Weil's Representation and the Spectrum of the Metaplectic Group

Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann

530 Stephen S. Gelbart

Weirs Representation and the Spectrum of the Metaplectic Group

Springer-Verlau Berlin.Heidelberg-NewYork 1976

Author Stephen Samuel Gelbart Department of Mathematics Cornell University Ithaca, N.Y. 1 4 8 5 3 / U S A

Library of Congress Cataloging in Publication Data

Gelbart, Stephen S 1946Well's representation and the spectrum of the metaplectic group. (Lecture notes in mathematics ; 530) Bibliography: p. Includes index. 1. Lie groups. 2. Linear algebraic groups. 3. Representations of groups. 4. Automorphic forms. I. Title. II. Series: Lecture notes in mathematics (Berlin) ; 530. QA3.I28 no. 530 [QA387] 510'.8s [512'.55] 76-45609

AMS Subject Classifications (1970): 10 D 15, 22 E 50, 22 E55

ISBN 3-540-07799-5 Springer-Verlag Berlin 9 Heidelberg 9 New York ISBN 0-38?-0?799-5 Springer-Verlag New York 9 Heidelberg 9 Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under w 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. 9 by Springer-Verlag Berlin 9 Heidelberg 1976 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr.

For Mary

CONTENTS Introduction w

Background

w

Metaplectic

w

w

w

w

and

summary

groups

and

2.1.

Local

2.2.

Global

theory

2.3.

Well's

metaplectlc

2.4.

A philosophy

2.5.

Extending

2.6.

Theta-functions

AutomorDhic

Connections

3.3~

The

3.4.

Odds

11

theory:

13

representation

to

28

. . . . . . . .

39

GL 2 . . . . . . . .

41

. . . . . . . . . . . . . . . . . . . .

with

the

classical

of a u t o m o r p h i c

KrouD

. . . . . . . . .

theory

. . . . . . . . .

46 51 51

forms . . . . . . . . . . .

of the m e t a p l e c t i c

ends

22

. . . . . . . . . . .

representation

on th~ m C t a o l e c t i c

spectrum and

representation

for W e l l ' s

forms

Factorization

group

58

. . . . . . . . .

62

. . . . . . . . . . . . . . . . . . . . . .

69

arehimedean

places

. . . . . . . . . . . . . .

72

representation

theory

. . . . . . . . . . . . . .

72

4.1.

Basic

4.2.

The

4.3.

Application

4.4.

The b a s i c

local

theory:

map

. . . . . . . . . . . . . . . . . . . . .

of W e i l ' s

Well

representation

representation

the p - a d i c

96

representation

Class

i representations

5.3.

Hecke

operators

5.4.

The

5.5.

The b a s i c theory

Well

theory

. . . . . . . . . . . . . .

96

. . . . . . . . . . . . . . . .

102

. . . . . . . . . . . . . . . . . . . .

I05

. . . . . . . . . . . . . . . . . . . . .

111

representation

. . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . .

6.1.

The

discrete

6.2.

Construction

6.3.

Open p r o b l e m s

86

places . . . . . . . . . . . . . . .

Basic

map

81 .

93

5.2.

local

in 3 - v a r i a b l e s

. . . . . . . . . . . . .

5.1.

Global

. . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . .

Well's

3.2.

Local

representations

1

theory . . . . . . . . . . . . . . . . . . . . . .

3.1.

Local

of r e s u l t s . . . . . . . . . . . . . .

non-cuspid~l of cusp

forms

spectrum . . . . . . . . . . . of h a l f - i n t e g r a l

weight.

. . . . . . . . . . . . . . . . . . . . .

118 121 122 125 132

References . . . . . . . . . . . . . . . . . . . . . . . . . . . .

134

Subject

138

Index

. . . . . . . . . . . . . . . . . . . . . . . . . .

Introduction These notes are expanded from my earlier paper graphed at Cornell in July 1974.

[8] mimeo-

In addition to including correc-

tions and revisions for [8], the present notes contain new results and insights obtained during the past year and a half.

Some of

these results have already been described briefly in [9] and [i0]. Others are due to Roger Howe and Pierre Cartier. parts of Subsections pondence with Howe,

In particular,

2.4, 5.5, and 6.2 are taken from corresand parts of Subsections 2.5 and 3.1 were sug-

gested by Cartier after his critical reading of the original manuscript.

My indebtedness to both goes beyond acknowledgement

of

their suggestions within the text. The goal of these Notes is a general theory of automorphic form for the metapleetie group.

I am indebted to Robert Langlands

for inspiring this project and giving freely of his ideas. also grateful to Paul Sally for his collaboration Subsections 4.4 and 5.5), Kenneth Brown,

and my colleagues

Robert Strichartz,

on [i0]

at Cornell,

I am (cf.

especially

and William Waterhouse,

for

many helpful conversations. The theory of automorphic

forms on the metaplectic group

described in these Notes is still in its infancy. of the results,

if not incomplete,

Moreover,

are in preliminary form.

many I am

grateful to all the people named above for helping me realize this.

The expeditious typing of these Notes was done by Joanne Lewis, Arletta Havlik and Esther Monroe. S. Gelbart November 1975

B a c k ~ r g u n d a n d ,Sum/nar~ o f R e s u l t s .

w

The m e t a p l e c t i c group was f i r s t

i n t r o d u c e d by Well i n [47].

His purpose was to reformulate Siegel's analytic theory of quadratic forms in group theoretic terms~

The motivation for our investigation

comes from more recent works of Shimura and Kubota.

Our purpose

is to describe the spectrum of the metaplectic group modulo its subgroup of rational points and to relate this spectrum to the theory of automorphic forms for

GL(2).

Our work relies heavily upon Weil's. suggested,

However,

as already

it is more closely related to important recent discoveries

of Shimura and Kubota which we shall now briefly describe. Fix

k

to be an odd positive integer,

divisible by 4, and N.

X

N

a positive integer

a character of the integers defined modulo

Put

to(N)

=

{[~ bd] ~ sL2(~):

c

9

O(N)},

and e(z) =

~

exp(2vi n2z).

Then

le(Yz)/e(z)i for all

Y c to(N).

= [cz+dj I/2

(Hecke [16], pp. 919-940.)

In [38] Shimura deals with cusp forms

f(z)

satisfying the

identity

f(~z) = ~(d) E~(~z)/6(z)]kf(z) for all

F ~ ~o(N).

cus~ forms of weight [~(u

Such functions comprise the space of classical k/2, character

-I/2.

X, and "6-multiplier system"

We denote this space by

Sk/2(N,X ).

Func-

tions in it arise naturally in number theory from the study of partitions and quadratic forms in an odd number of variables.

To study functions in N

operators

Tk,x(m )

weight).

These

However,

~,~(m)

square and

Sk/2(N,X )

one introduces certain linear

(following Hecke in the case of forms of integral

T(m)

operate in

Sk/2(N,X)

for all integers

operates as the zero operator if

(m,N) = 1.

m

m.

is not a

This curious fact seems to have discouraged

Hecke from beginning a systematic study of such forms along the lines of his already successful theory for forms of integral weight. theless,

Shlmura established

Suppose

the following provocative

f(z) = Z a(n)exp(2~i n z)

eigenfunction

for every

T N x(p2), k,

in

result.

Sk/2(N,X )

is an

say

T(p2)f : ~,(p)fo Suppose also that

k > 3o

Then

n:l

.

= ~[l_~(p)p-s P where to

L*(X,.)

- - ~ - -s) + ~(p)2pk-2-2s]-l,

is essentially the Dirichlet L-series associated

y; furthermore,

and more significantly,

the inverse Mellln

transform of A ( n ) n -s : H [ l - k ( p ) p -s + X(p)2pk-2-2S] -I,

i

p

namely F(Z) =

~ A(n)exp(2vi n z), n=l

satisfies

F(~z)

for all

y e ~0(No).

usually equals

N/2. )

= X(d)2(cz+d)k-iF(z),

(Here

NO

depends only on

N

and

X

and

None-

The signlflcance

of Shimura's result is that it establishes

a correspondence between forms in (actually even) weight

k-1.

Sk/2(N,X)

Note that

and forms of integral

T(p)F = k(p)F.

Thus this

correspondence preserves eigenvalues for the Hecke ring. The success of Shlmura's theory leads one to ask several important questions.

For example,

can Shimura' correspondence be

defined without recourse to L-functions? group theoretic interpretation?

In particular, what is its

Is the correspondence one-to-one?

]~qthe first part of this paper I shall interpret forms of halfintegral weight as irreducible representations of the metaplectlc group "defined over

Q."

Thus, the full weight of the representation

theory of this group can be used to discuss these questions. Kubota's results also concern forms of "half-lntegral weight". To describe them, fix an imaginary quadratic field F =

denote by

0

the ring of integers of

power residue symbol in subgroup mod r(N)

Q(J'Z'd),

N

of

F.

SL(2,0)

Let

F,

F(N)

and by

(~)

the quadratic

denote the congruence

and define the function

X(y)

on

by

1

otherwise .

The starting point for Kubota's investigation is his discovery that

~

is a character of

congruent to

0

law for

modulo the fact that

(See [19]

(~)

mod 4).

F(N)

(provided

N,

as before,

is

This result is equivalent to the reciprocity ~d

is totally imaginary.

for the original proof and Subsection 2.2 of this paper

for the case of arbitrary number fields. Now let

H

denote the three-dimensional quaternionic upper

half-space whose points are of the form (z e C, v > 0).

The operation of

Z u = (z,v) = iv

~ = [c a bd ] ~ SL(2,~)

-V

~]'

on

H

is

given by

% (identifying to

t e ~

SL(2,~)/SU(2)~

-- (au+b) (cu+d)-l

with the matrix The quotient

[0

])

so

H

is isomorphic

space

r(~)/H is of finite

volume.

In [21] Kubota to

P(N)

considers

and character

X;

E(u,s)

modular

function~

his special

:

Z

on

interest

•

H

with respect

is in

s+l

r XZ(Ni where F|

s

is a complex

is the upper The series

variable,

triangular defining

only for

continuation

to the whole

for all

form which

s-plane

%(u)

of

over

Rather

~(~T~.

One of Kubota's

u = (z,v),

E(u,s)

series.

and a pole

of the first order at

at

s = 1/2

~

a function

the metaplectic

satisfies

automorphic

defines

a theta-function. of

on a certain

GL(2) two-fold

group of w

results

is a computation

of ~(u)

generalizes

the fact that the eigenvalues

of the classically

holomorphic

Eisenstein

) O]

arithmetic

functions

As already

in

to Hecke's

Jim(z)

ring.

of the

elgenvalues

series

one purpose

This result defined

are given by

such as the sum of the divisors

suggested,

Although

itself has an analytic

to the adele group

with respect

and

F(N).

In fact

it defines

principal

if

a s~are-inte~rable

be lifted

of this group,

of

E(u,s)

is not a ~usp form~ can not

= v

is an Eisenstein

~ l,

and defines

This function

covering

Re(s)

The residue

~ e ~(N)

subgroup

E(u,s)

it converges

s = 1/2.

v(u)

of an integer.

of our investigation

is

to reformulate

and extend the results

of Shimura and Kubota in the

context

of a general theory of automorphic

forms for the metaplectlc

group.

In the context of this general theory,

and Kubota appear to be closely related. shed llgat on both of them~ a tensor product local groups computation

Gv

Kubota's

"extraordinary"

@(u)

becomes

representations

defined at each place of

of eigenvalues

of Shimura

Thus it is hoped we have

In particular,

of certain

the results

F.

of the

The corresponding

then becomes part of the spherical

function

theory of these groups. In addition to working over arbitrary ntumber fields, are new.

our methods

In that our point of view is representation-theoretic

follow Jacquet-Langlands

([18]) rather closely.

we

In fact, a second

gaal of cur program is to develop a theory for the metapleetlc

group

analogous

GL(2).

to Jacquet-Langlands'

The possibility

treatment

of Hecke theory for

of such a theory was already

suggested by Shimura in

[38]. To be more precise, and

~

let

F

its rang of adeleso

denote an arbitrary

Let

G

denote

algebraic

group over

F.

The meta~lectic

extension

of

by

Zn,

GLn(~ )

aL 2

group

global field

regarded as an G~

is a central

the group of square roots of unity.

Thus

1

i s e x a c t and

Z2

Gin(F)

~atur.a.l unitary

Gs2(

is a trivial

of this extension points

s2

)

1

GL2(#)-module.

The c r u c i a l

is that it splits over the subgroup

and the fundamental representation

of

property

of rational

problem is to decompose the G~

in

Ln(GF~).

Suppose we denote this representation

by

~o

Then

T(gO)~(g) = ~ ( g go ) for all

g,

gO ~ ~ '

~d

m ~ L2(GF~)

~

Note t~at

L2(GF\~),

6

L2(G~"~,~A/Z2)__ (which

as a T - m o d u l e ,

is the direct sum of

L2(GF'~DL2(~)))

and the space of functions

~(~g) : ~ ( g ) , ("genuine" functions on

~)o

--~s

)

is Just

satisfying

~ e Z2)

Thus

T=T| where constituents of

T

correspond to aut0morphic forms on

The constituents of Langlands'

T

GL(2J.

comprise the subject matter of Jacquet-

treatment of Hecke's theory of forms of integral weight.

The irreducible constituents of

T,

on the other hand, are what we

call "genuine automorphic representations of

the metaplectic group",

or "generalized automorphic forms of half-integral weight over

F".

This terminology is apt since the forms considered by Shimura correspond to special forms on

~

defined over

is the starting point for our theory;

Q.

(This observation

see Subsection 3.1 for details.)

The Eisenstein series considered by Kubota lead to forms which are defined over a totallz ima~inar[ field

F,

and which occur outside

the space of cusp forms~ The main emphasis of our theory is on relations between automorphic forms on the metaplectic group and automorphic forms on GL(2), i.e. between constituents of

~

Eventually we shall define a map

between arbitrar[

necessarily automorphic) and representations of

S

and constituents of

representations of GA.

~

(i.e. not

non-trivlal on

Z2

This map will be consistent with

ShimuraTs when restricted to automorphic forms on it will be one-to-one,

T.

G~.

Moreover,

and its definition will be entirely represen-

tation theoretic. A correspondence and

~

D3

between certain representations of

is constructed completely independently of

theory of the metaplectic group. S. Niwa

[30] and T. Shintani

S

GA

using Well's

Motivated by recent results of

[hl] we collect evidence for the

assertion that

D3(s(w)) whenever

S(~)

=

is in the domain of

D3o

The validity of this

h y p o t h e s ~ is one of the focal points of our theory.

We also describe

other features of our theory and explain its connection with the results of Shimura and Kubotao

All our results lend support to the

assertion that forms on

and the metaplectic

GL(2)

group are inex-

tricably connected. A more precise summary of the contents of this paper now follows. In Section 2 we analyze the metaplectic over local and global fields. follow Kubota,

then Weil~

coverings of

GL(2)

In constructing these groups, we first

Whereas Kubota's construction involves an

explicit factor set and is well-suited

for basic computations,

construction is entirely representation-theoretic larly crucial to our approach.

In Subsections

Weil's

and hence particu-

2. h and 2.5 we explain

how Well's construction leads to a general philosophy which not only yields the map

D

alluded to above but also a correspondence

between automorphic forms on

~

and ~utomorphic forms on

GL 1.

These matters are dealt with in detail in Sections 4,5, and 6. speaking,

to each Well representation

quadratic form Dq

q

and to each

between irreducible

q

r

of

~

r 9 q

constituents of

r

q

In particular,

q3(xl,x2, x3) = x I yields the map

D3

alluded to above, while

ql(x) relates to Kubota's results and

Roughly

there corresponds

a

there is attached a correspondence and constituents

natural representation of the orthogonal group of of

D1

= x2

DIo

q

of the

in the space

In Section 3 we introduce paper.

the subject matter proper

Some general features of the decomposition of

are sketched and the connections between constituents

of this

L2(GF~) of this

decomposition and the functions considered by Shimura and Kubota are described in detail.

Some miscellaneous

used later on are also collected here.

results which will be

Our basic observation in

Section 3 is that one can make sense out of the relation ~=|

for certain irreducible that

GA

"genuine" representations

~.

(Note

This result is important since it reduces

global questions to local ones. S

of

will not be a restricted direct product of the local

covering groups ~v. )

map

V

alluded to above,

In particular,

in describing the

one is lead to the study of certain local

maps Sv: ~v ~ ~v for each place

v

of

F.

cuspidal spectrum of series on

In this

L2(GF~)

Section we also isolate the non-

using the theory of Eisenstein

G~o

In Section 4 we treat the local map for the archimedean places in complete detail. of

Fv = ~

as

Gv

the map

or

~,

To certain pairs of quasi-characters irreducible representations

are described. Sv

If

~--v is attached

is defined by setting

representation

of

Gv

attached to

Sv(~v ) (~12,~)o

to

of

Gv

(~i,~2)

lowest weight Gv

as well then

equal to the irreducible This is consistent with

Shimura's map since a discrete series representation of k ~

(~I,~2)

~v

with

is mapped to a discrete series representation of

with lowest weight

k-lo

(It is consistent with Kubota since it

attaches the trivial representation of series representation of index

G@

s = i/2o)

to the complementary

In Subsection attached

4.4 we describe

to the quadratic

form

the problem of decomposing forms in an odd number it deserves.

This involves attached

the decomposition form

GR

S v. G~

of a map

In particular, corresponds

q3(xl~x2,X3)o

~k-i ~ ~i/2

inverse

to

S v.

r3

The orthogonal

group r3

of a certain regular representation to (most)

and t~as one obtains

to the discrete

of

representations an inverse

series representation

series representation

This result seems to be new~

result in

and the idea is to decompose

~

the discrete

to quadratic

of the Well representation

GR

of

In general,

attached

D~

The result is that one attaches

a representation

~.

The most significant

the construction

according to the decomposition

of

over

rI

has not yet received the attention

[52].)

of this form is essentially

itself~

decomposes

of variables

to the quadmatic

G~

q!

Well representations

(See, however,

Section 4 concerns

how the Weil representation

to

~k-i

~k/2

of

of

~.

A classical version of the correspondence

(using theta functions

was first obtained by T~ Shint~li

in place

of Well's

representation)

([41])o

In Section 5 we begin the local tDeory for non-archimedean places by describing tions

Sv

for a wide class of irreducible

(the non-supercuspidal

of quasi-characters

of

Fv

representations).

We also analyze

Hecke algebra~

showing that our map is consistent finite places preserved

as well,

i.eo,

(cf. Theorem 5.13).

the basic non-archimedean gether with the results correspondence

DI

we define

and inducing up to

the class 1 representations

for the generalized

After attaching pairs

to such representations

again by squaring these characters

representa-

and compute

These results

Sv

G v. eigenvalues

are useful in

with Shimura and Kubota at the

eigenvalues

for the Hecke ring are

In Subsection

Well representation

of Subsection

5.5 we describe rI

how

decomposes.

To-

~.h this leads to the global

alluded to above.

In Section 6 we attempt

to tie together the local threads

of

10 Sections

A and 5.

Our goal is a global description

of the space of automorphic

(genuine)

the theory of Well represenbations

forms on

expounded

locally with the theory of Eisenstein The result is a complete discrete non-cuspidal the implication

series

characterization

spectrum of

of the constituents

~.

Roughly speaking,

in Section 2 is mixed sketched

in Section 3.

in Subsection

L2(G~).

6.1 of the

In classical

is that any square-integrable

modular

terms

form of half-

integral weight which is not a cusp form must be a "translate" the basic theta-function be generated by residues kind of Siegel-Well In Subsection Here ignorance

6(z).

series,

(cf. [48] and

this amounts

essentially

dominates

6.3 describes

the situation.

global constituents

bute to the space of cusp forms~

Similar results

some speculations

to return to in future papers~

to a

[i0])~

6.2 we treat the cuspidal spectrum of

shown that all "non-trivial"

Subsection

Since such forms are also shown to

of Eisenstein

formula

of

of

L2(GF~).

Nevertheless rI

it is

indeed contri-

are discussed

for

and problems which we hope

r 3.

{2.

Metaplectlc

Groups and .Representations.

In [21] Kubota constructs of

GL2(g)

a non-trivlal

over a totally imaginary

number field.

I shall describe the basic properties arbitrary number field and complete struction

at the same time.

two-fold

covering group

In this Section

of such a group over an

some details of Kubota's

I shall also recall Well's

con-

construction

in a form suitable for our purposes. I start by discussing

the local theory and first collect

elementary facts about topological Let

G

denote a group and

as a trivial G-space. on

G

(2.1)

group extensions.

T

a subgroup of the torus regarded

A tvp-cocyc.le

is a map from

Ox G

to

T

some

(or multiplie.r,

or factor

set)

satisfying

~(glg2,g3~(gl, g2 ) = c(gl, g2g3)~(g2, g3)

and

(2.2)

~(g,e)

for all

g'gl

in

G .

=~(e,g)

in additlonl

if

= 1

G

is locally compact,

a

will be called Borel if it is Borel measurable. Following Moore 2-cocycles

on

G

cocycles

(cocycles

from

to

G

T).

dimensional represents G

by

T

~

let

and let

Z2(G,T)

B2(G,T)

of the form

equivalence

denote the group of Borel

denote

its subgroup of "trivial"

s(gl) s(g2)s(glg2 )-I

Then the quotient

cohomology

group of

G

group

H2(G,T)

with

a map

(the two-

with coefficients

classes of topological

s

in

coverings

T) groups of

which are central as group extensions.

To see this, class

[28],

in

let

H2(G,T). GX T

~

be a representative Form the Borel space

multiplication

in

by

(2.3)

[gl'~;1][g2'~;2)

= [glg2'~(gl'g2)r

of the cohomology GX T

and define

12

One can check that product for

G• T

of H a a r m e a s u r e s

G X T.

topology

G

and

invarlant

admits

a unique

compatible

with

the g i v e n

Borel

structure.

that

the n a t u r a l

from

locally

sequence

of

~/T

compact

class

of

The

with

~

measure

locally

and

latter map,

G.

that the

~

to

Borel,

compact

G

are

hence

moreover,

Thus we have

an exact

groups

as a group e

to

and

and o b v i o u s l y

continuous.)

is c e n t r a l

the e q u i v a l e n c e

g ~ (g, I)

T

(They are homomorphisms,

a homeomorphism of

maps

i ~ T -~ ~ - ~ G-~

~:

is an

G• T =~

they are a u t o m a t i c a l l y

This

T

group

[25]

continuous.

sequence

on

Borel

Thus by

Note

induces

is a standard

.

i .

extension

Its natural

and depends

Borel

only on

cross-sectlon

is

2. I. Local T heor[. Let

F

denote a local field oT zero characteristic.

is archimedean,

F

is

is a finite algebraic If of

F,

F U

or

extension

of

P .

Let

if

F

let P

0

denote

denote

F

Qp

the ring of integers

its m a x i m a l prime

q = lwl-I

F

is non-archimedean,

of the p-adic field

its group of units,

ideal,

the residual

and

characteristic

F . The local m e t a p l e c t l c

GL2(F) of

C;

is non-archlmedean,

a generator of

~q

If

which

involves

group

is defined by a two-cocycle

the Hilbert or quadratic

on

norm residue

symbol

F . o

The Hilbert F xx Fx

to

Z2

F(J~).

from

square.

(',')

which takes

(',')

Fx• Fx

to

(x,y)

itself

if

I

iff

x

in

is identically

is trivial on

Some properties use throughout

is a symmetric b i l i n e a r map from

(x,y)

In particular,

Thus

trivial on

sy~ibol

(FX) 2 • (FX) 2

Fx 1

Is a norm

if

y

is a

for every

F

and

F = ~ .

of the Hilbert

symbol which we shall repeatedly

this paper are collected below.

Proposition

2.1.

(2.41

(i) For each

(a,bl

= (~,-ab)

F,

(',.)

is continuous,

= (a,(l-a)bi

,

and

(2.5

(a,b)

= (-ab,a+b)

(ii)

If

q

is odd,

(u,v)

(ill)

If

q

is even,

and

then

(u,v)

is ident.lcally

1

is identically v on

The proof of this P r o p o s i t i o n of O'meara

[31] and Chapter

Now suppose or

d

according

c

in

U

is non-zero

i

on

is such that

U x U; vml(4),

U . can be gleaned from Section 63

12 of A r t i n - T a t e

ab s = [cd] 6 SL2(F) if

;

and set or not.

[i]. x(s)

equal to

c

14

Theorem 2.2.

(2.6)

The map

~: SL2(F) • SL2(F) ~ Z 2,

defined by

~(Sl, S2) :(X(Sl),X(S2))(-X(Sl)X(S2),X(SlS2) ) ,

is s Borel two-cocycle cohomologically

trivial

This Theorem preliminary

on

SL2(F ).

Moreover,

if and only if

this cocycle

F = @ .

is the main result of [20].

remarks

it determines

is

According

to our

an exact sequence of topological

groups

1 ~ z2 ~ s~2(F) ~ SL2(F) ~ I where

SL2(F)

is realized

plication given by (2.3). not the product suppose

N

The topology for

topology of SL2(F)

is a neighborhood

Then a neighborhood where

as the group of pairs

UeN

and

Proposition extension of

2. 3 .

SL2(F)

Z2

SL2(F)

F/@,

by

unless

is provided

is identically

If

SL2(F),

Z2

with multl-

however, F = @ .

basis for the identity

basis for

~(U,U)

and

[s,~]

in

is Indeed,

SL2(F).

by the sets

(U, 1)

one.

each non-trivial

is isomorphic

topological

to the group

SL2(F )

just constructed. Proof.

If

F =~,

nected Lie group, SL2(F)

isomorphic

obtained by factoring

on the other hand, shown

hence

each such extension

is that

that

F

is automatically

to "the" two-sheeted

its universal

cover by

is non-archlmedean.

H2(SL2(F),Z2) = Z 2.

a con-

cover of

2~.

Suppose,

What has to be

For this we appeal to a result

of C. Moore's. Let in

F .

EF

denote the (finite cyclic)

Consider

the short exact sequence

l~Z 2 ~ ~ / z The corresponding

group of roots of unity -

2~

l

long exact sequence of cohomology groups

is

15

'''~I(s~(F),~/Z2)

~2(SL2(F),Z 2) ~H2(SS2(F),S ;)

H2(SL2(F),EF/Z2) +H3(SL2(F),Z2) ~''' Recall that

HI(sL2(F),T ) =Hom(SL2(F),T).

its commutator subgroup. H2(SL2(F),Z 2)

Therefore

Moreover

. SL2(F)

equals

Hl(sL2(F),EF/Z2) =[1]

imbeds as a subgroup of

Theorem 10.3 of [29] it follows that

H2(SL2(F),EF).

But from

H2(SL2(F),EF) = E F .

deaired conclusion follows from the fact that

and

Thus the

H2(SL2(F),Z 2)

is

non-trlvial and each of its elements obviously has order at most two.

[] Remark 2.4.

of

SL2(F)

for

As already remarked, a non-trlvial two-fold cover F

a n on-archimedean field seems first to have

been constructed by Well in [47].

His construction, which we shall

recall in Subsection 2.3, is really an existence proof.

His general

theory leads first to an extension

where

on

T

L2(F).

is the torus

and

Mp ( 2 )

i s a group o f u n i t a r y

Then it is shown that

element of order two in Remark 2. 5 .

Mp(2)

operators

determines a non-trlvlal

H2(SL2(F),T).

In [20] Kubota constructs n-fold covers of

SL2(F ).

His idea is to replace Hilbert's symbol in (2.6) by the n-th power norm residue symbol in of unity).

F

(assuming

F

contains the n-th roots

In [29] Moore treats similar questions for a wider range

of classical p-adic linear groups. Now we must extend

~

to

a = as2(F) In fact we shall describe a two-fold cover of non-central extension of

SL2(F)

by

F x,

G

which is a trivial

i.e. a seml-direct product

16

of these groups. ab If g = [cd]

belongs to

G,

P(g) = [

ca

(2.7)

For

gl'g2

in

(2.8)

G,

write

i 0 g = [0 det(g) ]p(g)

where

~cSL2(F)

define

cz*(gl, g2) =ct(p(gl )det(g2) , p ( g 2 ) )

v (det(g2),p(gl))

where I 0 -I

(2.9)

I 0

sY = [0 y]

S[o y]

and

=f

(2. lO) if

v ( y , s)

s = [c

]'

Note t h a t

coincides with

e/O

otherwise

the restriction

If

{sY,~v(y,s)].

y c F x, Then

of

and

s-~s y

and the seml-direct product of isomorphic to the covering group Proof.

i~

c~~

to

SL2(F) X SL2(F)

a 9

Proposltion 2.6. equal to

1

\ (y,d)

~=[s,~]

c~2(F) ,

put

is an automorphism of

SL2(F)

and

~

G

of

Fx

~Y

SL2(F),

it determines

is

determined by (2.10).

It suffices to prove that

~ ( s l, s 2) = ~ ( s l Y , s2Y) v ( y , s 1) v ( y , s 2) v(y, s 1 s 2) and this is verified Remark 2.7.

in Kubota

[21] by direct computation.

We shall refer to

~

as "the" metaplectic

even though there are several (cohomologically extend [g,~]

~ with

to

G 9 g ~ G,

distinct)

group

ways to

We shall also realize it as the set of pairs ~ ~ Z 2,

and multiplication

described by

[gl,~l][g2,~2] = {gl, g2,~*(gl, g2)~l~2] Now let B,A,N, and K denote the usual subgroups of 9

Thus

G 9

17

A=

, ai~F

,

a2

a2

and

K

is the standard maximal

F =~,

0(2) If

inverse

if

H

F =R,

and

GL(2,0)

is an[ subgroup of

image in

~ .

isomorphic

to

H .

G,

Moreover,

is the direct product of

Z2

We shall denote

it is important

over the subgroups below

is useful

that if

x

by

~

splits over

H'

by

Proposition is a positive

H,

H'

H

even though

H'

H .

In particular,

field

~

splits

the Proposition

in constructing the global metaplectic

x=w~

then

of

to know whether or not

listed above.

if

will denote its complete

belongs to a non-archimedean

by the equation

G (U(2)

otherwise). H

if

subgroup of

and some subgroup

need not be uniquely determined In general,

compact

group.

(Recall

its order is defined

u eU.)

2.8.

Suppose

F/C

integer divisible

by

or h .

~

and

Then

N

~

(as usual)

splits over the

compact group =

More precisely,

ab [[c d ] 6K:

s([ c d]) -

Then for all

(2.12)

c ~0(mod

~)]

set

f(c,d

det(g))

if

cd~0

L

1

otherwise

and

ord(c)

is odd

gl, g 2 e K N ,

~*(gl, g 2) = S(gl)s(g2)s(glg2) -1

Proof. gl, g 2

ab g = [c d ] e G,

for

a b

(2.11)

a ~l,

in

Theorem e of [21] asserts ab [[c d ] e K:

ab [c d ] ~ 12(m~

that (2.12)

N)].

is valid for all

But careful

inspection

18

of Kubota's proof reveals that the conditions d i l(mod N)

are superfluous.

b--0(mod N)

and

Indeed Kubota's proof is computational

and the crucial observation which makes it ~ork is the Lemma below. We include its proof since Kubota does not. Lemma 2..9.

a b d ] c KN 9 k = [c

Suppose

s(k) =f(c,d(det k)) (2.13)

\

Proof.

1

and

d =0

-bc~U

c~0

~

[ac ~d]b

KN .

In particular,

U .

implies that

implies

c~U

otherwise.

belongs to

Suppose first that

c~U

c%0

Throughout this proof assume

dot(k) --ad-bc

Indeed

if

Then

and

c ~U

dot(k) =-bc,

Thus if

.

Then clearly

cd~0

.

a contradiction since

order (c) is odd,

s(k) = (c~d(det(k)) by definition.

On the other hand,

if

ord(c) = 2 n

(where

n~0

since

c~U)

it remains to prove that (c,d(det k)) = 1 . But

d(det k) = a d 2 - d b c ,

so

kcK N

implies that

d(det k) -~l(mod 4).

Thus (c,d(det(k))) = (w2nu, u ') = (u,u') = i by (iii) of Proposition 2.1. To complete the proof it suffices to verify that c~U

if

cd~0

and

since if

c cU,

then

ord(c)

is odd.

ord(c) =0,

c ~0

and

This is obvious, however,

a contradiction,

since

ord(c)

must be odd. Note that when the residual characteristic of

~ - - K : GL(2,0F).

F~ F

is odd,

19

Definitien 9.10. function on let

s(g)

G

KNX KN and

C,

let

s(g)

If

F

denote the is non-archimedean,

~(gl, gg)

denote

~*(gl,g2 ) s(gl) s(g2)s(glg 2) ~

determines a covering group of

G

isomorphic

But according to Proposition 2.8, its restriction is identically one.

KN

or

which is identically one.

Obviously ~ .

F =~

be as in (9.11), and in general, let

the factor set

to

If

Thus

lifts as a subgroup of

~N ~

is isomorphic to

via the map

k ~ {k,l].

this reason we shall henceforth deal exclusively with Lemma 9. II.

K~• Z 2 , For

~ .

Suppose

F

~i

gi =

xi] i=1,2

~ B,

o

Then

~(gl, g2) = (~i,~2) Proof.

Since

~i

xi

0

hi

=

0

~i

~i~DL 0

det(gi)

xi ~[l

P(gi)

,

it follows that

= (~{I,~I)(-WIIw2

-i

,

But using (2.4) together with the symmetry and billnearlty of Hllbert's symbol, this last expression is easily seen to equal Thus the Lemma follows from the identity

(Wl,~2).

20

(2.1~)

sIE~0 ~Jl = 1

valid for all

[

Corollary central in

~

Proof. ~.

Then

/3(y,y')

~] r B.

[]

2.12.

Suppose

iff y

is a square in

Suppose

Y = [~ 0].

V' = {Y',C']

Then

is

A. is a r b i t r a r y

= [[~0' O,],C,

V V' = Y' Y iff {YY',~(Y,Y')CC] = #3(y',y)

V = {Y,~

= [Y'Y,~(Y',Y)~']

in

iff

iff

(2.z5)

(~,~,)

= (~,,x) 4

So suppose

first that

are squares

in

one for all

Fx

y

and

by the n o n - t r i v i a l i t y

that

obtains by default

if

say,

= -i

(2.15)

Corollary

2.12'.

So does

~,

A2

2.13.

~' = I, say.

Thus

and the proof is complete.

The subgroup

N

of

G

The center of

z(g) (b)

F J g. ~

Then:

is

z o = {[[o z]'g]:

z ~

(F x) 2]

Suppose

-'2 G = [{g,(:] Then the center of

-J2 G

is

~'G:

det(g) c (FX) 2]

g = [[[~ 0z ] , ( ~ ] :

V

will not []

lifts as a subgroup

with

Fix

F x, then

h' ~ F x

Obvious.

Corollary (a)

and

symbol,

A 2 = {y ~ A: y = 8 2 , 6 ~ A] 9 Proof.

~

(both sides equal

is not a square in

for some

fails for

I 0 {[0 ~,],I}

commute with

~.

Then both

(2.15)

of Hilbert's

(~,k')

of

A.

~',k').

On the other hand,

This means

is a square in

z ~ F x]

9

21

if

Proof.

Since

g=[g,~]

eZ(~)

g = [~ oZ]

with

z eF x,

it suffices to prove that

commutes with ever[ Clearly

~(g,, [~ oz]),

only if

g' =([ca b ],~') e G

[[~ 0],~]

commutes with

iff g'

geZ(G),

i.e. only {[0z o],r z

z

is a square in iff ~([0z O z ]' g, ) =

Fx

But a simple computation shows that

a,(K

>=

= T(q(X,Y)),

The identity

(X,Y c V)

where

q(X,Y)

=

q(X+Y)-q(X)-q(Y),

establishes a self-duality for the additive group of equipped with its obvious topology. applicable

in particular

to

G = V

V

Thus Well's theory will be together with

(q,T).

This

example in fact will suffice for the applications we have in mind. For each

w = (u,u*)

the unitary operator in

u'(w)~(x)

~n L2(G)

G x G*

let

defin@d by

= <x,u*>~x~u).

U'(w)

denote

31

Then

U'(Wl)U'(w2) = F(Wl, W2)U'(Wl+W2)

where

(Wl, W 2) = if

W i = (Ui, U[).

That is,

U'

of

G • G*

with multiplier

I/F

U(w,t) comprises

(w e G ~ G, t e T) law given by

(wl, tl)(w2, t 2) = (Wl+W2, F(Wl,W2)tlt2).

In particular, extension G

and the family of operators

a group with composition

(2.38)

of

= tU'(w)

representation

is a multiplier

the multiplier

of

G x G*

by

and denoted by

exponentiation

A(G) o

0

i

x*

0

0

i

T

[U(w,t)]

The crucial fact is that in fact, U(w,t)

U'

G = ~, A(G)

group

is (the

group

A(G);

A(G)

may be viewed (2.34)

is an irreducible

is the unique irreducible

or as the

fixed.

Therefore

representation;

representation

of

at least the first

part of the result below is plausible. Theorem 2.20 automorphisms the normalizer

of of

(Segal). A(G)

Let

B0(G )

leaving

~

in

T

denote the group of

pointwise

L2(G),

fixed,

and let

Bo(a) be the natural projection. In other words,

an

(in which case it is denoted by ](G)).

U(w,t)

pointwise

determines

the Heisenberg

equipped with the group law

group of operators

T

Yn case

is the center of

G • G* • T

which leaves

which is cal~ed

of) the familiar Heisenberg

In general, either as

T

representation

Then

PO

is onto with kernel

there exists a multiplier

T.

representation

A(G)

32

of

B0(G)

in

L2(G)

whose range (choosing the constant in all

possible ways) coincides with

BO--~-~.

formula

defines an irreducible unitary

uS(w,t) = U((w,t))s)

representation of

A(G)

equivalent to

Therefore

U.

which fixed

some unitary operator a scalar in

Indeed if

A(G)

U((w,t)s)

r(s)

in

s c B0(G),

pointwise and hence is

: r-l(s)U(w,t)r(s)

L2(G)

(uniquely determined up to

r(s)

determines a multlp]!er representation of representation)

satisfying

P0(r(s))

B0(G )

(the Well

: So

This "abstract" Weil representation is relevant to B0(G)

is the semi-direct product of

abstract symplectic group which reduces to More precisely,

s~nce

s

in

B0(G )

of the form

(w,l) ~ (w~,f(w)).

form

where (w,t)s

a

Thus on

= (w,t)(a,f)

is an automorphism of

(GX G*)* SL2(F)

fixes

completely determined by its restriction to

(~,f)

for

T) and s ~

since

the

SL(2) and an

when

G = F.

(l,t), it is

G• G* • [i]

G • G* • T

where it is

it is of the

= (wa,f(w)t),

G • G*, f~ G • G* ~ T

is continuous,

and

f(w1+w2) F(Wl,W21 : f(wl)f(w7

F(Wl~,W2~)

(2.35) Conversely,

each pair

automorphism of

(o,f)

A(G)

fixing

satisfying (2.35) defines an T

pointwise,

so

B0(G):[(~,f)] ,

with group law

(~,f)(~,~,) : (~,,f") if

f"(w)

: f(w)f'(w~).

Now let

Sp(G)

automorphlsms of

denote the abstract symplectic group of

G ~ G*

which leave invariant the bicharacter

33

F(Wl,%) <Xl,X~> F(~2,~l) = ~ Using

(2.35)

one checks

(~i = (xi'x~))

that if

(a,f) = s e Bo(G)

then

a r Sp(G).

Our claim was that the exact sequence

(2.36)

1 + (GX G*)* + B0(G) ~ Sp(G) + 1

actually

splits.

To check this it is convenient m a t r i x form.

Since each

to describe

in

Sp(G)

is of the form

~ e A u t ( G x G*)

(x,x*) ~ (x,x~)(y ~) ~:G ~ G, t3:G ~ G*, u

where

+ G*, we s h a l l ,

following

Well,

write o=

[u

,

and define ~I = [ _ ~

where

~*

denotes

~ A u t ( G • G*)

-

the a p p r o p r i a t e

is symplectic

a bd ] = a ~ Sp(G), [c

Given

],

map dual to

~.

Then

iff a~ I = I. define

fq

on

G • G*

by

f~ (u, u~) = < 2 - 1 u ~ y 6 ~, u~> . Here we are assuming, automorphism

of

G.

as we do throughout, Then

f~

and

a

that

satisfy

x ~ 2x (2.35)

is an

and the map

-, (q, f) is a m o n o m o r p h i s m In short,

(2.37) and

Bo(G )

of

sp(G)

Theorem

into

B0(G )

2.20 produces

an exact sequence

1 ~ T-~ BO--(-C7, B0(G) + 1 contains

a copy of

which splits

Sp(G).

(2.36).

Example 2.21. in Example 2.19. Sp(G)

Suppose

Then

F

SL2(F)

is local and

obviously satisfies

(Since

F

to act on

~* = a

a~ i = i; note that

in

representation attached to

L2(V) (q~V)

via

~ e F, ~ = [a c ~]eSL(2,F)

Sp(G) = SL2(F)

one can associate to each quadratic form SL(F)

V

for each

but in all other cases properly contains it.)

representation of

is as

can be imbedded homomorphically in

by allowing each element of

scalar multiplication.

G = (V,q,T)

if

G = V

From this it follows that

(q~V)

a natural projective

which we shall call the Well and denote by

rq.

Before further analyzing this representation we need to adapt Weil'sgeneraltheorytothespecialcontext Recall that

V

is a vector space over

additive character of to the linear dual by

[X,X*].

V*

F

of Examples 2.19 and 2.21. F

and

~

is a non-trlvlal

(fixed once and for all). we shall denote its value at

The natural isomorphism between

V

and

If

X*

belongs

X

in

V

V*

is then

[X,Y] = q(X,Y). Using

[.,.]

in place of

the theory Just sketched, (a)

B(Zl, Z2) = [XI,X~]

(b)

the Heisenberg group

A(V)

(c)

the symplectic group

Sp(V)

in place of

(d)

the pseudo-symplectic

group

Ps(V)

a e Aut(VXV*)

(2.38)

Ps(V)

belongs to

in place of

is of the form Sp(V)

and

A(G); Sp(G); and

in place of (~,f)

f : V • V* ~ F

B0(G).

where satisfies

f ( z l + z 2 ) - f ( Z l ) - f ( z 2) = B(Zl~,Z2~)-B(zl, z2).

This relation, of course,

is the linearlization of (2.35):

of both sides of it yields (2.35). Ps(V)

in place of the

F(WI,W2);

A typical element of

f

one can now "linearlze"

introducing:

the bilinear form

blcharacter



is the subgroup of

quadratic;

B0(G)

Note that

taking

Ps(V) ~B0(G).

consisting of pairs

(o,f)

In fact with

cf. (2.39) below.

Note now that (2.38) associates to each and only one quadratic form on

VxV*.

o

in

Sp(V)

~We are assuming

one F

35

has characteristic to two.)

zero, in particular,

characteristic not equal

Therefore the homomorphism

(a,f) § f is an isomorphism between

Ps(V)

Since the same map from isomorphic to

(G•

and

B0(G)

Sp(V) to

Sp(G)

the analogy between the exponentiated

and unexponentiated

~heories breaks down here.

(2.39)

(~,f)

does imbed

Ps(V)

obtain an extension of

However,

the map

~ > (~,~~

homomorphically

1

has kernel

Ps(V)

> T

in

Bo(G)

so we can still

by pulling back (2.37) through

> Mp(V)

~ > Sp(V)

~:

> 1

I The group Mp(v) ={(s,~) ~ P s ( V ) x ~ ) :

p0(~) =~(s)]

is Well's general metaplectic group. of

sp(v)

by

It is a central extension

T .

Weil's results concerning the non-triviality of

Mp(V)

include

the following: (a) Z2,

Mp(V)

always reduces to an extension of

i.e. the cohomology class it determines in

order

Sp(V)

H2(G,T)

by is of

2 ; (b)

the extension

particular,

if

F~@,

Mp(V) and

a non-trivial cover of

V

Sp(V)

is in general non-trivial; is one-dimensional, by

Z2 .

These results yield a (topological) (2.4o)

l~

z2 ~ % 7 ( v )

Mp(V)

central extension

~ sp(v) + i

in is always

36

which by Proposition

2.3 must

coincide

with

1 ~ Z 2 ~ ~r2(F ) ~ S ~ ( F ) when

V=F.

plectic

I.e.,

cover of

Although explicit

Well's

metaplectic

group generalizes

SL2(F )

constructed

in 2.2.

Well's

factor

construction

set for

S-L2(F)

as a group of operators. for this group. explicit

These

of

F

whose

basis

XI,...,X n

= ~i x , Ti(Y)

with

If

F

of

Let

on

F,

is non-archimedean~

Consequently

over

F

and in

transform

T

is the canonical

fix an orthogonal

X :~xiXi, Ti

then

denote

and let normalized

character

q(X) =q(xl,...,Xn)

the character

diY

denote

the

as above.

the limit

= me~lim~pm~i(y2)diY

(by Well

[AT]

it is an eighth

root of unity).

the invariant

v(q,~) is well-defined. equal to

forms

(q,V),

if

i = l,...,n,

Y(Ti)

this group

y

Given

q(Xi'Xi)" F,

realize

such that the Fourier

that

such that

Haar measure

is known to exist

F

OF .

V

~i=5

= T(giY)

corresponding

of

on

Recall

is

yield an

a host of representations

to quadratic

= ~f(y)~(2xy)d

(x) =f(-x). conductor

it provides

the meta-

as follows.

dTy

~(x)

immediately

it does a priori

correspond

Fix a Haar measure

(f)

does not

In fact,

terms are realized

satisfies

~ 1

exp(88

If

F=R, sgn(%i) )

if

F =C,

set

Y(q,T)

on

L2(V)

is defined

=

and

n ~(~i )

i=l

~i(x)

and define

identically to be

= exp(~

equal

r

set

Y(q,T)

as above.

to i .

The Fourier

y(~i ) Finally, transform

37 $(X) = ~ %(Y)T(q(X,Y))dY V n

where

dY=

Z diY 9 i=l

Theorem of

F

2.22.

defined

Suppose

by

we use

For each Tt(x)

(q,V,T)

Then a cross-section provided

t eF x

= ~(tx)

S-p(V)

denote

the character

y(q,r

in

)

Sp(V)

SL2(F)

by

y(q,t).

(and

(c.f.

B0(V)).

(2.40))

is

by the maps

1 b r( 1 b ) ~ ( X ) [0 1 ] + [0 1 ]

(2.42)

[1

0 -i

(~ e L2(V)).

=

r( 0 -i

0] ~

i1

0 ])~(x)

More precisely,

to a multiplier

for each

representation

of

(X))~(X)

Tt(bq

= ~(q't)-l~(-X) these maps

t e F x,

SL2(F)

L2(V)

in

extend

whose associated

is of order two.

Remark respect

SL2(F)

over

(2.4l)

cocyele

Tt

and denote

to imbed

of

let

to

2.23. Tt

In (2.42)

the Fourier

and Haar measure A

transform

dtY = Itln/2dy.

(X) = ~ 9(Y)~t(q(X,Y))Itln/2dy

is taken with Thus

9

v Proof

of Theorem 2.22. Without loss of generality we assume n t :I. Since d Y : H d.Y it is easy to check that the operators t:l ~ in question are tensor products of the operators in L2(F) corresponding

to

~i'

namely

r([ol ~])f(•

= ~i(b~2)f~(xl,

and

r([ ~ -IO]If(x) =~(~)-l?(-x) i = l,...,n. q(x) = x 2,

Thus the Theorem precisely

We note that

Theorem

SL2(F)

is reduced

,

to the case

I.A.I of Sha!ika

is generated

V:F,

[35].

by elements

of the form

38

I b [0 1 ]

0 -I [I 0 ]

and

follow Shalika relations

subject to certain

Thus one could

directly and prove Theorem 2.22 by checking

are preserved

In any case,

relations.

the resulting m u l t i p l i e r

will be denoted

rq

to the quadratic

form

representation

r

and called q.

Corollary 2.24.

r

of

SL2(F)

representation

is an explicit

attached

realization

of the

2.21).

is ordinary

q

0 -I r([l 0 ])"

and

representation

"the" Well

(This

of Example

q

i b r([0 1 ])

by the operators

that these

if and only if

V

is even

dimensional. We conclude of operators

this Section with some remarks.

B0(G)

representations decomposition

is irreducible

rq

are never

forms and group

For the case when [45],

[36],

q and

form of a division algebra already

remarked

q

importance

their for the theory

is the norm form of a quadratic [18];

for the case when

in four variables

is a quadratic

q

field,

is the norm

see [37] and

no such complete

theory of the two-fold

In these Notes we shall describe one and three variables forms on

the m e t a p l e c t i c Subsection

In fact,

the

[18].

As

results have

form in an odd number of

This is because until recently no one seriously attacked

the r e p r e s e n t a t i o n

automorphic

T(G)),

the group

representations.

in the Section I,

yet been obtained when variables.

irreducible.

is now known to be of great

of automorphic

see [35],

(it contains

Although

2.4.

group.

decomposes

GL(1)

and

covering groups of

SL2(F).

how a We~l representation

and how its decomposition GL(2)

to automorphic

The general p h i l o s o p h y

relates

forms on

is explained

in

in

2.4. A p h i l o s o p h y

for Well's

The purpose which

of this Subsection

(though unproven)

Roughly

representation.

speaking,

the idea is that quadratic

b e t w e e n automorphic

and automorphic

forms

F

representation q.

V.

The resulting

SL2(F ).

F, and Let

This group acts on

H

L2(V)

Now fix

representation

F

to be local.

and

~ ~ S-L2(F)

r

H

in

L2(V)

the operator that

Suppose

Then the p r i m a r y constituents

pr~ary

constituents

A.

its n a t u r a l action on m a y be assumed

A(h).

is so.

of

group

From T h e o r e m 2.22 it follows

commuting algebras.

zero,

the corresponding

q

denote the orthogonal

of

In fact it seems plausible

each others

group

group.

through

to be u n i t a r y and we denote it by

rq(~).

forms index I-i

forms on the m e t a p l e c t i c

on the orthogonal

space over

of

of

h c H

of these Notes.

denote a local or global field of characteristic

(q,V) a quadratic

for

a simple principle

underlies most of the results

correspondences

Let

is to describe

A(h) rq

commutes

and

A

rq

In particular,

with

generate

for the m o m e n t

of

that

that this

correspond

i-I

the commuting

to

diagram

L2 (V)

SL 2

H

leads to a c o r r e s p o n d e n c e H

which occur in

which occur in D

rq.

for "duality".

forms on

H

A

b e t w e e n irreducible

and irreducible

representations

representations

F o l l o w i n g R. Howe, D

should pair together

which occur in

A

with automorphic

rq.

This is the c o r r e s p o n d e n c e

Since the existence rest on the hypothesis commuting algebras,

r

q

and

A

some further remarks

automorphic

forms on alluded

of this correspondence

that

SL 2

we call this correspondence

Globally,

which occur in

of

of

generate

D

SL 2

to above.

seems to each others

are in order.

The precise

40

formulation of this hypothesis, reductive pairs",

in the greater generality of "dual

was communicated to me by Howe;

as such,

it is

but one facet of his inspiring theory of "oscillator representations" for the metaplectic context of

group

(manuscript in preparation).

SL 2, at least over the reals,

suspected by S. Rallis and G. Schiffmann hand,

important

literature.

special examples of

D

In the

this fact had also been (cf. [52]).

On the other

had already appeared in the

In [36] Shalika and Tanaka discovered

that the Well

representation attached to the norm form of a quadratic F

yielded a correspondence between forms on

seems to have been the first published

SL(2)

example of

and D.

extension of S0(2).

This

(Actually the

correspondence of Shalika-Tanaka was not i-i since they dealt with S0(2)

instead of 0(2).)

For quadratic forms given by the norm forms

of quaternion algebras over

F

the resulting corresponding between

automorphic forms on the quaternion algebra and by Jacquet-Langlands Shimizu.

GL(2)

was developed

([18],Chapter III) following earlier work of

In both cases,

the fact that r

q

and A generate each others

commuting algebras was not established apriori;

rather it appeared as

a consequence of the complete decomposition of

rq.

One purpose of these Notes is to describe the duality correspondences belonging to two forms in an odd number of variables, namely 2 2--T gl(x) = x 2, and q3(xl, x2,x3) = X l - X l - X 3. The inspiration for our discussion derives directly from general ideas of Howe's, an initial suggestion of Langland's, [

], and Niwa [

T~eorem 6.3, above.

].

and earlier works of Kubota

Our results,

[

], Shintani

specifally Cor.4.18, Cor.~.20,

and

lend further evidence to the general principle asserted

However, as in earlier works,

each others commuting algebras, of D.

the fact that r q and A generate appears as a consequence ofthe existence

2.5

Extending Well's Well's

representation

construction

representation

rq

it is advantageous that Well's

of

to

GL 2

of the m e t a p l e c t i c SL 2 .

group produces

There are several

to extend this c o n s t r u c t i o n

representation

for

SL 2

often

character

natural

depends

analogue of

rq

for

In this Subsection representation an analogue when

q

depends

of

rq

GL 2

to

GL 2 .

on

for

T

q

By contrast,

the

~ .

only on

More precisely,

GL 2

which

One is

not only on

q .

I want to explain how Weil's

original

i want to define

is independent

is the norm form of a quadratic

discussed

reasons now why

depends

but also on the choice of additive

a multiplier

of

~

or quaternionic

The case space

is

I shall treat the cases

in [18].

ql(x) : x 2 and

2

q3.xl, x2,x3)( following

: xI -

x~- x~

a suggestion of Carrier's.

Throughout

this p a r a g r a p h

the following

conventions

will be in

force:

l)

F

2)

n=l

will be a local field of characteristic

rn 4)

or

3

according

will denote

[[a b],l]

notation

for

for emphasis, r

or

q =ql

rn

or

by

a b [c d ] "

given in T h e o r e m 2.22 depend on

I shall denote

will be reserved

q3 ;

rq3 ;

will be abbreviated

Note that the formulas Therefore,

rql

as

zero;

rn

by

rn(~ t) ;

for the r e p r e s e n t a t i o n

Tt "

the

eventually

n

introduced

for

GL 2 .

Proposition

2.27.

(Cf.

[18] Lemma

1.%.).

If

a ~ F x,

and

a

= [s,(] ~ST2(F), 2.6).

Then

define

~a

to be

is ,~ v(a,s)]

(cf.

Proposition

42

rn(Ta~(Y) = rn(T)(Fa) for all

a cF x

Proof. and

~

and

~

SL2(F ) .

We may assume without loss of generality that

-SL2(F). -

is a generator of

Suppose first that

n =I

s=w=[[

Then 0

2a : ~ a

:

a

o ~ {1,(a,~)]

a

[-a-lo]

:

[o a - z ]

and

= (a,a) V(qn, T)(rn(T)([O a a-o l

rn(T)(~a)~(X)

])~)(x)

Some tedious computations

with Hilbert symbols also show that -I a 0 i a i a -I -1 a [0 a -1] = W[o I ] ~ [0 1 ] ~ [0 I ][l,(a,a)]

Consequently

(2.~3)

rn(~)([a.

O1])%(X) = (a,a)

lal n/2 ~(qn'Ta)~ ~(aX)

0 a-

Y~qn 'Tj

and

rn(T)(~a)~(X But

laln/2y(qn, Ta)~(aX)

rn(Ta)(~),

iff

proof. a cF x. in

The analogous

The representation

rn(T )

Suppose

rn(T)(~a ) :

identity for

[]

rn(Ta)

is independent

extends to a representation of rn(Ta)

Then for each

L2(F n)

Therefore

is completely straightforward.

Corollary 2.28. a eF x

y(qn, Ta)~(ax).

: rn(Ta~)%(X).

as was to be shown.

~ [[0i bl ] ,I]

of

) = [al n/2

a eF x

is equivalent to

rn(T )

such that

s e SL2(F).

for all

there is a unitary operator

rn(T a)(~) = R a rn(T)Ral for all

G-L2(F).

Equivalently,

by Proposition 2.27,

Ra

Ol 0],I]. i

43

rn(T)(~a) = R a rn(Ta)R[l But

G-L2(F)

Thus

rn(T)

defining

is the semi-direct product of

Conversely,

if

rn(T )

rn

i 0 FX=[[ 0 a] ].

G-L2(F)

G-T2(F),

I 0 rn(T)([O a],l)

will []

a ~ (FX) 2,

and

by

Ra

rn(~a)

If

rl(Ta)

to be

extends to

with

Example 2.29. intertwines

and

can be extended to a representation of

rn(T ) ( [0i 0a ] , i)

intertwine

ST2(F)

rl(T);

say

a = 2,

then

in particular,

Ra~(X)=I~Ii/2h(~x)

rl(~)

extends

to a representation of G-L22(F) : Jig,c] ~ G-L2(F) : det(g) ~ (FX) 2] In general, rq(T)

rq(Ta)

will not be equivalent to

rq(T).

Indeed

will extend only to a representation of the subgroup [[g,~] ~ G-~2(F) : det(g)

fixes

To get around this problem we "fatten up" extend it.

rq(~)]

rq(~)

before attempting to

This way there is more room in the representation

space

for intertwining operators to act.

Definition 2.30. Haar measure on

F

to

of the representations

Let

dt

Fx .

Let

rq(T t)

Note that the space of

denote the restriction of additive r

q

denote the direct integral

with respect to

dt .

is isomorphic to L2(F n• FX). q Moreover, the methods of Corollary 2.28 imply that r q extends to a representation of G---L2(F) satisfying

(2.44)

rq([ol 0a])r

Proposition 2.31 9

r

= laI-i/2~ta_l(X)

The action of

rq

in

by the formulas (2.45)

rq([ 0I b1 ] )~(X,t) = rt(bq(X))~(X,t)

L2(F m• F x)

is given

44 A

r (w)@(X,t) q

(2.46)

= ~(q,t)9(X,t) : ~(q,t)~

~(Y,t)Tt(q(X,Y)dtY Fn

rq( [a0 a-O 1 ] ) ~ ( X , t ) =

(2.87)

a,a)laln/2

~~(q'Tat)

~(aX, t)

and

r q ( [ o1 0a] ) ~ ( X , t )

(2. ~8) Proof.

Apply

Proposition for

GL 2

2.31

of

r

and my Definition Note

the definition implies

using formulas

construction

2.30

simply

and (2.48).

In fact thi{

to me by Cartier

reformulates

~(q~,ta) = y(ql, t )

all

iff

g ~ GL2(F )

ral

0 r l ( [ 0 a a])

awhile

ago

his ideas.

-1/2~ (

aX, ta -

2)

commutes w i t h

det(g) c (FX) 2

This

. rl(g)

is consistent

for

with

2. 13(a).

Now it is natural

to ask what

in the present

is that the orthogonal of similitudes Suppose is

(2.46),

directly

rq

that

checks t h a t

2.Z~ takes

[]

integral.

we could have defined

(2.45),

Thus one e a s i l y

Corollary

of direct

was communicated

q

a o rl([ o a])r

(2.49)

= la -1/2%(y~,ta-l)

of

group

of

Roughly q

speaking,

is simply

of Subsection

the answer

replaced

by the group

q .

first

FX=GLI(F)

context?

shape the philosophy

that

q :ql

and its action

" in

The group

of similitudes

L 2 ( F x F x)

of

ql

is given by

(A(y)~) (x, t) = l al - 1 / 2 9 ( a x , ta -2) Note that

A(y)

clearly

Now suppose symmetric

matrices

q =q3"

commutes If

F3

with

rI .

is realized

with coefficients

in

F

as the space of

then

q(X) :det(X).

2x2

45

The group of similitudes precisely,

of

q3

is essentially

GL2(F ).

More

define go X = tg X g

when

g e GL2(F)

action of

and

GL2(F )

X c F 3.

in

Then

q(g~X) = (det g)2 q (x).

L2(F 3 • F x)

The

is given by

(A(g)~)(X,t) = I det gl 2 ~ ( g , X , t ( d e t g)-2) and

A

once again commutes with

The result now is that into irreducible representations GL2(F)

rI

(resp.

representations of

which occur

on the metaplectic

GLI(F) in

r3 .

of

(reap.

A ).

GL 2

A ).

5.5.

representations

GL I

rI

(resp.

(reap.

r3 )

automorphic

The global results

obtained or expected are described The local analysis

indexed by irreducible

irreducible

group which occur In

which occur in

should decompose

Globally this means automorphic

be indexed by automorphic forms on on

~

r3 )

forms

should forms

that can be

in Subsections 6.1 and 6.2.

is carried out in Subsections

of

4.3, 4.4, and

2.6,

Theta-functions Classically,

series

attached

lated in SL 2

a.utomorphic to quadratic

representation

forms forms.

This

theoretic

from theta-

is the procedure

terms by Well

reformu-

in [47].

For

the idea is this. Suppose

ratic

F

form in

is a number n

field and

variables. e(~)

Piecing of

are constructed

together

ST2(~ )

v

of integers

of

Fv

As in Subsection of matrices

place

2.1,

integer

Proposition

GL2(0v)

~(r

s

attached

rv(q )

to

define

0(rq(~)~)

= 0(~)

Fn

For

GL2,

F

let

0v

by

rq

is

for all

q.

Uv

place

of

its group

representation K vN

with

of units. of GL2(Fv)

is the subgroup a ~ I

divisible

by

and

s e SL2(F).

the point

of

denote

Let in

of

4.

Thus

the ring

rv(q)

denote

L2(F n x Fx).

GL2(0v)

c ~ 0 (mod N).

consisting Here

K~ = GL2(O v)

N

if

is Fv

characteristic. 2.32.

Suppose

is class

q = ql

I, i.e.,

or

q3"

Then for almost

the restriction

has at least one fixed vector. v,

on

a representation

that

~(rq(S)~

~

quad-

2.32 below.

and Weil

[~ bd]

has odd residual

v,

:

a non-archimedean

the corresponding

every

a distributuon

In particular,

is Proposition

a positive

Define

is defined with the property

This is the theta function

For

is an F-rational

local Well representations

SL2(F)-invariant.

departure

q

~v0 e L2(F n x Fx)

of

rv(q)

More precisely,

to

for each

by n

O .... Xn, t ) ~v(Xi,

(2.50)

Here

i0

denotes

=

(

~ i=l

the characteristic

the characteristic

) | v

1U N

function

v denotes

10

v of

OF

and

IuN v

function

of

Ur~v = { y e U v ' y i 1 (mod N ) ] .

47 For all odd

v,

(2.51) for

rv(q)(k)~o = go

k e K N. V

Proof.

The group

GL2(0v)

[oi b1 ]

(b

is generated by the matrices

e ~

W

and

i 0

(a

[o a ]

~ Uv).

Thus it suffices to check (2.51) for these generators. Recall that our canonical additive character 0v.

In particular,

t e U v.

~v(tbq(X)) = i

Therefore

if

rv(q)(l b)~~

the Fourier transform of

i0

q(X) e O v, = ~v~176

is

i0 .

V

Thus

o o rv(q)([ 01 a])~v(X,t) = lal-i/2~~

obvious since

a

To define

G~,

b e 0v, and Note also that

rv(q)(w)~~

) = ~v~

ta -I)

=

oix ' t)

is

U v.

rq

globally,

and to introduce theta-functions

we need first to define an appropriate • ~.

has conductor

V

V

The fact that

e

T

on

~

We shall say that

if

~(X,t) = N ~v(Xv, tv)

~

on

~

on

space of Schwartz-functions • ~

is "Sehwartz-Bruhat"

and:

V

(i)

~v(Xv, tv)

creasing on (ii)

is infinitely differentiable x

• Fv

for each archimedean

for each finite

v,

~v

and rapidly de-

v~

is the restriction to

of a locally constant compactly supported function on (iii)

for almost every finite v, ~v = ~o

~v x ~v

F~v+l;

(the function defined

V

by (2.5O)). Denote this space of functions by A ( ~ is dense in in

L2(~

L2(~ • F~)

• F~),

• F~).

Since ~ ( ~

we can define a representation

through the formula

rq

• FI) of

48

V

By virtue of Proposition least all

% e ~(~

2.32 this definition

x F~).

is meaningful

Indeed for almost all

v,

for at

~v e GL2(Ov)

O

and

%v = @v"

~(~

X F~).

rv(q)(~v)%v

By continuity,

The role of character

Thus

~v T

of

on

~

For each

Proposition

(unitarily)

in

to

L 2.

is now played by a non-trivial

corresponding ~ e ~(~

2.33.

Z

to

• F~)

q (or rq)

define

i. e. ,

8(~,g)

are on

.

y e GL2(F),

8(%,Yg) = @(~,~); e (~,~)

Proof. y

(rq(~))@({,D)

For each

(2.52)

with

operates

again belongs

by

8(~'g) :

with

and rq(~)%

F~.

defined as follows.

and

rq(~)

in the local theory

The theta-functions

a~

= %v

is

GL2(F )

We may assume without loss of generality that

is a generator of b s F.

T = H~

rq(~)%({,~)

invariant.

GL2(F)o

Suppose first that

~ = [I,i]

i b Y = [0 1 ]

Then

V

rq(y)~(X,t)

= T(btq(X)@(X,t)

trivial on

F.

= ~({,9)

Now suppose it follows that

for

y : wo

But

q

is F-rational.

({,~) c F n • F x

and (2.52) is immediate.

From Well's theory

y(q, Tt) -" 1.

Moreover,

(cf. Theorem 5 of [47])

Itl : i

if

Therefore, r~ rq(W)~(g,n)

= :

~1~1n'/2 ~

Thus

y(q, Tn)$(~g,~ )

$(~,~)

t e F x.

49

= z ~(g,~).

The last step above

i aO] Y = [0

with

= lal - l / 2

~(~,~a-1)o

But

rq(y)%({,9)

= Z 9({,q)

Observation 2.34. automorphic

form on

Nevertheless, zero.

odd function

a e F x.

in

Xo

formula.

By formula

lal -1/2 = 1.

(2oI$8)

Therefore

as d e s i r e d .

The f u n c t i o n

~,

i.e.,

for example,

For simplicity, that

summation

8(9,g)

8(~,g)

always defines

is always

it may often be the case that

Suppose,

follows

from Poisson's

suppose

Finally rq(~)~(y,~)

follows

Then

suppose

that rq(~)~

also that

rq([ -i0 - ~])~(X,t)

But by the GF-invariance

of

8(~)

[ o

invariant.

is identically

= -@(X,t),

i.e.

~

is also odd for all

~ e ~.

F = Q.

(2o~7)

From formula

= -~(X,t),

-1

e(~,

~(-X,t)

GF

an

is an

it

i e.

o

]7)

: - e(~,~).

8,

-i ~]~) Therefore

8(~,~)

We close the basic

i 0~

this paragraph

by demonstrating

how

8(~,~)

generalizes

theta function 2~in2z n ~

Proposition = n ~p

Itl -l/q

where

e- I t ! 2 ~ x 2

2~176 ~p = ~o P Then

Fix

n = i, F = ~, and

for all finite

if

8(%,g)

g = [[yl/2 0 = yl/As(x+iy)o

p

rq = r I.

and ~ (x,t) =

xy-l/2 -1/2 ]' 1 ], y

Suppose

50

Proof.

Since

[yl/2 xy-1/2 0 y-I/2 ] =

Z rq(~)%({,~). - .

I x

[yl/2

[0 1 ] 0

lyll/4

- =

o

y -I/2]'

e-2W7~ 2 e2~i~2x~.

({,~) Here we choose 0p

for

only if

T = ~ Tp

p < ~. { e ~

Thus and

with

({,~)

T (X) = e 2~ix contributes

9 = i (recall

and

Tp

trivial

to the summation

~2 = 102~176

l.e.,

e2~in ~ ( x + l y ) as was to be shown~ Note that for

a c (0,~),

a o r([ 0 a ] ) ~ ( ~ , ~ )

Thus

8(~,~)

is actually

and

=

-I/2

as above,

=

ai

=

al-i/2

=

tl-i/~ e-ltI2~x2

defined 7

@

z~

on

~(ax, ta -2)

Ita-21-1/4

e-ltl2~x2

above

on

w

AutomorPhic Forms on the Metaplectic Group: Global Theory.

3.$o Preliminaries. We start by describing the correspondence between classical forms of half-integral weight and automorphic forms on the metaplectic group of Section 2.2. For convenience, we shall assume that the ground field is Thus we shall consider functions in

f(z), defined and holomorphic

Jim(z) > 0], satisfying

(3.1)

f(yz)

for all N

Q.

~ e El(N).

=

f ~#az+b~ j = Y(~)(cz+d)k/2f(z)

Here, as before,

k

is an odd positive integer,

is a positive integer divisible by 4, and

plier system of dimension

1/2

We shall also assume that of

FI(N ).

cusps, i.e., [38]. )

is the multi-

described by (2.30). f(z)

is holomorphic at the cusps

In fact we shall assume that f(z)

X(?)

is a cusp form.

f(z)

vanishes at these

(For precise definitions,

see

Suffice it to say that cusp forms have Fourier expansions of

the type oo

(3.2)

with

f(z) =

Z a(n)exp(2wi n z) n=O

a(0) = 0, and for these forms,

(3.3)

If(z)IIm(z) k/$ < M~

Nhat we want to show is that such forms correspond to particularly nice functions on by

~

~

The space of these forms will be denoted

S(k/2, rl(N)). Recall that

K

denotes the maximal compact subgroup of

whose connected component is K* = S0(2)

=

Jr(8)

~cos@ - sin@ = Lsin@ cos@ ]]

GL2(~)

52

0 ~ ~ < 2~.

with

is still abelian. realize with

--# K

r(e)

.~

Although

To describe

as

~/4~ ~

~s above.

K*

with

0 ~ e ( 4w.

K-~

must make

elements ~*

its character

rather

[~(e)

=

~sine

of pairs

to

[[r(e),~}]

cose~

The isomorphism

~(0),-i}

Z2

between

correspond

to

these

7(2~)

(and at most

can have this property). on

than as the group

it is convenient

rCOSe - s i n e ] ] =

are of order two

non trivial

group

Thus

-~ (3.#)

S0(2),

does not split over

realizations

since both these

one non-trivial

Consequently

of

element

each character

of

of --# K

must be of the form k

(3.5) with

~k/2(~(@)) k Next

: e

odd. recall

the Casimir

operator

for

S--~2(~)

in terms

of the

local p a r a m e t e r i z a t i o n

2 = (x,y, e).

Here

~ =

yl/2 [ o

The Casimir

x y-z/2 y-l/2]

operator

,

Proposition to a function

(1)

on

~(yg) : ~(g)

homomorphically (2)

3.1.

~f

~({~)

in

y > O,

is the differential

2. 8 2 ~2 ~ = -y ( - - ~ + T )

(3.6)

on

T(e)

S--L2(~)

for

for

0 ~

e < &~.

operator

~2 ~x~---e

Each cusp form

SL2(~)

= {~(~)

- y

x ~ ~, and

f

in

S(k/2,FI(N))

corresponds

such that:

y ~ SL2(Q) ( r e c a l l

y ~ { y,S~(y)}

via the map ~ ~ Z 2,

SL2(Q )

i.e.,

~

imbeds ;

is a genuine

function

G--L2(~);

(3)

~(~k0) = ~(~)

(4)

~(g ~(9))

for all

ko

= ~k/2(T(6))~(g)

in

~o n SL2(~);

for all

T(B)

in

K-~ ,

i.e.,

53

_

transforms under (5)

_

w

K*~

according to the character

viewed as a function of

satisfies the differential

S-L2(~) alone,

ek/2; ~

is smooth and

equation

A~ = - ~k (k~ - 1)~; (6)

~

is square integrable;

\,~

I,~(~)1 2 ~

SL2(Q) (7)

~

more precisely,

< 0%

2 (~)/Z 2

is "cuspidal" on

NQ~ ~ ~(E~I

S-~2(~), i.e.

l]~)dx = 0

(This expression is meaningful

(for

a.e.

since

N~

g)

lifts as a subgroup of

'~2 (~).) The significance

of (6) and (7) ~s that

the space of square-integrable The significance

~f

will belong to

cusp forms for the metaplectic

of (5) is that

-k/~( ~k -

i)

group.

will be the

eigenvalue of the Casimir operator for the representation of

S--L2(~) with lowest weight vector

k/2

(~k/2

in the notation

of Section ~)o To define

~f

and establish Proposition 3.1

we use a few

Lemmas. Lemma 3.2.

Suppose

= ~k

with

y =

-~2(~)

~y~S~(y)]

in

~ = (g,~)

belongs to

S-~2(~).

Then

o

SL2(Q ),

k 0 =[k0,1 ]

determined up to left multiplication

in 4 n by

SL2(~), and g~ in

[Y0, S~(Y0)]

in

r l ( N ) = SL2(Q) n SL2(R).~~ 9 Proof.

By "strong approximation"

SL2(Q)SL2(~)~ ~.

More precisely,

for

SL(2),

g = yg k 0 ,

with

SL2(~) = g~

in

SL2(~)

54

determined up to left multiplication by elements of [g,C] = {y,S~(~)} [g~,r

r

with

r

:

~&(Y,g~)~&(Yg~,k0)S&(Y)"

FI(N ).

Thus

{ko, l] If

yO:[Yo, S&(YO)]

r

straightforward but tedious computation shows that [g,{] = {YYo I, S~(yY01)]

(~og~,r

=

For

[y0g~,{'][ko, l},

{Yo,S~(Y0)~{g~,r

g : [ac db ]

in

with

[]

does not define a factor of automorphy for choose

w I/2

so that

~ = y

in

(We agreed ~o

Indeed by the

(Hecke [i~], pp. 919-940),

x(~)(e~+d)I/2

FI(N ) .

Now we can define

and fo~

FI(N ).

-v/2 < arg(w I/2) ~_ ~/2.)

functional equation for the theta-function

for

J*(g,z) = (cz+d)i/2

SL2(~), recall that

f(z)

(3.7)

in

~f~

For

S(k/2,~l(~)),

g-- = (g~,{)

in

ST2(~ ),

set

set

~f(g) = f(g~(i))j*(g~,~)-k.

Note that in (3.7), g~(z)

if

g~ = ([ca bd ]'{)"

(3.8)

az+b : c--Yg~

Thus (3.1) rewrites

itself as

f ( V ( z ) ) = j* (-{, ~ ) k f ( ~ )

for all

-{ -- { ~ , S ~ ( y ) ]

in

rl(~).

(By P r o p o s i t i o n

2.16,

S~(u

= x(Y):) Lemma 3.3. S-~2 (~), i.e.

(a)

J*(~,z)

defines a factor of automorphy on

55

(3.9) for all

g,g'

in

SL2(~); (b)

a character of Proof.

K*~

(a)

The restriction of non-trivial on

takes its values in F

Z2, namely

to

K~

defines

~k/2"

The function

F(g,g,) =

Since

J*(g,i) k

j~(g g,,z) j~(g,g,(z))J(g,,z)

Z2

and is obviously a Borel map on

SL2(~)xSL2(~).

can also be shown to satisfy the factor set relations

and (2.2) (cf. Maass

[24], pp. 115-116),

J*

automorphy on the extension of

SL2(~ )

computations

is the cocycle

then show that

F

(2.1)

defines a factor of

determined by F. Further ~

described in

Theorem 2.2. (b)

the

By (a), and the fact that ~(e)

restriction

of

(j.)k

--g.~ K But by definition,

follows

to

-'g" K

stabilizes

clearly

defines

J*((l,C),i) k = ~k = C.

from the('definition

of

i e Jim(z) >0],

a character

That

of

J* = ~i/2

~1/2" ~

Proof of Proposition 3.1. Note first that

~f(~)

is well defined even though

(3.7) is determined only modulo

TI(N).

g-~ in

Indeed by (3.8) and (3.9),

j~ (~o,~(i)kf (~(i))j~ (yo,~(i))-hj~ (~, k)-k = for each

YO

~f(7) in

Properties

TI(N ). (i) - (4) are trivial.

analyticity assumption for tion (applied to

f(z)

To prove (5), recall our

Then use straightforward

~f(x,y, 0) = yk/4f(x+iy)e-i(k/2)6)

computa-

to verify that

56

ik ~-- (9

ik

A0 and

; ~ :I

H-

are completely determined by their restriction is a Hilbert space with inner product given by

the inner product in That

r =0

for all

and

vanishes outside

Note that such H~ 9

where

such that

~2

k-3 g(tX) =g(txl, tx2,tx 3) :t 2 g(X) (iii) g(-X) =(-i) r g(X)

V

L2(H~)

realizes an essentially irreducible unitary repreSL2(~)

can be deduced from recent work of Strichartz

([44]) and Ehrenpreis ([53])Proposition 4.12. the representation

The precise result is:

The natural action of

~k-I ~ k - I

SL2(~)

in

Gk

realizes

"

This follows from a careful study of the K-types appearing in [44] (pulled back from If we let

S0(1,2)

L2(k,H -)

to

SL2(R)).

demote the subspace of functions in

L2(V)

which can be approximated by linear combinations of functions of the form a(x) with

g e Gk

and

f(m) e L2((-~,0),

Corollary 4.13. for

H(g)

and (as an _

many copies of

~ f(-lq(x)ll/2)g(X)

L2(k,H -)

Imlk-ldm),

~k_l~k_l

9

we have:

is an invariant subspace of

SL2(~ ) -module) +

,

L2(V)

is equivalent to infinitely

9O

Now using a series of non-trivial Lemmas we shall establish the following:

for

Theorem 4.14.

L2(k,H -)

~

S-L2(R) -module)

and (as an

copies of

(a)

L2(V)

is equivalent to infinitely many

~k/2 "

Lemma 4.15. L2(k,H -)

is an invariant subspace of

with

Suppose f(m) ~ ( ~ )

rq([o1 ~])a(x)

:

G(X) =f(-lq(X) II/2)q(x)

belongs to

(the Schwartz space of

~ ).

Then:

f,(-rq(X)ll/2)g(x)

where f'(m) = e~ibm2f(m];

(b)

rq([ 0l ol])a(x) : e -~ i/4y( -[q(X)[ 1/2) g(x)

where

k-2 ~(m-lt) 2 cos ~ [ ~ + r

f(m) = e~ir

Jk_2(2mt)f(-t)t

0 Proof.

Part (a) is completely straightforward.

the other hand, (Theorem I).

dt .

5

results from formulas appearing

Roughly speaking,

Strichartz'

Part (b), on

in Strichartz

[43]

theorem describes the

Fourier transform of a function on

V

natural action of

SL2(R)

according to a given irreducible

representation of

G.

weight vector that

(~• r

k-i

on

V )

which transforms

If this representation

condition

has a highest or lowest

(3) in the definition of

on p.509 of [43] must be odd.

(under the

Gk

implies

Thus the formulas there

simplify considerably and an application of them to our situation proves the lemma.

[]

Cprollar[ 4.16. the restriction of

L2(k,H -)

rq

Furthermore,

to L2(k,H -) is equivalent to infinitely q many copies of the ~-representation of SL2(R) generated by the operators

r

is invariant for

91

[0

: f-~f'

and [01 -01] : f-~e-~i/4 ~ .

To complete the proof of Theorem %.14, observe that L2((-~,0),Imlk-ldlml)

is isomorphic to

L2((0,~),mdm)

via the

map k-i

f(r)

-~ f ( - r )

l rl 2

Thus Theorem 4.14 follows from Lemma 4.17 below coupled with some straightforward manipulations.

(A useful identity here is the

following:

~-r k-1

cos 2L 2 +.c] =e

_ _~2 ( k - l )

~i~/2

e

.)

~k/2

(resp.

~--

Lemma 4.17. is realizable in (a)

The representation L2((0,~),mdm)

the operator corresponding to

1 [o

(resp.

~]

S--~2(R)

is

the operator corresponding to

Proof.

( resp.

e ik~/4 )

Explicit realizations of the representations of

are to be found in Pukansky [32] and Sally [33]. corresponds to

R•

in [33] with

follows from Lemma 0.1.5 of [33]. Corollary 4.18.

S--L2(~)

In particular,

h =k/~.

Thus our Lemma

[]

There is a natural correspondence (the "duality

correspondence") which associates to the representation G*

of

e-~ibm2f(M)) ;

f(m) ~ e -IkT/% ~ Jk_2(2mt)f(t) t dt 0 2

~k/2

~k/2 )

so that

f(m) ~ e~ibm2f(m) (b)

--3 c

the genuine representation

~--k/2 of

~*.

correspondence inverts (part of) the local map

~k-I

of

In particular, this SR

of Section 4.2.

92

Proof. from r

q

rq

Let

r*

denote

q

the representation

( v i e w e d as a r e p r e s e n t a t i o n

depends on

~,

r* q

~k-l"

Its restriction

indexes the primary

when induced up to

contains

--g G

r* q

(Cf.

Lemma

will contain

continuous

series

The required Fourier the formula The result

is that

+ ~k-I @ ~ k - I

is

component

of

r

correspondence

which

q

~v~u/2" [ ]

to class i

as well since these representations

of

L2(V)

transform formula

~(~i,~2 )

speculations

will appear

(albeit

continuously).

is now more complicated

than

the representation

is mapped to

about how this

r3

The representation discussed

denote the restriction representation to

-~ 1/2 1/2~ ~i '~2 J"

correspondence

More inverts

in Section 6.

Concluding Remark.

isomorphic

G

in Lemma 4.15 but again it can be derived from [43].

comprehensive

the trivial

this

representations

also occur in the decomposition

to

infinitely many copies of

We note that one can extend

r'q

Although

does not.

In particular,

and this representation

let

induced

~k/2 = Ind(~'G--g'~/2)"

So now consider

S: ~ + ~

--V G

G = SL2(N)).

it can be shown that

4.3 and the proof of Lemma 4.%.) copies of

of

of

~xr3(Tt)dt.

of

of

r* q

in Section 2.5. r*q

.~-~2(~) _ ~

to in

above is essentially More precisely,

S-T21~)_ tensored with L2((O,~)).

Then

r'q

is

4.4

The basic Well representation The Well correspondence

(Joint with P. Sally)

discussed in Corollary 4.18 pairs

together discrete series representations

of

GL2(~ )

Z

with genuine discrete series representations

on

Z O.

trivial on

of

G-L2(~)

Its domain actually extends to a larger set.

irreducible

"tempered" representations

Equivalently,

of

GL2(~ )

consider all possible constituents

representation

of

GL2(~ )

in

L2(~3 • ~x).

to above pairs together constituents with constituents

of

Consider all

trivial on

GL---2(~) r I.

The correspondence

of

of this adjoint representation

modulo

(but

relating a subset of the dual group of

to the dual group of

~-L2(~).

alluded

r 3.

correspondence

Actually,

~I.

of the "adjoint"

Our purpose in this Subsection is to describe a similar simpler)

trivial

GLI(~ ) .

for convenience,

Thus we deal with

In place of

r3

we deal with

we deal with the representation G--L2(~) modulo its center and

rl(~

GLI(~ )

(0,~).

Fix

T(x) = e wix.

Then the action of

rl(m )

in

L2(~)

is

g i v e n by t h e o p e r a t o r s I b ~ibx 2 rl(m)([ 0 l])~(x ) = e ~(x) and rl(m)(Q)~(x ) : e ~i/4 ~(x).

Theorem 4.19. (a)

With the notation of Subsection 4.1,

rl(~) : ~ 1 2 R e c a l l that

sentation

~i/2

| ~s

"

is a subquotient of the principal series repre-

p(i/2,1/2)~

it has lowest weight

1/2; ~3/2

is a subi

representation (b) to

Let

of

p(i/2,-I/2);

rl(m i )

m_l(X ) = e -~ix

it has lowest weight vector

denote the Well representation

Then

-3/2.

corresponding

94

rl(T-1) = ~ / 2 Here

Y~/2

has highest weight vector

weight vector

and

--+ ~3/2

has highest

It is easy to see that the subspace of

consisting of even (resp. odd) functions is irreducibly

invariant for

rl(T+l ).

representations of

To identify the resulting irreducible

S-L2(~) one uses the models for

in Chapter IV of [33].

rI

Let

r~ (resp. r~)

m

presented

denote the restriction

to the space of even (resp. odd) functions in

irreducible representation tion

m[/2

[]

Proof (2) (Sketch). of

-1/2

3/2.

Proof (i) (Sketch). L2(~)

| ~3/2 "

r~

L2(~).

The

is intertwined with the subrepresenta-

-

~i/2

of

p(-i/2,1/2)

via the operator

~(x) ~ rl(g)~(0)~ the representation

r~

is intertwined with the subrepresentation ~3/2

d (x) = ~(rl(q)~)l x=o Proof (3) (Sketch). A basis for L2(~) is provided by the 2 functions 9m(X) = e -~x H n ( ~ x) when Hm is the Hermite polynomial of degree

m, m ~ 0.

= i(m+i/2)@ m, i.e., rE 90)

and

r~ =~i~2

r~ and

has lowest weight

has lowest weight

3/2

1/2

rl(U)9 m

(corresponding to

(corresponding to

91).

Thus

r~ : ~3/2 "

Corollary %.20. induced from

But by differentiation,

rl(T )

Let on

r~

denote the representation of

-~2(~)

~-L2(~). Then

rl = ~i/2 ~ ~3/2 " Note that

r~

restricted to

is essentially the representation

rI

of Section 2.5

S-L~(~).

Corollary ~.20 associates to the trivial representation of GLI(~)

the representation

~--i/2 of

G-L2(~) and to the representa-

9S

tion

sgn(x)

of

GLI(~ )

the representation

this is the sought-after correspondence Concluding remark. Moreover,

rl(~ )

If

~3/2"

Modulo

(0,~),

W I.

F = @, rl(T)

is independent of

defines an ordinary representation

of

T.

SL2(@ ).

In

fact Kubota has shown in [23] that the "even" piece of

rl(~)

the complementary

s = 1/2.

series representation

of

SL2(~ )

Thus this complementary

series representation

the analogue of

The "odd" piece of

~I/2"

equivalent to the representation character

z ~ z/Iz I

SL2(@ )

once again appears as

rl(T )

for

F = @

is

induced from the

of the subgroup

operator is constructed fact that pieces of

of

at

is

rI

[[z W_l]~" The intertwining 0 z just as in Proof (2) of Theorem 4.19. The (over arbitrary fields)

could be identified

in this way was pointed out to Sally and myself by Howe; cf. his Zentralblatt

review of [i0].

w

Local Theory:

the p-adic places.

Throughout this Section, field of characteristic acteristic

of

F

F

zero,

will denote a non-archimedean

For simplicity,

will be assumed to be odd.

see the remarks after Lemma 5.6.

the residual charFor dyadic fields,

The metaplectic

cover of

G = GL2(F) (constructed

in Section 2.2) will be denoted by

~.

The purpose of this Section is to describe the local map S:T + for certain irreducible unitary representations non-supercuspidal 5.1.

~,

namely the

representations.

Basic Representation Theory. Suppose

of

of

~

~

is a (not-necessarily preunitary)

on a complex vector space

admissible if (i) for every is an open subgroup of group

~'

of

~,

~,

V.

Then

~

representation

is said to be

v c V, the stabilizer of

v

in

and (ii) for every open compact sub-

the space of vectors stabilized by

~'

is

finite dimensional. If

~

is irreducible and admissible,

if for every vector

v

in

(5.z)

we say

Y

is supercuspidal

V,

~ Y(u)vdu

= o

U for some open compact subgroup meaningful

V

of

N,

Note that

(5.1) is

since

~

splits ove#

Now suppose

Y

is an irreducible admissible non-supercuspidal

representation of in

U

satisfying

~.

Let

N,

V(~,N)

(5.1) for some

denote the set of vectors U

as above~

union of all its open compact subgroups a subspace of Since obtain in

B

Since

U, V(~,N~

N

v

is the

is actually

V. normalizes

V/V(~,N)

V, ~(~) preserves

a representation

~'

of

V(7, N). B

Thus we

(actually

~/N,

97

since ~'

~(n)v-v

e V(~,N)

will be n o n - t r i v i a l The significance

is that

V/V(Y,N)

cam be imbedded of

for all on

Z2

v e V

and

n e N).

if and only if

of our a s s u m p t i o n

(hence ~')

that

~ ~

is genuine

induced

on

~.

not be s u p e r c u s p i d a l

is then non-zero.

in the G-module

Note that

Consequently,

from an irreducible

quotient

V/V(~,N). To be more precise,

is irreducible,

v

the vectors

is an open compact is a finite

fix

a non-zero

7(~)v,

subgroup

set of r e p r e s e n t a t i v e s

V/V(~,N)

is a ~ - m o d u l e

subspace

V'

of

~

in

B\G/~',

w h i c h contains ~

let 6

of

define,

If

v,

Since ~'

and

{gi ]

under

also span

V.

That is,

Thus there is an invariant

in

and is such that the

V/V'

denote the modular

Ind(B,~,~),

V.

transforms

V(~,N)

B/N

V.

of

v i = ~(~i)v

of finite type.

representation

As always, imbed

V

span

in the stabilizer

of the finite n u m b e r of vectors

natural

g e ~,

vector in

for each

= W

is irreducible.

function v

in

for

B.

To

V,

fv(~) = T(~)v where

v

denotes

fv:~ ~ W

the class of

v

in

W = V/V'

Clearly

satisfies 1/2

(5.2)

where

T = 5 -1/2 ~.

between

~

That is,

and ~

subgroup If

of 7

= 6

Thus

v + f

Ind(~,~,T)

v

Ind(~,~,T)

(5.2)

is an i n t e r t w i n i n g

(non-zero

since

of

~

in addition

Ind(~,~,~).

to being right

operator

is not supercuspidal).

as the space of functions

(We are f:~ ~

invariant by some open

~.) is trivial

on

Z2

a general t h e o r e m of Jacquet irreducible

(~)T(~)fv(~)

is a. s u b r e p r e s e n t a t i o n

interpreting satisfying

fv(~)

admissible

this conclusion and H a r i s h - C h a n d r a

non-supercuspidal

is a special case of classifying

representations

the

of a p-a.dic

98

reductive

group.

Now let

T

denote an arbitrary

representation irreducible

of

B/N.

unitary

irreducible

(finite dimensional)

Henceforth we shall deal exclusively with

representations

of

~

which are equivalent

to

Ind(~,~,T) o r some u n i t a r i z a b l e representations, Let

B0

subrepresentation

have even v-adic order

subgroup

of

To a n a l y z e

such

the theorem below is crucial.

denote the subgroup

modulo units).

thereof,

(i.e.,

B0

In particular,

and its irreducible

B

whose diagonal

the diagonal

By Lemma 2.11,

B.

of

entries

entries are squares

lifts homomorphically

]0 = B0/N = A0 x Z 2

(finite-dimensional)

as a

is abelian

representations

are of the

form x

aI

(5,3)

TO(~I,~2)([O

with

~i,~2

quasichara.eters

Theorem 5.1. representation

of

is trivial on

a1

0 a2]'~)

~([0

other hand, and

[

a,2],C) = C#l(al)P2(a2) of

Suppose

~

~ = B/N

on a complex vector space

Z2,

V

T

is an irreducible

is one-dimensional,

= ~l(a')~2 (b)

if

F x.

for some choice of

is non-trivial

is equivalent

on

Z2,

V

finite dimensional V.

If

and

~i,~ 2,

On the

is four dimensional

to

r(ZI,U2 ) : Ind(go, g,~4D(~l,Z2 ) with

rO(~l,~2 )

a,s

in

(5.3)).

The first part of this Theorem is obvious. we shall exploit the theory of representations mal subgroup Clifford

of finite index.

for groups with nor-

This theory goes back to Weyl and

and has already been described

in Lemma 4.3.

To prove the second

for unitary representations

99

As before, If

L0

let

H

suppose

denote

the subgroup

H = G

Lemma

if

5.2.

representation

L0

of

L*

H(Lo)

(i)

Ind(H,G,L*)

(2)

the restriction

(3)

of

subgroup

irreducible h

in

G

representation

restriction

of

G. N,

(Lo)h = L O.

and equals

is any irreducible

whose

of index two in

such that

is self-conjugate

Suppose

is selfconjugate

N

otherwise.

finite-dimensional

to

N contains

L O. Then

is irreducible, of

Ind(H,G,L*)

and equivalent

is irreducible

L 0 | L h0

to

al__~ifinite-dimensional

if

L0

otherwise;

irreducible

representations

of

are so obtained. This

of

[3].

assumes

lemma is essentially Thus we shall not

that

G contains

this assumption

matrices units

whose

the content

include

the normal

diagonal

Thus we can prove

B+

of

units.

B

N

consisting

Clearly

5.1 by applying

properties

aI = ([0

and

wL(x)

but

o

a'

],~)(

[~0:~+]

[o I

symbol

o a~

it follows

)-I

],~'

0 a2]'~)(L2'(ai'a2)(a2'al))

al, a2, a~,a ~ e F x.

In particular,

fixing

in

= (m,x)~i(x).

L 0 = TO(~I,~2),

then

In other words, H(Lo)

= T 0.

= [~+:A] = 2.

Lemma 5.2 to these pairs.

of Hilbert's

a2

of

modulo

~(~i,~2 )h = ro(~i,~ ~) with

III

(In [3] Boerner

are either both squares

modulo

([ai o a1 o a~],C')([o

= ([o

12 of Chapter

of order two outside

subgroup

elements

Theorem

From elementary

for arbitrary

a proof here.

an element

or both non-squares

(5.~)

of Section

is not necessary.)

Now consider

h

is a normal

is any finite-dimensional

Clearly

G

N

if

N = ~0'

Consequently

G =

that

100

is irreducible of

~+

and all i r r e d u c i b l e

non-trivial

on

Z2

from

and the r e s t r i c t i o n ~i,~ ' 2'

(with of

of

(T:) h

Fixing h = ([0

is not equivalent

Z'

to

~0

is

to

Indeed

~nd(~+,~,~(~l,~

2)

representation

and Theorem 5.1 is proved.

to an appropriate, basis, to

~0

the

takes the form

0 0

0

0

o

(~,ala2)~

o

0

0

0

0

o

-rO(~l,~2)([O

T'.

],i) in

TO(~l,~2)|

Hence every irreducible

Ind(~0,B, T0(~l,~2))

a1

~'.

al 0

With respect

Co.r011ary 5.3.

(5.5)

representations

a2],C) = (w,a2)rO(~l, P2)([ 0 a2],~),

as above).

of

and

aI 0

~ is of the form

restriction

A+,A,

(5.4) that

zO(~l'~2)h([o

dimensional

are so obtained.

Now apply Lemms 5.2 to it follows

finite

a2],~)

8

~

(w, a,2) L~

0

o

(|

We close this S u b s e c t i o n with some Lemmas w h i c h will be useful in our analysis definition

of the representations

of the local map for

Lemma 5.4. representation of

A

Suppose of

~.

Clearly

~

in

is a f i n i t e - d i m e n s i o n a l

A

and in our

F.

Then its c h a r a c t e r

whose p r o j e c t i o n s Proof.

~

Ind([,~,T)

X

T

genuine

vanishes

on elements

are not squares.

is a class function

satisfying

T L for all of

~

~ c A in

A

and

C ~ Z 2.

is not a square,

L On the other hand, by Corollary 2.12

if the p r o j e c t i o n

101 for some

~ c ~.

Thus our claim is immediate.

Lemma 5.5.

The character of

T(~I,~2)

is computable

from

2 Corollary 5.3.

Indeed suppose

X

(5.6)

(a)

:

[ = [(~ 0

~0 2]'~)"

Then

11-~1(c~2)~2(,82 )

T(P-1, ~ 2 ) Proof.

HllbertTs symbol is trivial on squares.

Lemma 5- 6.

Suppo s e (i = 1,2)

~i(a) : ~i(~) for all squares Then

T(~I,~2 )

[]

a c FX~ equivalently, is equivalent to

(~i)2 = (vi)2

Fx .

on

T(Vl,V2) ~ in particular

multiplication by characters of order 2 is irrelevant. Proof.

Compare characters~

Concluding Remarks

[]

(the case of a dyadic field).

Suppose the residual characteristic of

F aI

is even, say

2 n.

Let

0

A0

denote the subgroup of matrices

[0

a2

have even

has unit part congruent to a

square modulo

v-adic order and 40.

Then

T

aI

splits over

irreducible representations

T

with the non-trivial character of in

~.

Thus

(cf. Theorem 5.1 where still valid.

Z

where both

(~l,m2)

?2).

But

has dimension

[FX:(FX) 2] = &).

aI

and

A O, and each of its

(non-trivial on

from a one-dimensional representation

[FX: (FX) 2]

a2 ]

Z2) of

]0

is induced ~0

(tensored

has index

~(2n) ~

Moreover,

Lemma 5.6 is

Since a detailed discussion of the representation

theory of the metaplectic group over a dyadic field will appear elsewhere, henceforth we shall deal exclusively with the case of odd residual characteristic remarks).

(except possibly for a few parenthetical

5.2. Class i representations. In this paragraph we begin a careful analysis of the induced representation Y(Ul,~2 ) = ZndG,~,T(~I,U2))o Denote its space by is non-trivial on Lemma 5.7. of

B(~I,~2 )

~(~l,U2).

The restriction of

Let

B(UI,~2 )

is admissible. space of

V(T)-valued functions on

aI x

tion

to the dense subspace

Then the dense subspace of K-finite vectors in

~(([o all

~(Ul,U2 )

V(L) denote the four-dimensional

consists of locally constant

for

~(UI,~2)

Z2.

~-finite vectors in Proof.

and keep in mind that

aI

aI x ( [ 0 a.2],C)

for

I/2T ( a I

a2]'~)~) = I ~22 I e ~.

~

B(UI,~2 ) satisfying

x

[o a2]'~)~(~)

On t h e o t h e r

hand, by Zwasawa's d e c o m p o s i -

G,

~ = ~ N K.

(5.6)

Thus this space is naturally isomorphic to the space of locally constant V(L)-valued functions on

K

satisfying

a.I x a.2]k) = T ( [ 0 a2]'l)M(k)

aI x

M([0

for all

a1

x

[0

a2 ] e B N K.

(Since

aI

and

a,2

are units in

F x,

aI

I~1 = l . ) To prove admissibility, let

H(~')

~' of

G

denote the subspace of K-finite vectors in

stabilized by

~'.

Since each

compact open subgroup

~' A ~,

values on the finite set finite-dimensional, proved.

fix an open group

~

in

H(~')

each such

K/~' n K.

~

and

~(~i,~2 )

is left fixed by the is determined by its

Consequently

H(~')

is

and the non-trivial part of admissibility is

103

Definition 5.8. class i

A genuine representation

of its restriction to

Y0: ( k , r

[

of

~

will be called

contains the representation

-~ C

at least once; equivalently,

its restriction to

K

contains the

identity representation.

and

2 Theorem 5.9. ~(~i,~2 ) is class i if and only if both 2 i D2 are unramified. In either case the identity representation

occurs at most once. Proof.

Each

~

in

B(~I,~2 )

aI x M([O a2]'l) :

(5.7)

for all

aI x [0 a 2] e B.

a I 1/2 I~I

Moreover,

(hence identically equal to

satisfies

~(e)

if

aI 0 r(~l, P2)([ 0 a.2],l)M(e)

e0 on

is right K-invariant K),

aI 0 T(~I,~2)([ 0 a2],l)~(e ) = ~(e)

(5.8) for all

aI 0 [0 a2] e A n K.

Thus

space of K-finite vectors in) satisfying

(5.8).

~(~i,~2 ) B(~I,~2 )

In particular,

is class i only if (the contains some function

by (5.8)

if and only if (5.8) obtains for some such

~(~I,U2)

is class i

~.

From the Corollary to the proof of Theorem 5.1, recall that aI 0 the eigenvalues of T(~I,U2)([O a2],l ) are

~i = ~l(al)~2(a2 ) h 2 = (~,ala2)~ I h 3 = (|

I

h 4 = (|

1.

and

Thus

~(~!,'~2)

is class I if and only if one of these

hi

is

1

104

for all F x.

al, a 2 c 0 x = U.

Hence

(5.8)

But

(~,.)

is impossible

On the other hand,

suppose

Then by Lemma 5.6 we may assume In this case,

~i ~ i

Note finally Indeed

these

the space

eigenvalues

one-dimensional space

consists

constant V(T)

Analogous

uI

and

in

of

those

a ramified 2 o__rr ~2 2 ~2

and ~2

are unramified.

is class h~

i.

are non-trivial.

(5.8)

of the theorem whose

of

F x.

comprises

Thus a

is complete.

restrictions

in the one-dimensional

to

subspace

(This

K

are of

~i. )

results

hold for dyadic

of

is ramified.

characters

satisfying

~

character

are also unramified.

P(Ul,~2 )

ramified

~(~i,~2 )

and the proof

2 ~i

~2, h3, and

then define

precisely

to

2 ~i

both

~i = i,

and take their values

belonging

GL 2 (0).

space

if either

and consequently

that when

of functions

defines

fields with

KN

replacing

5.3. Hecke operators. Let

H(~,~ 0) denote the Hecke algebra of continuous

supported functions

~

(5.9)

on

r

for all

[,~T

in

G

compactly

satisfying

g ~') : Yo(g)r

K.

Multiplication

in

H(G,~0)

is given by the

convolution product (5.i0)

r

Note that if r