127 60 6MB
English Pages 156 [146] Year 1976
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 Heidelberg 9 New 9 York ISBN 0-38?-0?799-5 Springer-Verlag New York Heidelberg 9 Berlin 9 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 9Heidelberg 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
46
KrouD
. . . . . . . . .
51
theory
. . . . . . . . .
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)
where F|
s
is a complex
is the upper The series
:
defining
continuation
to the whole
The residue
~ e ~(N)
form which
s-plane
%(u)
of
can not
over
Rather
~(~T~.
One of Kubota's
of
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,
= v
E(u,s)
is not a ~usp form~
with respect
is in
is an Eisenstein
~ l,
and defines
This function
covering
Re(s)
H
s+l
v(u)
subgroup
on
interest
•
E(u,s)
only for
for all
Z
r XZ(Ni
triangular
function~
his special
variable,
it converges
s = 1/2.
modular
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
q
constituents of
r
q
9 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) = [
ac
(2.7)
For
gl'g2
in
(2.8)
G,
i 0 g = [0 det(g) ]p(g)
write
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
(2. lO) if
v ( y , s)
s = [c
]'
Note t h a t
coincides with
a
1
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~
\ (y,d)
c~~
to
SL2(F) X SL2(F)
9
Proposltion 2.6. equal to
=f
~=[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 g ~ G,
distinct)
group
ways to
9 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.
s(k) =f(c,d(det k))
\
(2.13) Proof.
1
c%0
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 Indeed
if
9 Then
Throughout this proof assume
dot(k) --ad-bc
c~U
a b d ] c KN k = [c
Suppose
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)
~
determines a covering group of
G
isomorphic
But according to Proposition 2.8, its restriction Thus
lifts as a subgroup of
~N ~
is isomorphic to
via the map
Lemma 9. II.
K~• Z 2 ,
k ~ {k,l].
this reason we shall henceforth deal exclusively with
For
~ .
Suppose
F
~i
gi = Then
denote
~*(gl,g2 ) s(gl) s(g2)s(glg 2)
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
xi]
i=1,2
~ B,
o
~(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
Corollary
~,
is not a square in
and
h' ~ F x
~' = I, say.
Thus
V
and the proof is complete. of
G
A 2 = {y ~ A: y = 8 2 , 6 ~ A]
9
A2
2.13.
The subgroup
N
The center of
F J g. ~
lifts as a subgroup
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 ] , ( ~ ] :
will not []
with
Fix
F x, then
symbol,
for some
fails for
2.12'.
z(g) (b)
say,
~
(both sides equal
Obvious.
Corollary (a)
if
= -i
(2.15)
So does
Proof.
obtains by default
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
>=
(~,~~
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
If
exp( 88
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 i=l
Theorem of
F
2.22.
defined
Suppose
9
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 introduced
n
for
or
q3 ;
rq3 ; by
a b [c d ] "
given in T h e o r e m 2.22 depend on
I shall denote
rn
by
rn(~ t) ;
for the r e p r e s e n t a t i o n
Tt "
the
eventually
GL 2 .
= [s,(] ~ST2(F), Then
rn
will be reserved
Proposition
2.6).
for
for emphasis, r
or
q =ql
will be abbreviated
Note that the formulas Therefore,
rql
as
zero;
2.27.
(Cf.
define
~a
[18] Lemma to be
1.%.). a
If
is ,~ v(a,s)]
a ~ F x, (cf.
and
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 2a : ~ a
:
0
[-a-lo]
a
o ~ {1,(a,~)]
a
:
[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)
0 a-
lal n/2 ~(qn'Ta)~ ~(aX) 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,
R
a
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 rq
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
r
= laI-i/2~ta_l(X)
Proposition 2.31 9 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)~
rq( [a0 a-O 1 ] ) ~ ( X , t ) =
(2.87)
Fn
~(Y,t)Tt(q(X,Y)dtY
a,a)laln/2
~~(q'Tat)
~(aX, t)
and
0 )~(X,t) r q ( [ o1 a]
(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
r l ( [ 0 a 0a])
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.
ratic
theoretic
F
form in
is a number
is the procedure
terms by Well
n
field and
variables. e(~)
Piecing
together
ST2(~ )
reformu-
in [47].
For
v
of integers
of
Fv
As in Subsection of matrices
GL2(0v) place
2.1,
integer
Proposition v,
rv(q )
~(rq(S)~
attached
to
define
Uv
place
i0
v
denotes
0(rq(~)~)
= 0(~)
Fn
For
GL2,
F
let
0v
by
rq
is
for all
q.
of
its group
representation K vN
with
of units. of GL2(Fv)
is the subgroup a ~ I
divisible
by
s e SL2(F).
the point
of
and
of
denote
Let in
Thus
rv(q)
denote
L2(F n x Fx).
GL2(0v)
c ~ 0 (mod N).
4.
the ring
consisting Here
K~ = GL2(O v)
N
if
is Fv
characteristic. 2.32.
Suppose
is class
q = ql
I, i.e.,
or
=
(
n
~ i=l
function
Then for almost of
rv(q)
More precisely,
to
for each
by
the characteristic
the characteristic
q3"
the restriction
~v0 e L2(F n x Fx)
O .... Xn, t ) ~v(Xi,
(2.50)
on
a representation
that
has at least one fixed vector. v,
denotes
~(r
s
~
quad-
2.32 below.
and Weil
[~ bd]
has odd residual
every
:
a non-archimedean
the corresponding
a positive
a distributuon
In particular,
is Proposition
For
is an F-rational
Define
is defined with the property
This is the theta function departure
q
local Well representations
SL2(F)-invariant.
Here
This
from theta-
the idea is this. Suppose
of
are constructed
10
v
) |
1U N
function of
v of
OF
and
IuN v
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
[o a ]
(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 rv(q)([ 01 oa])~v(X,t) = lal-i/2~~
obvious since
a
To define
G~,
rq
globally,
on
~
if
~(X,t) = N ~v(Xv, tv) ~v(Xv, tv)
creasing on (ii)
Note also that rv(q)(w)~~ ta -I)
=
oix ' t)
is
~
on
~
• ~
is "Sehwartz-Bruhat"
and:
is infinitely differentiable x
• Fv
for each archimedean
for each finite
on
space of Schwartz-functions
v,
~v
and rapidly de-
v~
is the restriction to
of a locally constant compactly supported function on (iii)
) = ~v~
and to introduce theta-functions
We shall say that V
(i)
b e 0v, and
U v.
we need first to define an appropriate • ~.
has conductor
V
V
The fact that
e
T
for almost every finite v, ~v = ~o V
~v x ~v
F~v+l;
(the function defined
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 ~(~ O
and
%v = @v"
~(~
X F~).
x F~).
Thus
~v T
rv(q)(~v)%v
defined as follows.
on
~
For each
Proposition
i. e. ,
(unitarily)
in
to
L 2.
corresponding ~ e ~(~
to
• F~)
q (or rq)
define
8(~,g)
are on
2.33.
e (~,~)
y
(rq(~))@({,D)
For each
.
y e GL2(F),
8(%,Yg) = @(~,~);
Proof.
with
again belongs
is now played by a non-trivial
Z
(2.52)
with
operates
~v e GL2(Ov)
by
8(~'g) :
and
rq(~)
and rq(~)%
v,
for at
F~.
The theta-functions
a~
= %v
in the local theory
of
is meaningful
Indeed for almost all
By continuity,
The role of character
2.32 this definition
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$8)
Therefore
as d e s i r e d .
The f u n c t i o n
~,
i.e.,
for example,
For simplicity, that
summation
8(9,g)
8(~,g)
always defines
is always
it may often be the case that
Suppose,
follows
from Poisson's
suppose
Finally rq(~)~(y,~)
follows
Then
suppose
that rq(~)~
also that
rq([ -i0 - ~])~(X,t)
e(~,
But by the GF-invariance
of
~(-X,t)
8(~)
-1
invariant.
is identically
= -@(X,t),
i.e.
~
is also odd for all
~ e ~.
F = Q.
(2o~7)
From formula
= -~(X,t),
[ o
GF
an
o
]7)
is an
it
i e.
: - e(~,~).
8,
-i ~]~) Therefore
8(~,~)
We close the basic
i 0~
this paragraph
by demonstrating
how
8(~,~)
generalizes
theta function 2~in2z n ~
Proposition = n ~p
Itl -l/q
where
e- I t ! 2 ~ x 2
2~176 ~p = ~o P Then
Fix
n = i, F = ~, and
for all finite
if
8(%,g)
g = [[yl/2 0 = yl/As(x+iy)o
p
rq = r I.
and ~ (x,t) =
xy-l/2 -1/2 ]' 1 ], y
Suppose
50
Proof.
Since
[yl/2 xy-1/2 0 y-I/2 ] =
Z rq(~)%({,~). - .
Here we choose 0p
for
only if
T = ~ Tp
p < ~. { e ~
Thus and
- =
({,~)
with
({,~)
I x
[yl/2
[0 1 ] 0
lyll/4
e-2W7~ 2 e2~i~2x~.
T (X) = e 2~ix contributes
9 = i (recall
o
y -I/2]'
and
Tp
trivial
to the summation
~2 = 102~176
l.e.,
e2~in ~ ( x + l y ) as was to be shown~ Note that for
a c (0,~),
a o r([ 0 a ] ) ~ ( ~ , ~ )
Thus
8(~,~)
is actually
and
=
-I/2
as above,
=
ai
=
al-i/2
=
tl-i/~ e-ltI2~x2
defined 7
@
z~
on
~(ax, ta -2)
Ita-21-1/4
e-ltl2~x2
above
on
w
AutomorPhic Forms on the Metaplectic Group: Global Theory.
3.$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)
~s above.
-~ K*
with K-~
=
0 ~ e ( 4w. must make
elements ~*
~/4~ ~
its character
rather
[~(e)
=
than as the group
~sine
of pairs
to
[[r(e),~}]
cose~
between
correspond
to
these
7(2~)
(and at most
can have this property). Z2
it is convenient
rCOSe - s i n e ] ]
The isomorphism
~(0),-i}
on
group
Thus
are of order two
non trivial
S0(2),
does not split over
To describe
as
(3.#)
.~
Although
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
Here
~ =
yl/2 [ o
The Casimir
x y-z/2 y-l/2]
operator
T(e)
,
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
2 = (x,y, 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)
_
K*~
w
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
SL2(Q ),
k 0 =[k0,1 ]
r l ( N ) = SL2(Q) n SL2(R).~~ By "strong approximation"
SL2(Q)SL2(~)~ ~.
Then
o
determined up to left multiplication
Proof.
S-~2(~).
More precisely,
for
in 4 n by
SL2(~), and g~ in
[Y0, S~(Y0)]
in
9 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
Av fw,
in
fv, l
as H
of
w i t h the required
~ Hv
to
H
( ~ v Hw) | H$ invariant
a unit vector
H~
is invariant
for
are both irreducible
from
w~v Hw = Hv ~ ( ~ v
the only vector in
So choose an ortho-
V
it will suffice to construct
Thus for each
large,
v ~ v O.
Finally we need to check that
for each
~') contains
Hv)'
for each v.
for
in H'
invariant
under
(w~v Hw)
into
compatibility
Kv
is of
invariant
V
and
com-
fv, l K v. H
is
for
(up
Thus for (namely
conditions
satisfied.
3.3
The S p e c t r u m Let
GF\GA
L2(~)
of the M e t a p l e c t i c
denote
which are
the Hilbert
trivial on
ZO
Group
space of m e a s u r a b l e
functions
and square-integrable
on
on
= ZO GF~A,
a quotient
s p a c e of f i n i t e
volume.
The m e a s u r e on
~
is such that
f(~)d~ = ~ [ Z f((g,~))~dg x ~z 2
if
Let L2(X)
L2(X)
and
for future
Set
denote the subspace of
~2(~)
decomposition
(A~
GF\a~2(~)/Z ~
x : x/z 2 :
of
its
orthoeomplement.
L2(X)
are w e l l - k n o w n
reference.
B~
For details,
see
denote the non-unimodular
is the group
complex number
of positive
s,
let
diagonal
e sH(b)
b =
denote
Z2-invariant
functions
Basic facts
concerning the
but we recall [i~
and
them here
[17].
subgroup matrices
in
~A~ in
of
A .)
B~
For each
the character
-> a2
of
B~.
Here
~(b) : log
(3.12) with
a I 1/2
I=-I
In general,
~2
suppose
g = nh t akz nc N~ ' h t =
k c K= K *~ Let on
(3.13)
G~
[e 0
K v , and H(s)
t
[a I 0 0 -t ], a c A~ = [ ] ~ A~:]all e 0 a2
z c Z ~0 9 Then
denote the space
=
la21
= i~,
H(g) = t. of c o m p l e x - v a l u e d
such that
~ I~(k) l 2 dk
irreducible if
~lU21(x)
= Ixl I/2 .)
For
v
finite,
will almost always be the class 1 quotient of the principal v series representation pv(Ixl I/4,
~xj-i/4)
Now suppose at
s = 1/2
E(g,~,s)
generates
is the Eisenstein series whose pole
~ .
Then the eigenvalues of
with respect to Hecke operators in
H(G-L2(Fv),O O)
E(g,~,s)
can be computed
in terms of the spherical function theory of the local representation ,
i.e., from Theorem 5.13.
Kubota's results for
E(u,s)
The results are in agreement with (ef. Section ! and [21], P.53).
6.~
Ce.nstructlon of cusp forms of half-lntegral weight Fix
F~Q
and
~ =AQ.
Let
(~A)gen
denote the set of
equivalence classes of irreducible unitary genuine representations of
~
and
G~
the set of equivalence classes of irreducible
uultary represeutatlons of certain
~
weight
in
k/~ ,
(GA)gen
G&
9 From Section 3 it follows that
correspond to classical modular forms of
It is also well-known that certain
correspond to classlc~l modular forms of weight
~
A
in
G~
k-l.
Motivated by the correspondence between forms of weight ~nd
k-I
discovered by Shlmura it is natural to introduce a general
correspondence between
(Gg)gen
and
4
G~
following the local theory
developed in S~ctlons h and 5.
Thus we let
of
(irreducible unitary genuine
l-1
maps between
represeutatlons of of
k/2
G~ )
(Gp)gen
~p )
Gp^
and
~Sp]
denote a family
(irreducible unitary representations
with the following three properties:
(i)
for each
~
A
(~&)gen'
S(~) =@ Sp(~p) determines a well-
deflued irreducible unitary representation of (ii)
for almost every
~,
S(~)
G~ ;
is automorphic if
~
is
autemorphlc~ (iLl)
if
~p
is not supercuBpidal,
in which case
~p
is
naturally ~s~oclated to some pair of quasl-characters (Wl,~2) x of Qp~ Sp(~p) is ~Imply ~(W~,~), the irreducible representation of
Gp
associated to the pair
2 (~1,~2).
Implicit lu (1) is the fact that
Sp
preserves class 1
~e~reseut~tlon~ (Proposition 5.16) and the fact that f~ctored as
| ~p
with
~p
class I for almost every
~
can be p
(Theorem
B.~), The necessity of the word "almost" in (ii) results from our having defined "automorphic representation" rather rigidly. we not required automorphic forms to occur in have to worry about
S
L2
Had
we would not
sometimes taking square'integrable cusp
126
forms on
~&
to slowly increasin~ non-cuspidal forms on
G~
See Conjecture 6.2 below and the remarks directly following it. Note that if
~
and
S(~)
ar~both
cusp forms then the "strong
multiplicity one theorem" of Casselman [4] and Miyake [27] implies that the family
[Sp]
is actually uniquely determined by conditions
(i)-(lil). To prove that at least one such family
[Sp]
exists one has
only a finite number of "bad" primes to worry about. every component ~%)
S(~p)
%
of
~
is non-supercuspidal and (for such
is defined as in (iii).
form is used.
To prove (ii) the Selberg trace
We shall treat these matters elsewhere,
in the near future.
In the meanwhile,
why constituents of
--*T O should occur in
we use r3
Note first that the restriction of forms must agree with S
or
Sp
of
below to explain r1
G
Indeed by (iii)
~k/2
of
Zp
for
corresponding to
T(p 2)
S(~)
to the
Furthermore,
~p
and
S(~p)
are class
corresponds to a classical modular form of weight
elgenvalue
~
according
preserves eigenvalues for the zonal
spherical functions as soon as ~
Vk_ 1
hopefully
to holomorphic cusp
takes the discrete series representation
to Proposition 5.17,
for
S
S
Shimura's correspondence.
discrete series representation
if
Indeed almost
1 . k/2
So with
(cf. Shimura [38]) the "new form"
is of weight
k-i
and has eigenvalue
P
T(p). Now recall the correspondence D3:~ ~ ~ .
This correspondence is defined on a subset of in ~*.
(G~]gen
For convenience, ~
For each place
rp(q3,Tp)
sentation of
S--L2(%)
let in
and takes values
we restrict our attention again to
In fact we reformulate p
G~
L2(~)
in the context of
S--L 2
as follows.
denote the Wei! repre-
attached to
q3"
We assume in
127
advance
we have fixed a character
T (x) = e ~ix it follows is class
Since
Tp
is trivial
from the discussion
one with respect
According expounded
T = ~ ~ on
to
SL2(0p)
{\A
with
for almost
of Subsection
to the philosophy
in Section
of
P 0p
2.6 that
for almost
every
P
rp(q3,T p)
every
p.
of the Well representation
2 the representation r3(T ) = | rp(q3,T p)
is a well-defined algebra
is generated
PGL2(~)
in
representation
by the adjoint
L2(~).
one obtains of
genuine
We denote
and irreducible
By inducing constituents
~
map we call
~.
action
between
Z~ \ G ~
of
A
action by
irreducible of
we obtain a map
and constituents
The non-trivial
whose
commuting
of the orthogonal
representations
to
S-L2(A)
this adjoint
a i-i correspondence
r3(~ )
of
~
GA
fact about
A.
which occur in
r 3.
D3
Thus
constituents
(no longer of
group
i-I) between
This is the
we use below
is
that
(6.1)
D3(s(~))
whenever
the left-hand
=
side makes
sense,
i.e.
S(~ ~)
occurs
in
A . The analogue section 4.3. p .
of (6.1)
One can also check
In general,
one wants
8(~,~)(g)
generate Here
:
a non-tzivial
~ e .