΩbe a quasi-character ofK × SinceK × = F × × F ×we may identifyΩwith a pair(ω1, ω2)

of quasi-characters ofF × ThenrΩwill be the representation defined by

rΩ(g) = |detg| 1/2

ω1(detg)r(g).

Ifxbelongs toK ×andν(x) = 1thenxis of the form(t, t −1)withtinF × IfΦbelongs toS(K)

forbinB F The applicationsΦ→ WΦ andΦ→ ϕΦare no longer injective

Ifµ0is the quasi-character defined by

µ0(a) = Ω((a, a)) = ω1(a) ω2(a)

Again we introduce in a purely formal manner the distribution

to the whole complex plane as meromorphic functions and there is a meromorphic functionC(µ)which

is independent ofΦsuch that

§2 Representations of GL(2, F ) in the non-archimedean case. In this and the next two paragraphsthe ground fieldF is a non- archimedean local field We shall be interested in representations π of

G F = GL(2, F )on a vector spaceV overCwhich satisfy the following condition

(2.1) For every vector v in V the stabilizer of v in G F is an open subgroup of G F

Those who are familiar with such things can verify that this is tantamount to demanding that themap(g, v) → π(g)vofG F × V intoV is continuous ifV is given the trivial locally convex topology inwhich every semi-norm is continuous A representation ofG F satisfying (2.1) will be called admissible

if it also satisfies the following condition

(2.2) For every open subgroup G  of GL(2, O F ) the space of vectors v in V stablizied by G  is finite-dimensional.

O F is the ring of integers ofF.

LetHF be the space of functions onG F which are locally constant and compactly supported.Letdgbe that Haar measure onG F which assigns the measure 1 toGL(2, O F) EveryfinHF may beidentified with the measuref (g) dg The convolution product

f1∗ f2(h) =



G F

f1(g) f2(g −1 h) dg

turns HF into an algebra which we refer to as the Hecke algebra Any locally constant function

on GL(2, O F) may be extended to G F by being set equal to 0 outside of GL(2, O F) and thereforemay be regarded as an element of HF In particular if π i, 1 ≤ i ≤ r, is a family of inequivalent

finite-dimensional irreducible representations ofGL(2, O F)and

is an idempotent ofHF Such an idempotent will be called elementary

Letπbe a representation satisfying (2.1) Iff belongs toHF andvbelongs toV thenf (g) π(g)v

takes on only finitely many values and the integral

(2.3) For every v in V there is an f in HF such that πf (v) = v.

In factf can be taken to be a multiple of the characteristic function of some open and closedneighborhood of the identity Ifπis admissible the associated representation ofHF satisfies

(2.4) For every elementary idempotent ξ of HF the operator π(ξ) has a finite-dimensional range.

We now verify that from a representationπofHF satisfying (2.3) we can construct a tationπofG F satisfying (2.1) such that

withv iinV andf iinHF If we can show that

πwill clearly be a representation ofG F satisfying (2.1)

Suppose that (2.3.1) is satisifed and choosef inHF so thatπ(f )w = w Then

w = r

It is easy to see that the representation ofG F satisfies (2.2) if the representation ofHF satisfies(2.4) A representation ofHF satisfying (2.3) and (2.4) will be called admissible There is a completecorrespondence between admissible representations of G F and of HF For example a subspace isinvariant underG F if and only if it is invariant underH F and an operator commutes with the action

ofG F if and only if it commutes with the action ofHF

>From now on, unless the contrary is explicitly stated, an irreducible representation ofG F orHF

is to be assumed admissible Ifπis irreducible and acts on the spaceV then any linear transformation

of V commuting withHF is a scalar In fact if V is assumed, as it always will be, to be differentfrom0there is an elementary idempotentξsuch thatπ(ξ) = 0 Its range is a finite-dimensional space

invariant underA Thus Ahas at least one eigenvector and is consequently a scalar In particular there

is a homomorphismωofF ×intoCsuch that

If χ is a quasi-character ofF × theng → χ(detg) is a quasi-character ofG F It determines aone-dimensional representation ofG F which is admissible It will be convenient to use the letterχtodenote this associated representation Ifπis an admissible reprentation ofG F onV thenχ ⊗ πwill bethe reprenentation ofG F onV defined by

(χ ⊗ π)(g) = χ(detg)π(g).

It is admissible and irreducible ifπis

Letπbe an admissible representation ofG F onV and letV ∗be the space of all linear forms on

V We define a representationπ ∗ofHF onV ∗by the relation

v, π ∗ (f )v ∗  = π( ˇ f )v, v ∗ 

wheref ν(g) = f (gˇ −1) Since π ∗will not usually be admissible, we replaceV ∗by V = π ∗(HF )V ∗.The spaceV is invariant underHF For eachf inHF there is an elementary idempotentξ such that

ξ ∗f = fand therefore the restrictionπofπ ∗toV satisfies (2.3) It is easily seen that ifξis an elementary

idempotent so isξ To show thatˇ πis admissible we have to verify that



V (ξ) = π(ξ) V = π ∗ (ξ)V ∗

is finite-dimensional LetV (ˇ ξ) = π( ˇ ξ)V and letV c=

1− π(ˇξ)V V is clearly the direct sum ofV ( ˇ ξ),

which is finite-dimensional, andV c MoreoverV (ξ) is orthogonal toV cbecause

v − π(ˇξ)v, π(ξ)v = π(ˇξ)v − π(ˇξ)v, v = 0.

It follows immediately that V (ξ) is isomorphic to a subspace of the dual of V (ˇ ξ) and is therefore

finite-dimensional It is in fact isomorphic to the dual ofV (ˇ ξ)because ifv ∗annihilatesV cthen, for all

vinV,

v, π ∗ (ξ)v ∗  − v, v ∗  = −v − π(ˇξ)v, v ∗  = 0

so thatπ ∗ (ξ)v ∗ = v ∗

πwill be called the representation contragradient toπ It is easily seen that the natural map of

V intoV∗is an isomorphism and that the image of this map isπ ∗(HF) V ∗so thatπmay be identifiedwith the contragredient ofπ

IfV1is an invariant subspace ofV andV2= V1\ V we may associate toπrepresentationsπ1and

π2onV1andV2 They are easily seen to be admissible It is also clear that there is a natural embedding

ofV2inV Moreover any elementv1ofV1lies inV1(ξ)for someξ and therefore is determined by its

effect onV1( ˇξ) It annihilates

I −π(ˇξ)V1 There is certainly a linear functionvonV which annihilates



I − π(ˇξ)V and agrees withV1onV1( ˇξ). vis necessarily inV so thatV1may be identified withV2\  V.

Since every representation is the contragredient of its contragredient we easily deduce the followinglemma

Lemma 2.5 (a) Suppose V1 is an invariant subspace of V If  V2 is the annihilator of V1 in  V then

V1 is the annihilator of  V2 in V

(b) π is irreducible if and only if π is.

Observe that for allginG F

π(g)v, v = v, π(g −1)v

Ifπis the one-dimensional representation associated to the quasi-characterχthenπ = χ −1 Moreover

ifχis a quasi-character andπany admissible representation then the contragredient ofχ ⊗πisχ −1 ⊗π.

Let V be a separable complete locally convex space andπ a continuous representation of G F

onV The spaceV0= π(HF )V is invariant underG F and the restrictionπ0ofπtoV0satisfies (2.1).Suppose that it also satisfies (2.2) Then ifπ is irreducible in the topological senseπ0is algebraicallyirreducible To see this take any two vectorsvandwinV0and choose an elementary idempotentξsothatπ(ξ)v = v vis in the closure ofπ(HF )wand therefore in the closure ofπ(HF )w ∩ π(ξ)V Since,

by assumption,π(ξ)V is finite dimensional,vmust actually lie inπ(HF )w

The equivalence class of π is not in general determined by that of π0 It is, however, when

π is unitary To see this one has only to show that, up to a scalar factor, an irreducible admissiblerepresentation admits at most one invariant hermitian form

Lemma 2.6 Suppose π1 and π2 are irreducible admissible representations of G F on V1 and V2 spectively Suppose A(v1, v2) and B(v1, v2) are non-degenerate forms on V1× V2 which are linear

re-in the first variable and either both lre-inear or both conjugate lre-inear re-in the second variable Suppose moreover that, for all g in G F

SinceSand T are both linear or conjugate linear with kernel 0 they are both embeddings Both take

V2 onto an invariant subspace ofV1 SinceV1 has no non-trivial invariant subspaces they are both

isomorphisms ThusS −1 T is a linear map ofV2which commutes withG F and is therefore a scalarλI.

The lemma follows

An admissible representation will be called unitary if it admits an invariant positive definitehermitian form

We now begin in earnest the study of irreducible admissible representations ofG F The basicideas are due to Kirillov

Proposition 2.7. Let π be an irreducible admissible representation of G F on the vector space V (a) If V is finite-dimensional then V is one-dimensional and there is a quasi-character χ of F × such that

π(g) = χ(detg) (b) If V is infinite dimensional there is no nonzero vector invariant by all the matrices 1

If π is finite-dimensional its kernel H is an open subgroup In particular there is a positive

do also Since the matrices generateSL(2, F ) the group H contains SL(2, F ) Thus π(g1)π(g2) =

π(g2)π(g1) for all g1 and g2 in G F Consequently each π(g) is a scalar matrix and π(g) is dimensional In fact

be enough to show that it containsSL(2, F ) In fact we shall show thatH0, the stabilizer ofv, contains

SL(2, F ) H0is open and therefore contains a matrix

a b

c d



withc = 0 It also contains

also belongs toH0 As before we see thatH0containsSL(2, F )

Because of this lemma we can confine our attention to infinite-dimensional representations Let

ψ = ψ F be a nontrivial additive character ofF LetB F be the group of matrices of the form

(i) The set V  is a subspace of V

(ii) Let X = V  \ V and let A be the natural map of V onto X If v belongs to V let ϕ v be the function defined by

(iii) If b belongs to B F and v belongs to V then

Lemma 2.8.1 Letp−m be the largest ideal on which ψ is trivial and let f be a locally constant function

on p with values in some finite dimensional complex vector space For any integer n ≤  the following two conditions are equivalent

(i) f is constant on the cosets of p−n in p

p− ψ( −ax) f(x) dx

is zero for all a outside of p−m+n .

Assume (i) and let a be an element of F × which is not in p−m+n Then x → ψ(−ax) is anon-trivial character ofp−nand

f (x) =



p−m+n ψ(−xy) f  (y) dy.

Ifybelongs top−m+n the functionx → ψ(−xy)is constant on cosets ofp−n It follows immediatelythat the second condition of the lemma implies the first

To prove the second assertion of the proposition we show that ifϕ vvanishes identically thenv

is fixed by the operatorπ1

The restriction off to an ideal inF takes values in a finite-dimensional subspace ofV To show that

f is constant on the cosets of some idealp−n it is enough to show that its restriction to some idealp

containingp−nhas this property

By assumption there exists an n0 such thatf is constant on the cosets ofp−n0 We shall nowshow that iff is constant on the cosets ofp−n+1it is also constant on the cosets ofp−n Take any ideal

p containingp−n By the previous lemma

Thus it will be enough to show that for some sufficiently largethe integral vanishes whenais taken

to be one of a fixed set of representatives of the cosets ofU1in the set of generators ofp−m+n−1 Sincethere are only finitely many such cosets it is enough to show that for eachathere is at least oneforwhich the integral vanishes

By assumption there is an ideala(a)such that

Ifp−ncontainsywe may change the variables in the first integral to see that it equals the second

It will be convenient now to identifyv withϕ v so thatV becomes a space of functions onF ×

with values inX The map Ais replaced by the mapϕ → ϕ(1) The representation πnow satisfies

Proposition 2.9 (i) The space V contains

V0=S(F × , X)

(ii) The space V is spanned by V0 and π(w)V0.

For everyϕinV there is a positive integernsuch that

implies thatϕ(α) = 0if the restriction ofψtoαpn

is not trivial Letp−mbe the largestideal on whichψis trivial Thenϕ(α) = 0unless|α| ≤ |C| −m−nifCis a generator ofp.

LetV0be the space of allCinV such that, for some integerdepending onϕ, ϕ(α) = 0unless

|α| > |C| To prove (i) we have to show thatV0=S(F × , X) It is at least clear that S(F × , X)contains

V0 Moreover for everyϕinV and everyxinF the difference

is identically zero forx = 0and otherwise vanishes at least onx −1p−m Since there is no function in

V invariant under all the operators

the spaceV0is not0

Before continuing with the proof of the proposition we verify a lemma we shall need

Lemma 2.9.1 The representation ξ ψ of B F in the space S(F × ) of locally constant, compactly ported, complex-valued functions on F × is irreducible.

sup-For every characterµofU F letϕ µbe the function onF ×which equalsµonU F and vanishes off

U F Since these functions and their translates spanS(F ×)it will be enough to show that any non-trivialinvariant subspace contains all of them Such a space must certainly contain some non-zero functionϕ

which satisfies, for some characterν ofU F, the relation

ϕ(a@) = ν(@) ϕ(a)

for allainF ×and all@inU F Replacingϕby a translate if necessary we may assume thatϕ(1) = 0.

We are going to show that the space containsϕ µ ifµis different from ν Since U F has at least twocharacters we can then replaceϕby someϕ µwithµdifferent fromν, and replace νbyµandµbyνtosee it also containsϕ ν

The characterµ −1 ν has a conductorpnwithnpositive Takex to be of order−n − m The integral,

which can be rewritten as a Gaussian sum, is then, as is well-known, zero ifais not inU F but differentfrom zero ifais inU F Sinceϕ(1)is not zeroϕ must be a nonzero multiple ofϕ µ

To prove the first assertion of the proposition we need only verify that ifubelongs toXthenV0

contains all functions of the formα → η(α)uwithηinS(F ×) There is aϕinV such thatϕ(1) = u.

Takexsuch thatψ(x) = 1 Then

1− ψ(x)u Consequently every uis of the formϕ(1)for someϕinV0

Ifµis a character ofU F letV0(µ)be the space of functionsϕinV0satisfying

ϕ(a@) = µ(@)ϕ(a)

for allainF ×and all@inU F.V0is clearly the direct sum of the spaceV0(µ) In particular every vector

uinXcan be written as a finite sum

u =

ϕ i(1)

whereϕ ibelongs to someV0(µ i)

If we make use of the lemma we need only show that ifucan be written asu = ϕ(1)whereϕis

inV0(ν)for someνthen there is at least one function inV0of the formα → η(α)uwhereηis a nonzerofunction inS(F ×) Chooseµdifferent fromνand letpn

be the conductor ofµ −1 ν We again consider

The properties of Gaussian sums used before show thatϕ is a function of the required kind

The second part of the proposition is easier to verify LetP F be the group of upper-triangularmatrices inG F SinceV0is invariant underP F andV is irreducible underG F the spaceV is spanned

byV0and the vectors

withϕinV0 But

ϕ ={ϕ  − π(w)ϕ} + π(w)ϕ

and as we saw,ϕ  − π(w)ϕis inV0 The proposition is proved

To study the effect ofwwe introduce a formal Mellin transform LetCbe a generator ofp Ifϕ

is a locally constant function onF ×with values inXthen for every integernthe function@ → ϕ(@C n)

onU F takes its values in a finite-dimensional subspace ofXso that the integral



U F

ϕ(@C n )ν(@) = ϕn (ν)

is defined In this integral we take the total measure ofU F to be1 It is a vector inX ϕ(ν, t) will be

withc > c0then the integral (2.10.1) is absolutely convergent Consequently allthe seriesϕ(ν, t) have positive radii of convergence

Ifψ = ψ F is a given non-trivial additive character ofF,µany character ofU F, andxany element

The integral is taken with respect to the normalized Haar measure onU F Ifgbelongs toG F,ϕbelongs

toV, andϕ  = π(g)ϕwe shall set

where the inner sum is taken over all characters of U F

(iii) Let ω0 be the quasi-character defined by

for all@inU F Ifϕ  = π(w)ϕthen

whereC n (ν)is a linear operator onX.

We introduce the formal series

C(ν, t) =

t n C n (ν).

We have now to verify the third formula of the proposition Sinceϕis inV0the product on the right

is defined Since both sides are linear inϕwe need only verify it for a set of generators ofV0 Thisset can be taken to be the functions defined by (2.10.2) together with their translates of powerC For

functions of the form (2.10.2) the formula is valid because of the way the various seriesC(ν, t)weredefined Thus all we have to do is show that if the formula is valid for a given functionϕit remainsvalid whenϕis replaced by

Consequently there is ann0such that C n (ν)u = 0forn < n0 Of coursen0 may depend onu and

ν This observation together with standard properties of Gaussian sums shows that the infinite sums

occurring in the following proposition are meaningful, for when each term is multiplied on the right

by a fixed vector inXall but finitely many disappear

Proposition 2.11 Let p be the largest ideal on which ψ is trivial.

(i) Let ν and ρ be two characters of U F such that νρν0 is not 1 Let pm be its conductor Then



σ η(σ −1 ν, C n )η(σ −1 ρ, C p )C p+n (σ)

is equal to

η(ν −1 ρ −1 ν0−1 , C )z0 νρν0(−1)C n (ν)C p (ρ)

for all integers n and p.

(ii) Let ν be any character of U F and let ν = ν −1 ν −1

0 Then



σ η(σ −1 ν, C n )η(σ −1 ν, Cp)C p+n (σ)

for all integers n and p.

The left hand sums are taken over all characters σ of U F and δ n,p is Kronecker’s delta The

The termπ2(w)ϕis equal toν0(−1)ϕ.

We compute the Mellin transforms of both sides

so that the Mellin transform of the right side is

ν0(−1)t n 

η(σ −1 ν, −C n )η(ρ −1 σ −1 ν0−1 , −C p )z0−p C p+n (σ) ϕp (ρ) (2.11.1)

On the other hand

These two expressions are equal for all choice ofn, p, ρ, and ν

Ifρ = νand the conductor ofνρ −1ispmthe gaussian sumη(ρν −1 , C r)is zero unlessr = −m−.

Replacingρbyρ −1 ν0−1we obtain the first part of the proposition

Ifρ = νthenδ(ρν −1) = 1 Moreover, as is well-known and easily verified,η(ρν −1 , C r) = 1if

Proposition(2.12) (i) For every n, p, ν and ρ

C n (ν)C p (ρ) = C p (ρ)C n (ν)

(ii) There is no non-trivial subspace of X invariant under all the operators C n (ν).

(iii) The space X is one-dimensional.

Supposeρνν0= 1 The left side of the first identity in the previous proposition is symmetric in

the two pairs(n, ν)and(p, ρ) Since(η −1 ρ −1 ν0−1 , C )is not zero we conclude that

for all choices ofnandp The first part of the proposition is therefore valid in ρ = ν.

Now supposeρ = ν We are going to that if(p, n)is a given pair of integers andubelongs toX

then

C n+r (ν)C p+r(ν)u = C p+r(ν)C n+r (ν)u

for allr inZ Ifr  0both sides are0and the relation is valid so the proof can proceed by induction

onr For the induction one uses the second relation of Proposition 2.11 in the same way as the first was

also takes values inX1 Therefore all we need to do is show that ifϕis inV1thenπ(w)ϕtakes values

inX1 This follows immediately from the assumption and Proposition 2.10

To prove (iii) we show that the operatorsC n (ν)are all scalar multiples of the identity Because

of (i) we need only show that every linear transformation ofXwhich commutes with all the operators

C n (ν)is a scalar SupposeT is such an operator Ifϕbelongs toV letT ϕbe the function fromF ×to

at least forϕonV0and forϕ = π(w)ϕ0withϕ0inV0 We have already seen that the identity holds for

ϕinV0 Thus ifϕ = π(w)ϕ0the left side is

Theorem 2.13 Suppose an equivalence class of infinite-dimensional irreducible admissible tations of G F is given Then there exists exactly one space V of complex-valued functions on F × and exactly one representation π of G F on V which is in this class and which is such that

ifbis inB F andϕis inV  LetAbe an isomorphism ofV withV such thatAπ(g) = π  (g)Afor allg.

LetLbe the linear functional

Thus we need the following lemma

Lemma 2.13.2 If L is a linear functional on V satisfying (2.13.1) there is a scalar λ such that

L(ϕ) = λϕ(1).

This is a consequence of a slightly different lemma

Lemma 2.13.3 Suppose L is a linear functional on the space S(F × ) of locally constant compactly supported functions on F × such that

for all ϕ in S(F × ) and all x in F Then there is a scalar λ such that L(ϕ) = λϕ(1).

Suppose for a moment that the second lemma is true Then given a linear functional LonV

satisfying (2.13.1) there is aλsuch thatL(ϕ) = λϕ(1)for allϕinV0=S(F ×) TakexinF such that

which implies thatL(ϕ) = λϕ(1).

To prove the second lemma we have only to show thatϕ(1) = 0impliesL(ϕ) = 0 If we set ϕ(0) = 0thenϕbecomes a locally constant function with compact support inF Letϕ be its Fouriertransform so that

ϕ(a) =



F ψ(ba) ϕ (−b) db.

LetΩbe an open compact subset ofF × containing1 and the support ofϕ There is an idealainF

so that for allainΩthe function ϕ (−b)ψ(ba)is constant on the cosets ofainF Choose an idealb

containing aand the support ofϕ  IfSis a set of representatives ofb/aand ifcis the measure ofa

then

ϕ(a) = c

b ∈S ψ(ba)ϕ (−b).

Ifϕ0is the characteristic function ofΩthis relation may be written

Theorem 2.14 (i) For any ϕ in V set

W ϕ (g) =

π(g)ϕ(1)

so that W ϕ is a function in G F Let W (π, ψ) be the space of such functions The map ϕ → W ϕ is

an isomorphism of V with W (π, ψ) Moreover

= Aπ(g)ϕ(1) = ρ(g)Aϕ(1) = Aϕ(g)

the mapAis determined byL Also

ConsequentlyAϕ = λW ϕandW = W (π, ψ).

The realization of π onW (π, ψ) will be called the Whittaker model Observe that the sentation ofG F onW (ψ)contains no irreducible finite-dimensional representations In fact any suchrepresentation is of the form

containsW2and the representation of the quotient spaceW1/W2is equivalent toπ.

Proposition 2.15 Let π and π  be two infinite-dimensional irreducible representations of G F realized

in the Kirillov form on spaces V and V  Assume that the two quasi-characters defined by

for all ν then π = π 

Ifϕbelongs toS(F ×)then, by hypothesis,

π(w) ϕ(ν, t) = π  (w) ϕ(ν, t)

so thatπ(w)ϕ = π  (w)ϕ SinceV is spanned byS(F ×)andπ(w) S(F ×)andV is spanned byS(F ×)

andπ  (w) S(F ×)the spacesV andV are the same We have to show that

π(g)ϕ = π  (g)ϕ

for all ϕ in V and all g in G F This is clear if g is in P F so it is enough to verify it for g = w.

We have already observed that π(w)ϕ0 = π  (w)ϕ0 if ϕ0 is in S(F ×) so we need only show that

π(w)ϕ = π  (w)ϕifϕis of the formπ(w)ϕ0withϕ0inS(F ×) Butπ(w)ϕ = π2(w)ϕ0= ω( −1)ϕ0and,sinceπ(w)ϕ0= π  (w)ϕ0,π  (w)ϕ = ω (−1)ϕ0

LetN F be the group of matrices of the form



withx inF and letBbe the space of functions on G F invariant under left translations by elements

ofN F Bis invariant under right translations and the question of whether or not a given irreduciblerepresentation π is contained in B arises The answer is obviously positive when π = χ is one-dimensional for then the functiong → χ(detg)is itself contained inB

Assume that the representationπwhich is given in the Kirillov form acts onB Then there is amapAofV intoBsuch that

satisfies the relationAπ(g) = ρ(g)Aand takesV intoB Thusπis contained inBif an only if there is

a non-trivial linear formLonV which satisfies (2.15.1)

Lemma 2.15.2 If L is a linear form on S(F × ) which satisfies (2.15.1) for all x in F and for all ϕ

LetΩbe an open compact subset ofF ×containing the support ofϕand letp−nbe an ideal containing

Ω There is an idealaofF such thatϕ (−b)ψ(bx)is, as a function ofb, constant on cosets ofafor allx

inp−n Letbbe an ideal containing bothaand the support ofϕ  IfSis a set of representatives for thecosets ofainb, ifcis the measure ofa, and ifϕ0is the characteristic function ofΩthen

Thus any linear form onV verifying (2.15.1) annihilatesS(F ×) Conversely any linear form on

V annihilatingS(F ×)satisfies (2.15.1) because

is inS(F ×)ifϕis inV We have therefore proved

Proposition 2.16 For any infinite-dimensional irreducible representation π the following two erties are equivalent:

prop-(i) π is not contained in B.

(ii) The Kirillov model of π is realized in the space S(F × ).

A representation satisfying these two conditions will be called absolutely cuspidal

Lemma 2.16.1 Let π be an infinite-dimensional irreducible representation realized in the Kirillov form on the space V Then V0=S(F × ) is of finite codimension in V

We recall that V = V0+ π(w)V0 LetV1 be the space of allϕinV0 with support in U F Anelement ofπ(w)V0may always be written as a linear combination of functions of the form

Lemma 2.16.2 For any character µ of  U F there is an integer n0 and a family of constants λ i ,

Lemma 2.16.3 There is a finite set S of characters of U F such that for ν not in S the numbers

C n (ν) are 0 for all but finitely many n.

If µis not inS the functionη µ is inV0 ChooseµinS and letV µ be the space spanned by thefunctions

Ifϕis inV µ thenϕ(ν, t) = 0 unlessν = µand we may identifyϕwith the sequence{  ϕ n (µ) }.

The elements ofV µ ∩ V0are the elements corresponding to sequences with only finitely many nonzeroterms Referring to Proposition 2.10 we see that all of the sequences satisfying the recursion relation

forn ≥ n1 The integern1depends onϕ.

Lemma 2.16.1 is therefore a consequence of the following elementary lemma whose proof wepostpone to Paragraph 8

Lemma 2.16.4 Let λ i , 1 ≤ i ≤ p, be p complex numbers Let A be the space of all sequences {a n },

n ∈ Z for which there exist two integers n1 and n2 such that

Takep = −andn > − Then δ(n − p) = 0and

which, since almost all of the coefficientsC (ν)in the sum are zero, is the relation required

If ν = νtakep ≥ −and n > p Then η(σ −1 ν, C n) = 0unlessσ = ν and η(σ −1 ν, C p) = 0

There is certainly at least oneifor whichC i(ν) = 0 Takep − 1 −  ≥ i Then from (2.16.5) we deduce

a relation of the form

whereris a fixed integer andnis any integer greater thanp.

Lemma 2.16.3 is a consequence of the following more precise lemma Ifpm

is the conductor of acharacterρwe refer tomas the order ofρ.

Lemma 2.16.6 Let m0 be of the order ν0 and let m1 be an integer greater than m0 Write ν0 in any manner in the form ν0 = ν1−1 ν2−1 where the orders of ν1 and ν2 are strictly less than m1 If the order m of ρ is large enough

C (ρ) = ν2−1 ρ( −1)z0

η(ν1−1 ρ, C )

η(ν2ρ −1 , C )

and C p (ρ) = 0 if p = −2m − 2.

Suppose the order of ρ is at least m1 Then ρν1ν0 = ρν2−1 is still of order m Applying

Proposition 2.11 we see that



σ η(σ −1 ν1, C )η(σ −1 ρ, C )C (σ)

is equal to

η(ν1−1 ρ −1 ν0−1 , C )z0 ν1ρν0(−1)C n (ν)C p (ρ)

for all integersnandp Choose nsuch thatC n (ν1) = 0 Assume also that m + n +  ≥ − or that

m ≥ −2 − n Then η(σ −1 ν1, C ) = 0unlessσ = ν1so that

η(ν1−1 ρ, C )C (ν1) = η(ν2ρ −1 , C z0 ν1ρν0(−1)C n (ν1)C p (ρ).

Sinceν1−1 ρis still of ordermthe left side is zero unlessp = −2m − 2 The only term on the right side

that can vanish isC p (ρ) On the other hand ifp = −2m − 2we can cancel the termsC n (ν1)from bothside to obtain the relation of the lemma

Apart from Lemma 2.16.4 the proof of Lemma 2.16.1 is complete We have now to discuss itsconsequences Ifω1andω2are two quasi-characters ofF ×letB(ω1, ω2)be the space of all functions

abso-We take πin the Kirillov form SinceV0is invariant under the group P F the representationπ

defines a representationσ ofP F on the finite-dimensional spaceV /V0 It is clear that σ is trivial on

N F and that the kernel ofσis open The contragredient representation has the same properties Since

P F /N F is abelian there is a nonzero linear formLonV /V0such that

for alla1,a2, andx µ1andµ2are homomorphisms ofF ×intoC×which are necessarily continuous

Lmay be regarded as a linear form onV Then

onG F Ais clearly an injection ofV intoB(µ1, µ2)which commutes with the action ofG F

Before passing to the next theorem we make a few simple remarks Suppose π is an dimensional irreducible representation ofG F andωis a quasi-character ofF × It is clear thatW (ω ⊗

infinite-π, ψ)consists of the functions

g → W (g)ω(detg)

withW onW (π, ψ) If V is the space of the Kirillov model ofπthe space of the Kirillov model ofω ⊗ π

consists of the functionsa → ϕ(a)ω(a)withϕinV To see this takeπin the Kirillov form and observe