JOURNAL OF THE
AMERICAN MATHEMATICAL
SOCIETY
Volume 17, Number 3, Pages 679-722
S 0894-0347(04)00455-2
Article electronically
published on April 1, 2004
OF THE CENTRAL VALUE
ON THE NONVANISHING
OF THE RANKIN-SELBERG
L-FUNCTIONS
DAVID GINZBURG, DIHUA JIANG, AND STEPHEN RALLIS
to Ilya
Dedicated
I. Piatetski-Shapiro
with
on
admiration
the occasion
of his
75th
birthday
1. Introduction
Let
of
tti and 1T2be irreducible unitary cuspidal automorphic
representations
a
is
to
A
and
where
the
of
adeles
attached
ring
respectively,
GLm(A)
GLn(A),
number field k. The basic analytic properties
continuation
and
the
(meromorphic
x 7^) have been
functional
of the Rankin-Selberg
L-functions
equation)
L(s,tt\
established through the work [JPSS83], [CPS04]and also [Shd88]and [MW89]. It
x
from [Shd84] that the L-functions
L(s,tt\
method
and
the
method
by
Selberg
Langlands-Shahidi
The problems related to the value of the (complete)
=
L(s,7Ti x 7T2) at the center of symmetry
(i.e., s
|)
For instance, when both 7Ti and 1x2 are self-dual, the
to be nonnegative.
is expected
See [Lp03] and [LR03]
see
and
for
relations
of this problem
problem
[ISOO]
in characterizing
We are interested
the nonvanishing
L-functions
Rankin-Selberg
L(s,-K\ x tt2) in terms of
is known
1x2) defined by the Rankin
are the same.
L-function
Rankin-Selberg
are often very interesting.
x 7^)
central value
L(^,i\\
for a recent account of this
to analytic number theory.
of the central value of the
the nonvanishing
of certain
period
integrals.
of L(s, 7r? x 7^) ([CPS04]), the nonvan
Using the global integral representation
x
is
to
of
to the nonvanishing
be
of certain
ishing
equivalent
7T2) expected
L(|,7Ti
period
of automorphic
integrals
forms
in 7Ti and
^2
over
a general
linear
group.
How
to
ever, as remarked in [H94, ?5], one likes to relate the central value of L-functions
an
nature.
of
From
the
arithmetic
of
and
periods
Langlands principle
functoriality
we search for periods which
the philosophy
of the relative trace formula method,
are potentially
x 7^). It will
and related to the central value of
arithmetic
L(^, tti
be a very interesting problem to look for integral representations
L
of automorphic
functions based on period integrals of an arithmetic nature ([H94, ?5]). This should
lead to expressions of the central values of automorphic
in terms of the
L-functions
relevant periods which have much deeper arithmetic
From
this per
implications.
our
can
a
as
in
this
be
viewed
towards
such
paper
spective,
study
step
preliminary
intrinsic
relations
Received
by the
2000 Mathematics
between
editors
May
arithmetic
geometry
and automorphic
forms.
8, 2003.
22E55.
Classification.
11F67,
Primary
11F70,
22E46,
form.
phrases.
Special
value, L-function,
period,
automorphic
is partially
second author
the Sloan
supported
by the NSF grant DMS-0098003,
and the McKnight
Land-Grant
of Minnesota).
Fellowship,
Professorship
(University
Key
The
words
Subject
and
?2004
679
American
Mathematical
Research
Society
DAVID
680
GINZBURG,
DIHUA
JIANG,
AND
STEPHEN
RALLIS
In order to explain our work in a precise way, we recall the Langlands
functorial
lifts from the classical groups to the general linear group. Let tt be an irreducible
Then the Rankin-Selberg
of GLn(A).
representation
unitary cuspidal automorphic
? 1 if and
=
x
a
s
tt
n
at
has
if
L-function
pole
only
7rv, the contragredient
L(s,
tt)
of 7T. In this case the pole is simple. Now assume that tt is self-dual; i.e., tt = 7rv.
We
have
=
L(s, 7Tx 7r) L(s, 7T,A2) L(s, 7T,Sym2)
is the exterior square L-function
and L(s,7r,Sym2)
L(s,7r,A2)
tt
to
It follows that
attached
square L-function,
respectively.
is the sym
for any given
one
of GLn(A),
irreducible unitary self-dual cuspidal automorphic
representation
a
and only one of the two L-functions
and
has
simple
L(s,7r,A2)
L(.s,7r,Sym2)
= 1. If the exterior
square L-function
pole at s
L(s,tt, A2) has a simple pole at
s ? 1 (which implies that n is even [K00]), we say that tt is symplectic,
and if
=
a
s
we
at
the symmetric
L-function
has
square
1,
say
simple pole
L(s,7r,Sym2)
is clearly compatible with the Langlands
This terminology
that 7T is orthogonal.
functorial principle and with the recent work on the Langlands
functorial lifts from
where
metric
irreducible
generic
cuspidal
automorphic
representations
of classical
groups
to the
general linear group ([CKPSS01], [CKPSS], [GRS99c], [GRS01], [S02],and [JS04]).
is generic if it has a
We say that an irreducible cuspidal automorphic
representation
a
nonzero
to
coefficient
with
Whittaker-Fourier
respect
generic character.
(global)
of the automorphic
The following is the main consequence
descent constructions
in [GRS99c], [GRS01],and [S02].
Let tt be an irreducible unitary self-dual cus
(Ginzburg-Rallis-Soudry).
representation
of GLn(A).
pidal automorphic
If the exterior square L-function
=
s
an
a
at
exists
then
there
has
irreducible generic cuspidal au
1,
pole
L(s,7T,A2)
=
a
tomorphic representation
2r) such that tt is a weak Langlands
(n
ofSO2r+i(A)
a.
square L-function
lift from
If the symmetric
functorial
L(s,tt, Sym2) has a pole
Theorem
at
s =
1,
then
if n
=
21
is
even,
there
exists
an
irreducible
generic
cuspidal
au
a of SO21 (A) such that tt is a weak Langlands functorial
tomorphic representation
?
n
21 + 1 is odd, there exists an irreducible generic cuspidal
lift from a; and if
a of Sp2? (A) such that it is a weak Langlands functorial
automorphic
representation
a.
lift from
that at almost all
We recall that tt is a weak lift from a means by definition
of tt is a local Langlands
functorial
lift from the
local places, the local component
a.
Now
it
is
known
lifts in the
of
that
the
weak
local
component
corresponding
lift
functorial
with the local Langlands
theorem are strong; i.e., it is compatible
and [CKPSS] for split classical
at every local place (see [JS04], [K02] for S02n+i
lift
the Langlands
functorial
From now on we simply do not distinguish
groups).
lift. We remark that by the local converse theorem (which
D.
Jiang and D. Soudry for S02r+i
by
([JS03] and [JS04])
in
for
classical
is
work
other
which
their
and
progress
groups), one can prove that
a
in
is
the theorem
the representation
uniquely determined
by tt. From the
given
one
deduces
above theorem
from the weak
has been
functorial
established
Let tt be an irreducible unitary self-dual cuspidal automorphic
represen
Corollary.
then n = 2r is even and tt is the functorial
tation of GLn (A). If tt is symplectic,
lift
a o/S02r+i(A).
representation
from an irreducible generic cuspidal automorphic
then if n = 21 is even, tt is a functorial
lift from an irreducible
// tt is orthogonal,
ON
generic cuspidal
tt is a functorial
THE
NONVANISHING
OF
THE
CENTRAL
VALUE
681
a o/S02/(A);
and if n ? 2/ + 1 is odd,
representation
automorphic
lift from an irreducible generic cuspidal automorphic
representation
a of Sp2l(A).
Let tt\ and 7T2be irreducible unitary self-dual cuspidal automorphic
representa
tions of GLm (A) and of GLn (A), respectively. When
tt\ and 7T2are both orthogonal
or both symplectic, the Rankin-Selberg
x 7t2)may have
product L-function
L(s,tt\
a pole at s = 1. The existence of the pole of L(s,7Ti x 7r2) is equivalent
to the
=
seems
that
to
It
characterize
the
7T2
very mysterious
property
tt?.
nonvanishing
of the central value of L(s, tt\ x 7r2) at s =
in this case in terms of period integrals.
\
is to provide a characterization
Our objective
for the nonvanishing
of the central
value of the Rankin-Selberg
product L-function
L(s,7Ti x TT2)when one of the two
one is orthogonal.
is
and
and
the
other
In terms of
7T2 symplectic
tt\
representations
case
the global Langlands
this
be
should
the
where tt\ ? 7T2
reciprocity conjecture,
is symplectic, and the previous should be the case where tt\ (g)TT2is orthogonal. One
on the terminology
and the tensor product
lift
may find more detailed discussion
in [R94, ?3]. The symplectic case leads to one of the following two cases.
Case
1. m = 2/ + 1 is odd and tt\ is orthogonal;
n = 2r is even and 7T2is symplectic.
Case
2. m = 21 is even and tt\ is orthogonal;
n = 2r is even and 7T2 is symplectic.
In Case
that tt\ is
1, we know from the above theorem of Ginzburg-Rallis-Soudry
a Langlands
functorial lift from an irreducible unitary generic cuspidal automorphic
a of Sp2? (A) and 7r2 is a Langlands
functorial lift from an irreducible
representation
r of S02r+i(A).
unitary generic cuspidal automorphic
representation
By the global
we know that if the standard L-function
theta correspondence,
7^ 0, then
L(^,r)
r is a global theta lift (with respect to a given character
the representation
if) from
an irreducible
r of Sp2r(A),
cuspidal automorphic
representation
where Sp2r
double cover of Sp2r. In this case, we call 7T2a if
transfer
of r from Sp2r to GL2r The main result of this paper is to characterize
x
the nonvanishing
of L(
in terms of the nonvanishing
of the period attached
|, tt\ 7r2)
to either (a, r, Sp2/) if r > / or (a, f, Sp2r) if / > r. From [F95], one knows that if
=
0, it is expected that the global theta lift of r to Sp2r should be zero and
L(|, r)
case here.
the theta lift to
Sp2r+2 is cuspidal. We will not discuss this
In Case
lift from an irreducible unitary generic cuspidal
2, tt\ is a functorial
a of S02/(A)
and 7r2 is a functorial
lift from an irre
automorphic
representation
r of S02r+i(A).
ducible unitary generic cuspidal automorphic
The
representation
x 7r2) will be characterized
of
in terms of the nonvanishing
nonvanishing
L(^,tti
of the period attached
to either (a, r, S02?) if r > / or (a, r, S02r+i)
if / > r. By
unitary generic
is the metaplectic
the Langlands
from generic cuspidal automorphic
of
functoriality
representations
special orthogonal groups to the general linear groups ([CKPSS]), one might define
L(s,a
x r)
:= L(s,7Ti
x 7r2).
on the characterization
of the nonvanishing
of the central value
>
in
terms
r
of
to
attached
either
if
/ or (a, r, S02r+i)
periods
(a, r, S02/)
L(|, axr)
if / > r is a conjecture of Gross and Prasad
(see [GP92] and [GP94] for the global
= Z or / ? 1 and for the local
in general). When
conjecture when r
conjecture
r = 1 and 1= 1, the assertion was proved by Waldspurger
in [W85]. When
/= 2
and r = 1, it is a conjecture of Jacquet on the relation between the nonvanishing
Then
the assertion
682
DAVID
of the central
GINZBURG,
DIHUA
AND
JIANG,
STEPHEN
RALLIS
and the nonvanishing
of the triple product L-function
of the
was
in
For
it
the
and
proved
periods.
split period case,
[Jng98b]
[JngOl].
in [HK91] and [HK04]. When
For general period cases, it was proved completely
r ? 2 and 1 = 2, some special cases were studied
in [HK92] and [BFSP04].
In
value
trilinear
general this will be the subject matter
be omitted here.
The main
results
of this paper
of our work
(dealing
with
[GJR], the detail
Case
1) can be
of which
will
formulated
as
follows.
Main
Theorem.
representation
Let tt\ be an irreducible unitary cuspidal orthogonal automorphic
and let 7r2 be an irreducible unitary cuspidal automor
of GL2?+i(A),
Assume
that the standard L-function
representation
symplectic
o/GL2r(A).
a
an
Let
be
irreducible
0.
repre
unitary generic cuspidal automorphic
L(\,TT2) t^
to tt\ and let r be an irreducible unitary
sentation of Sp2? (A) which lifts functorially
phic
generic
cuspidal
automorphic
representation
o/Sp2r(A)
which
has
the ip-transfer
7T2
(1)
(Theorem
If the period
5.1)
integral
Vr,r-l(<t>U4>r,<fl) (r > I) Or Vl,l-r(4>r, <t>UVr)(r < I)
to (a,r,ip)
is nonzero for some
(see ?2 for an explicit definition)
choice of data, then the central value of the Rankin-Selberg
product L
x 7r2) is nonzero.
function L(|,7Ti
that Assumption
(FC) (see the remark below)
6.3) Assume
(2) (Theorem
hold for the pair (r, a). If the central value of the Rankin-Selberg
product
x 7r2) is nonzero,
then there exist an irreducible unitary
L-function
L(^,tt\
a\ of Sp2? (A) which is nearly
representation
generic cuspidal automorphic
a
an
to
and
irreducible
represen
unitary cuspidal automorphic
equivalent
tation T\ of Sp2r (A) which is nearly equivalent to r, such that the period
attached
integral
Vr,r-i(<Pi,
attached
choices
4>r, <Pi) (r >
to (<7i, t\,
of data.
I)
or
Vij-r(4>r,
ip) (see ?2 for an explicit
(pi, <pr) (r <
definition)
I)
is nonzero
for some
in the above theorem are generalized Gel
Remark.
First, the periods considered
are
called
and
fand-Graev model
type, which will
periods of Fourier-Jacobi
integrals
of the periods
defined in ?2. The nonvanishing
be explicitly
implies the implicit
a
the
character
for
and
character
the
relation between
ip for the -0-transfer
generic
Secondly, part (2) of the above theorem has been proved based on Assumption
of a residual
of certain Fourier coefficients
is about the nonvanishing
(FC) which
in detail in ?6. In
to the pair (a, 7r2) and will be discussed
attached
representation
=
=
1, while for the case I
0, it
?7, we verify Assumption
(FC) for the case of I
of t.
in [GRS99a].
In other words, part (2) of the above theorem has been
> I = 1,0. Finally,
r
case
it is expected
that t\ in part (2) is
the
for
completed
we
can
holds.
6.1
it
when
but
prove
Conjecture
only
generic,
is proved
The paper is organized as follows. In ?2, we introduce the notation which will
In ?3, we recall the basic facts of Eisenstein
be used in the rest of the paper.
series. ?4 starts with the
of
location
the
series and determine
poles of Eisenstein
truncation
which
needs
Arthur's
of
Eisenstein
series,
study of periods of residues
ON
THE
OF
NONVANISHING
THE
CENTRAL
683
VALUE
to justify the convergence. We provide the details for the case r > I. The
method
is an
other case is treated similarly. The main result here is Theorem
4.4, which
an
to
series
the
of
the
residues
of
Eisenstein
'outer'
the
period
identity relating
series. The idea was used to
'inner' period of the cuspidal datum of the Eisenstein
of the central value of the third symmetric power L-function
study the nonvanishing
of GL2 in [GJR01], which is a similar, but lower rank, case. In general such an idea
has been used to treat many cases in [JR92], [FJ93], [Jng98a], [JLR04], [GJR03]
is Theorem
5.1,
[GJR]. In ?5, we prove part (1) of the Main Theorem, which
same
as
in
treats
Main
of
the
part (2)
Theorem,
argument
by using the
[GJR01]. ?6
is Theorem
6.3. Here we use some ideas from the nonvanishing
of Fourier
which
to unipotent
of automorphic
forms attached
orbits
coefficients
[GRS03] and the
construction
of automorphic
descent maps
[GRS]. In ?7, we verify the important
?
1.
Assumption
(FC) for the case of I
and
2. Periods
In this section, we give a formal definition of the periods for automorphic
forms
in this paper. The remaining sections are devoted to the study of these periods
in the discrete spectrum.
for automorphic
forms occurring
a
number field and A the ring of adeles of k. Let
be the fc-split
Let A:be
Sp2p
form given inductively
group of rank p, which preserves the symplectic
symplectic
used
by
?
J2p
J2p-2
?y
V-i
Let {ol\ ,OL2, - - ,OLp]be the set of simple roots, which determines
the Borel subgroup
B = TU with U upper-triangular.
We have to consider two kinds of parabolic
in two different symplectic
groups. To unify the notation we are going
subgroups
to use, we introduce one standard parabolic
subgroup
of
Sp2p
unipotent
with
the Levi
radical
VPjP-i
*p,p?i
Lip,p?i
Wp,p
part LPjP_?
to GL\
isomorphic
x
Sp2(p-i)'
We
write
the
as
y
(2.1)
K.
:=
v(n,x,y,z)
o
ip
x'
0
eSP: 2p
n*J
V
n G iV? is the standard (or upper-triangular)
maximal
unipotent
we may
there is no confusion with the indication of
GL?. When
Sp2p,
where
*%p?i
Another
standard
parabolic
*i,p?i
=
Ltp?iVp?i'
of
subgroup
subgroup of
simply use
Sp2p,
we will
consider
is denoted
by
=
^I%,p?i^i,p?i
the Levi part M?)P_? is isomorphic to GL?
Here the indices {i,p?i}
xSp2(p_?).
form a partition of the rank p of
Since there will be two different symplectic
Sp2p.
to avoid unnecessary
groups occurring in the rest of the paper, we use such notation
where
confusion.
684
DAVID
DIHUA
GINZBURG,
AND
JIANG,
STEPHEN
RALLIS
It is clear from the definition
is a normal subgroup
that Vp-i+\
in Vp-i.
Let
=
onto
be
the
from
the
which
projection
quotient Vp-i/Vp-i+i,
lp-i
Vp??
?p,p-i
?
is the Heisenberg
of dimension
group Hp-i
2(p
i) -f 1. We may also identify
with the subgroup
(section) in Vp-i, which has the form
tp-i(Vp-i)
\
A-i
(2.2)
cp ?2
?
Lp l
(?)
T
eSP;
'2p
/i-l/
V
and z is the
x, y are the i-th (the last) row of x, y in (2.1), respectively,
z
in
of
(?, l)-th entry
(2.1).
For a given nontrivial
to be the generic character of
character
ip of fe, define ^
N? of the form
Here
(2.3)
We
:=
ipi(n)
also view
tpi as a character
U^p-i
is a normal
+ rii-i,,).
of V^_j by composing
VrP?I
Clearly,
+
^(ni,2
y
subgroup
.
yp?iI
of
/tt.
vi,p?i
.~?
with
the projection
TV.
?yi'
Vp-{.
To motivate
the choices of unipotent
subgroups above and the characters below,
one has to consider the unipotent
conjugacy classes in Sp2p(C), where C is the field
It is known that the conjugacy
classes of unipotent
of complex numbers.
elements
or
are
classes
of
called
param
unipotent
unipotent
(often being
orbits)
Sp2p(C)
that the odd parts of the partitions
eterized by the partitions with the property
occur with even multiplicity
(Theorem 5.1.3 in [CM93]). We will simply call such
The set of all unipotent
partitions
symplectic partitions.
(adjoint) orbits of Sp2p(C)
or simply by U. Let
is denoted by
be
the
orbit attached to
unipotent
Op?
?/(Sp2p)
to be the normal subgroup
We define
the symplectic
partition
((2i)l2(p~^).
Vopi
of Vp-i of the form
(2.4)
voPti
=
{v(n, ^5 2/,z) G Vp-i
For any a G kx, we define
a character
|Xij
ipQ
=
(2.5) *l>bPti(v) ^(nWKi)
=
yid
=
of Vq
=
0,
l<j<p-i}.
. to be
*l>iAv)
z. It is easy to check
v = v(n, x, y, z) G
Vp-i and z^\ is the (i, l)-th entry of
to a fc-rational orbit in Op?.
that the character ipQ . is the one corresponding
(See
subgroup Vqv? is denoted by U".)
[MW87], where the unipotent
double cover of
We follow [GRS02] and
be the metaplectic
Let
Sp2(? (A).
Sp2q (A)
of
and the
of basic structures and representations
[194] for the discussion
Sp2g (A)
related Jacobi groups. We consider the (adelic) Jacobi group
where
(2.6)
J,(A) := Sp2q(A) k Hq(
ON
where Hq
of Hq by
THE
OF
NONVANISHING
is the Heisenberg
THE
group of dimension
2q +
(la
(ai,---
,aq;bi,--
Iq
b
0
0 h
,bq;c)
1. We
ded
in
,aq) and b =
(a\,
shall denote
elements
eSP:
2<?+2
i/
,6q). The
(&i,
685
c\
b*
a*
\
a =
where
VALUE
CENTRAL
symplectic
group Sp2(? is embed
Sp2(?+2 by
For a given character
there is a unique, up to equivalence,
irreducible
ip of A/k,
of
is
in
in
which
realized
the
of
Bruhat-Schwartz
space
representation
uj^
Hq(?),
functions S(Aq), by the Stone-Von Neumann Theorem
in Chapter
II of [MVW87].
We
here
elements
of
kq
with
,
,0; 0) of Hq.
identify
(a\,
aq; 0,
(a\,
,aq)
One can extend u;^ to the Weil representation
of
and
define the
uj^
Jq(A) ([194])
theta function
9^ (Kg)by
(2.7)
0tq(Kg)=?ru>i,(hg)<pq(O
where ipq G S(?q).
Let (ppbe an automorphic
form on
Sp2p(A).
We
define
the (generalized)
Fourier
Jacobi coefficient of <PP([194], [GRS99c])by
(2.8)
Note
T?P_Mv)?)~
that because
I
Jvp-i(k)\Vp-i(A)
0t-?*P-i(v)9)Mv9)Mv)dv.
of the nature
of the Weil representations,
the automorphic
form
over
an
becomes
integrating
automorphic
(pp
Vp-i(k)\Vp-i(?),
Sp2p(A),
form on
This can be observed clearly from the discussion
below on
Sp2(p_?)(A).
the relation between
and
the
Fourier-Jacobi
in
coefficients
defined
(2.8)
[194].
Recall from (2.2) that Vp-i+i
is a normal subgroup of Vp-i and the projection
can be identified with the subgroup
given in (2.2). Note that Ni
ip-i(Vp-i)
Hp-i
is a subgroup of T/p_?+i, so that the generic character
ipi defined in (2.3) can be
viewed as a character of Vp-i+\.
the ipi Fourier coefficient of <pp
Consider
on
(2.9)
after
4>f{hg)
L Vp-i+i(k)\Vp-i+1(A)
h ? Hv
and g G
Sp2(p_i)(A).
on
the
group Hp
automorphic
0 * sP2(p-i]
where
(2.10)
r$p_MG)=
i
JH.
,-i(k)\Hp
p(uhg)ipi(u)du
It is easy to check
(A). Now we have
that
(pp(hg)
is
e^ihg^fihgjdh.
is a Fourier-Jacobi
coefficient of
and is an
[GRS99c], f$p_i((pp)(g)
(pp
form on
automorphic
\). For simplicity, we call T^ _i(4)p)(S) a Fourier
Sp2(p_
Jacobi coefficient of <i>p.
By
[194] and
DAVID
686
Let
Jacobi
DIHUA
GINZBURG,
JIANG,
form on
(pq be an automorphic
coefficient of (pqby
JV^iik^V^iik)/
STEPHEN
We
Sp2g(A).
:=
(2.11)
f+q_?q)(g)
AND
RALLIS
define
similarly
the Fourier
^(?q-i(v)^q(vg)iPi(v)dv,
Note here that g is a preimage of
is an automorphic
form on
Sp2(g_?)(A).
it is independent
is no longer genuine,
of the choice of the
the product
for
g.
any
g
preimage
and let (pq-i be an automorphic
form on
Let (pp-i be an automorphic
Sp2(p_?) (A)
We formally define two period integrals:
form on
Sp2(q_?)(A).
which
g.
Since
(2.12)
I
Vp,iCpv-i,(pp,Wp-i):=
$p-i(lf)^p-Mp)(9)d9
JSP2(P-i)(k)\SP2(P-i)W
and
(2.13)
:=
Vq4(Pq-i,jq,ipq-i)
(Pq-i(g)J^ ?q)(g)dg.
/
?/Sp2(q_i)(fc)\Sp2(g_i)(A)
sense be
This makes
in (2.12) that g is any preimage of g in
Note
Sp2/p_^(A).
cover
cause the product
forms over the metaplectic
of two genuine automorphic
in (2.12) is taken over
is no longer genuine, and hence the integration
Sp2(p_i)(A)
the linear group.
is a cuspidal automorphic
It is clear that if (pv-i (or (pq-i, respectively)
form, then
iswell defined;
the period Vp?((pp-i,
(pq, (fq-i), respectively)
(pp, ipp-i) (or Vq?((pq-i,
we
to justify the
in
have
the
converges
However,
general,
i.e.,
absolutely.
integral
of
these
convergence
problem
integrals.
Let (a, Va) be an irreducible unitary generic cuspidal automorphic
representation
of Sp2/(A) and let (r, V?) be an irreducible unitary generic cuspidal automorphic
? I
?
of Sp2r(A). When p = I and i = I r (I > r) or q = r and i ? r
representation
(pi, <Pr) (I > r) or Vr^r-i((pi, (pr, </?/) (r > I) is said
(r > I), then the period Vi?-r((pr,
if (p G Va and t ? V?, and ip is a given
to (a,r,ip),
to be attached
respectively,
additive character of k\?.
nontrivial
To simplify the notation, we use
[H] :=H(k)\H(A)
(2.14)
for any algebraic group H defined over k. We
if it does not cause any confusion.
also use H
for the A:-rational points
of H
3. Some
families
of
residual
representations
We consider here the symplectic group Sp4r+2? (p = 2r+
3.1. Symplectic
groups.
series associated
in ?2) and the family of cuspidal Eisenstein
l if using the notation
in
to the standard maximal
parabolic
subgroup P2r?/ of Sp4r+2Z. By the definition
?
=
x Sp2Z.
k
with
is
GL2r
?2, the Levi decomposition
M2r,i
M2r,i
^2r,\
P2r,i
P2r,i
The elements ofM2r?? will often be written as m = m(a, b) with (a, b) G GL2r x Sp2Z.
For
simplicity,
we
may
also
(3.1)
write
m = m(a,
b) G GL2r
x Sp2?.
represen
(tt<S>(t,Vngxr) be an irreducible generic unitary cuspidal automorphic
tation of GL2r x Sp2?. For the given cuspidal datum (P2r,u n ? cr), one may attach
Let
ON
an Eisenstein
THE
OF
NONVANISHING
THE
CENTRAL
VALUE
687
s, (p^^a) on Sp4r+2?(A) with (pni&o-? K-?o-- More
tt?ct can be realized in the space
representation
functions
where
integrable automorphic
L2(ZM2r>z(A)M2r?z(fc)\M2r?z(A)),
the center of M2r,h Let K be the maximal
of
compact subgroup
Sp4r+2/
=
P2rji(A)K
Sp4r+2;(A)
series E(g,
automorphic
the cuspidal
precisely,
of square
is
ZM2rjl
such that
is the Iwasawa decomposition.
Let (p^a be a K HM2r%/ (A)-finite automorphic
form
in Vn^a, which
is extended as a function of
so
that
for
([Shd88, ?2]),
Sp4r+2/(A)
= umke
g
Sp4r+2?(A)
=
<?Wr(#)
(pn?a(rnk)
and for any fixed k G K, the function
m
is a K DM2r.5/(A)-finite
(3.2) $(#,
*-+
:=
>k)
form in K-?^.
automorphic
s, (p??*)
<l>ir?a(
We
define
(/>ir?<r(g)exp(s + pp2rl, Hp2rl
(g))
for g G
s is identified with s?2r,
As in [Shd88, ?1], the parameter
Sp4r+2?(A).
where a2r is the co-root dual to the simple root a2r. Note that a2r determines
the
standard parabolic
subgroup P2r>?.
In our
case
we
have
(3.3) exp(s + pp2r>l,Hp2rJ(g))
where we write g = um(a,b)k
G
Sp4r+2/(A)
Eisenstein
series is given by
(3.4)
E(g,s,(p^a)=
=
\deta\s+2-r???1
and m(a,b)
G GL2r
x Sp2?. Then
the
^g,s,(p^a).
]T
7eP2r,?(fc)\Sp4r+2Z(fc)
term of the Eisenstein
series E(g, s, (p^^a) along a standard parabolic
=
zero
is
P
unless
of the
subgroup
always
P2r?? ([MW95, II. 1.7]). Because
tt
a
one
in
datum
this
has
?
case,
cuspidal
The
constant
P
(3.5) Ep^Xg^s,^?*)
=
j
E(ug,s,(p^a)du
J[U2r,i]
=
$(p, s, (p*?*) + M(w2r,u
?)(*(-, s, (pir?*))(g)
=
where
[C/2rj?]
?/2r>/(fc)\i72r.}?(A) (as in (2.14)). We denote here by W2r,i the
in the representatives
element
of the double coset decomposition
longest Weyl
of the Weyl groups. The intertwining operator
s)
M(w2r,u
WM2rti\Wsp4r+2l/WM2r,i
is defined by the following integral
M(w2r,hs)(^(',s,(p7T^a))(g)
:= /
^(w^^ug^^^^du
J[U2r,i\
which
(3.6)
is an
Sp4r+2? (A)-mapping
l(s, TT?a)
from the unitarily
=
lnd%^{A)
induced
representation
(tt0 a ? exp(5,Hp2rl ( )?.
tO I(-S,W2r,l(7r?<T))'
It follows from the Langlands
series that the Eisenstein
series
theory of Eisenstein
= s0 if and
in
>s,^?^))
only if the term M(w2r,i,
E(g,s,(p7T^)(T) has a pole at s
s)($(
= s0 for some
(3.5) has a pole at s
holomorphic
(or standard) section <fr(g,s, (p^?a)
in I(s, 71-00-). Since factorizable
sections generate a dense subspace in l(s, TT<g>a),it
DAVID
688
DIHUA
GINZBURG,
JIANG,
AND
STEPHEN
RALLIS
sections for the existence of poles
to consider the factorizable
stein series E(g, s, (p^^a), or for the existence of poles ofM(w2r,i,s)($(-,
When
the section $(-,5,07r2(g)(T) is factorizable;
i.e.,
suffices
, S, (pK?a) = ?v$v(-,
$(
(3-7)
of the Eisen
s, (p^^a))
S, (p^?^),
at almost all
$v(',s,<p7rv?<Tv) is a section in 1(s,ttv ? av) and is unramified
as an
local places v, the term M(w2r,h
s)(^(-, s, (p^^a)) can be expressed
infinite product
where
finite
(3.8)
=
M(w2r,hS)(^(',S,(p7T(S)(T))
Y[Mv(w2r,l,s)($v(-,S,(p7Tv?(Jv)).
v
By the Langlands-Shahidi theory ([L71], [Shd88])we have
?
(3-9)
/o ^x
M^S)
where Mv(w2r,us)
=
n>/
x
L(s,tt
a)L(2s,TT,
"i-r
A2)
L(s + l,?><*)L(2s+J,Ai)
is the normalized
=
Nv(w2r,l,s)
intertwining
?,-r
4r ,
n^(^.?.')
operator
Mv(w2r,Us)
r{S,TTv,av,W2r,l)
which
defines
ttv, av,
r(s,
a mapping
W2r,i)
L(s +
is equal
from 1(5, ttv<S)cfv) to I(?s, W2r,i[nv?o-v\).
Here
the function
to
x o~v)L(2s,ttv,
L(s,ttv
1,TTVx av)L(2s + 1,ttv, A2)e(s,ttv
A2)
x av,ip)e(2s,ttv,
'
A2,ip)
local intertwining
3.1 ([K00, Proposition
op
Proposition
3.4]). The normalized
nonzero
the
real
and
is
erator Mv(w2r,u
part of s greater than or
for
holomorphic
s)
section ^v(-,s,(p7Tv^)CTv) in 1(s,ttv <S>
o~v), as a
equal to |; i.e., for any holomorphic
nonzero
and
is
in s, Nv(w2r,u
for the real
function
s)(^v('-> s, (P<kv?ctv)) holomorphic
or
s
part of
equal to \
greater than
11.1
can be replaced by 0 in Proposition
3.1 according to Theorem
one has
is enough for the work in this paper. As a consequence,
\
can possibly have a simple
series E(g,s,(pn^)(7)
The Eisenstein
3.2.
Proposition
=
= 1. The existence
=
or s = 1 of E(g, s, (pn?a)
s
or
s
at
s
the
at
pole
of
pole
\
\
=
or s = 1 of the product of
to the existence
is equivalent
of the pole at s
\
Note
that
in [CKPSS].
\
But
L-functions
L(s,tt
x
cr)L(2s,7T,A2),
respectively.
terms of Eisenstein
series, the Eisen
theory of constant
Proof. By the Langlands
=
terms
constant
of the Eisenstein
s
a
if
the
stein series has
so if and only
pole at
=
to
s
the
series has a pole at
so- By (3.5), it is equivalent
property that the global
=
>
a
s
From identity (3.9), if
at
Sq
pole
operator M(w2r,i-> s) has
intertwining
\.
=
s
a
at
the global intertwining operator M(w2r,u
sq, then the quotient
pole
s) bas
must
have a pole at s =
L(s,tt x cr)L(2s,7r,A2)
L(s + 1, tt x a)L(2s + 1, tt, A2)
sq since, by Proposition
3.1, the product
WMv(w2r,us)
THE
ON
THE
OF
NONVANISHING
CENTRAL
Now both L-functions
does not vanish for s = so >
L(s,tt
|.
are nonzero for the real part of s greater than one. It follows
x
L(s,tt
689
VALUE
x a) and L(s,tt,
A2)
that the product
cr)L(2s,7T,A2)
bas a pole at s =
must have a pole at s = so >
if
\ M(w2r,i,s)
if the product of L-functions
so > \. Conversely,
L(S,TT X Cr)L(2s,7T,A2)
has a pole at s = so, then the global intertwining
in (3.9), we can always choose
at s = so, because
as
in (3.7), so that the product
$(-,s,^a)
S, (pnv?av))
J\Nv(w2r,l,s)($v(-,
v
at s =
is holomorphic
and nonzero
3.1.
Proposition
operator M(w2r,i,
s) bas a pole
a particular
section
factorizable
sq. Note
that
for this part we do not need
7.2 in [CKPSS], the irreducible generic cuspidal automor
Finally, by Theorem
a
is an
lift tt(cf), which
of
functorial
phic representation
Sp2?(A) has a Langlands
of
is
deter
irreducible unitary automorphic
representation
uniquely
GL2/+i(A),
mined by a, and is of isobaric type. Hence one has
x a) = L(s,tt
L(s,tt
x 7r(cr)).
at a real
and nonvanishing
It follows that the L-function
L(s, tt x a) is holomorphic
= 1 if and
tt
if
is
point so > 1, and L(s,tt x a) has a pole at s
isomorphic
only
to one of the isobaric summands of tt(g). By Theorem
3.1 in [K00], the complete
at any real
exterior square L-function
and nonvanishing
L(s,tt, A2) is holomorphic
=
a
s
even
1.
If
at
is
>
i
3.1 in
has
then
Theorem
so
1,
point
pole
by
L(s,tt, A2)
s= 1
one
a
in
at
A
knows
Theorem
that
has
pole
L(s,tt, A2)
[K00]. By
[GRS01],
if and only if tt is the image of an irreducible unitary generic cuspidal automorphic
r of SO?+i(A).
in
The uniqueness
of tt in terms of r is proved
representation
>
s
one
and
Hence
knows
that
for
the
both
L-functions
real
value
[JS03]
[JS04].
\,
can have possible poles only at s = 1. The proposition
L(s,tt x a) and L(s,7r,A2)
follows. D
We
denote
the residue at s =
\
(3.10) E^(g,(p^a)
of E(g,
s, (p^^a) by
:=Ress=iE(g,s,(p7T^a).
3.2. Metaplectic
We consider here the metaplectic
groups.
group Sp6r(A) and a
to
of
series
Eisenstein
associated
the
standard
maximal
family
parabolic
subgroup
P2r;r(A) with Levi decomposition
(3.11)
Note
P2r,r(A)
that
=
M2r,r(A)
X
cover
?/2r,r(A)
=
over
(GL2r(A)
X
S^2r(A))
X
tr2r,r(A).
the metaplectic
the GL2r(A)-part
and the unipotent
splits
so
we
that
in [/^(A)
its
with
may
subgroup t/2r,r(A),
identity C/2r,r(A)
preimage
in GL2r(A).
and GL2r(A) with its preimage
Let tt be an irreducible unitary self-dual cuspidal automorphic
of
representation
that tt is symplectic,
Assume
i.e., that the exterior square L-function
GL2r(A).
DAVID
690
L(s,7T,A2)
has a pole at s =
AND
JIANG,
STEPHEN
1. Let r be an irreducible
RALLIS
genuine
generic
cuspidal
so that
of Sp2r(A),
representation
automorphic
DIHUA
GINZBURG,
(3.12)(GL2r(A) x ??>2r(A),tt? r)
is a generic
Eisenstein
I(s, it? t) :=
(3.13)
where
the parameter
As
s is normalized
as in ?3.1, so that
=
an Eisenstein
in (3.4), one can define
series
E(g,s,4>n?r)=
(3.15)
form an
? ?exp(S,
H^J-)))
Indg^?Tr
exp(S + pp2rr,Hp2Jg))
(3.14)
As
we
in the case of
Sp4r+2?,
to a section $(g, s, (p^?r) m
cuspidal datum of Sp6r(A).
series E(g,s,
(p^??) associated
\deta\s+2r+k*.
([MW95])
?(19,s,4><k?t)
^2
In this paper we only consider this Eisenstein
series for the special case when
tt of GL2r(A)
is the
irreducible unitary cuspidal automorphic
representation
=
tt
in
is
It
that
this
of
r;
i.e.,
proved
-0-transfer enjoys
^-transfer
[GRS02]
tt^(j).
that
the property
x p) =
p)
Ls(s,TT7p(r)
L%(s,rx
the
p of GLm (A) with all pos
representations
cuspidal automorphic
Here the set S consists of all infinite local places and the finite
at v 0 S.
of r, p and ip are unramified
local places such that the local components
for r x p depends on the choice of ip ([GRS02]).
Note that the partial L-functions
If one takes p ? tt^ (t) , then one knows that the partial L-function
for all irreducible
itive integers m.
L^(s,t
has a simple pole at s ?
1. Since
x tt^(t))
the partial
exterior
square L-function
LS(s,TT^(T),k2)
has a simple pole at s =
1, one knows
that the partial
square L-function
symmetric
Ls(s,7ty(r),Sym2)
is holomorphic
for the real part of s greater than one.
series works as well for
the
theory of Eisenstein
Langlands
[MW95],
we
can
a
to (3.9), with com
time
have
formula
similar
This
groups.
and nonzero
Following
metaplectic
plete L-functions
The intertwining
operator
by the partial L-functions.
to the
the
induced
from
which
0
maps
representation
r)
I(s,tt^(t)
M(w2r,r,s),
induced representation
&>?))> can be expressed as
I(?s,W2r,r(^i)(j)
M(w2r,r,
replaced
S)
=
[rS(s, TT,a, W2r,r)
Y[ ?v(w2r,r,
s)]
X
r
N
(s,TT,a,W2r,r)
s_
=
LI(s,tt^(t)
L^(s
+
1,7ty(r)
J| Mv(w2r,r,
ves
vgs
XT)Ls(2s,TTi)(7),Sym2)
x r)Ls(2s
+
1, tt^(t),
Sym2)
*
s),
ON
THE
NONVANISHING
OF
THE
CENTRAL
VALUE
691
we only consider
the normalized
local intertwining
operators
for
unramified
local
For
unramified
local places, the normal
places.
Nv(w2r,r,s)
ization of local intertwining
is well known and the normalized
local in
operators
are
nonzero
at
the
sections.
This
is
unramified
the essential
tertwining operators
in ?3.1.
difference between this case and the case we considered
Since one can always choose the local sections <&vin I(s, tt^(tv)?tv)
such that the
=
are
nonzero
s
at
it
of
follows
if
that
the
$v
1,
images A4v(w2r,r, s)($v)
partial L
x t) has a pole at s = 1, then the global intertwining operator
function
L^(s, tt^(t)
= 1. Hence the Eisenstein
series ??g^,^??)
must
s) has a pole at s
M(w2r,r,
have a pole at s = 1, whose residue is denoted by
Note
that
(3.16) E?(g,
for some holomorphic
4>^?r)
section $(g,
of
4. Periods
:=
Ress=1?(g,
s, (pn??) in I(s,
residual
s, 4>^?r)
tt0 r).
representations
We recall that a is an irreducible unitary generic cuspidal automorphic
rep
resentation
of Sp2?(A) and r is an irreducible unitary genuine generic cuspidal
of Sp2r (A). Let tt^ (t) be the image of r under the ip
automorphic
representation
transfer with respect to a given character ip,which by definition
is the composition
of the ^-theta
and
with
the
functorial
correspondence
Langlands
S02r+i
lifting
to GL2r. The existence and basic properties
from S02r+i
of this ^-transfer
have
been proved through
[CKPSS01],
[JS03], [JS04], and [GRS01]. We assume that
can be found in
[S02]. It follows from
7T^(r) is cuspidal. More results about tt^(t)
that the residue
is nonzero. We
?3.2, formula (3.16) in particular,
Ei(h,(p7Tip^^)
are going to study the period integrals of the residues Ei
m (3.10) and
(g, (^(t)?^)
Ei(h,(pn^^)?r)
in (2.12) and
We
in (3.16). Following
(2.13), we have
from the formal definitions
(r >
I)
^3r,r-i(i?i(,?07rv,(r)<8)a)5^l(-,07rv,(r)?r),^2r+z),
(r <
I)
V2r^l,l-r(E1(-,^{r)?r),Ei(',(pn^^)0a),ip3r).
shall only
study
in detail
(4-1) V3r,r-l(Ei
the case
(r
>
(', (pir^{r)?<j),#i(-, ^(r)??),
^2r+/)
in a similar way.
By
(4-2)
^(5,^(f)?a)^'2p+I(Ei(-,^(f)?f))(?/)d?/
/
=
as in (2.14). The
Sp4r+2/(fc)\Sp4r+2?(A)
cient
of the residue
?i(-,^(f)0f)
Jr^2r+l(E1(-,^{r)?r))(g)
[Sp4r+2?]
integrals
I), i.e., the period
in the following, and the other case (r < I) can be treated
the period in (4.1) is given by the integral
definition,
where
of period
gral (as in (2.11))
(4-3)
I [V2r+ l]
Fourier-Jacobi
coeffi
is given by the inte
692
DAVID
It is clear
that
regularization
this period.
DIHUA
GINZBURG,
AND
JIANG,
STEPHEN
RALLIS
the period integral in (4.2) may not converge and needs certain
to make it well defined. We follow the argument
in [GJR01] to study
4.1. Truncation
of Eisenstein
series. We recall a special case of the Arthur's
our study of the period integral as defined in
to
truncation method
and apply it
we
use
is standard and can be found in [Jng98a] or [GJR01].
(4.2). The notation
=
=
we
set
In this section
G = Sp4r+2? and P = MU
P2r,?
M2r?/t/2r??. We
h->
a.
a
to
Then
identify up with R via aa2r
regular element in ap will correspond
a real number c G M>i, where we denote by R>c the set of all real numbers greater
than c. We denote
:= exp(l,HP(g)) = |detm(ff)|
(4.4) H{g)
for g = um(g)k G G (A) (the Iwasawa decomposition
given in ?3.1). Let rc (c G R>i)
be the characteristic
function of the subset M>c.
series E(g,s,(p)
of the Eisenstein
Following
[A78] and [A80], the truncation
=
<pn ,(r)?a) is defined as follows:
(where (p
(4.5)
=
AcE(g, s, 0) E(g, s,(P)-
EP(19, s, (P)rc(H(ig)).
^
7eP(k)\G(k)
constant
term Ep(g,s,(p)
of the Eisenstein
as
expressed
(see (3.5) for the definition)
The
=
s, (P) ?(g,
EP(g,
+ M(w,
s, </>)
series E(g,s,(p)
s)($(-,
can be
along P
s, <?>))(g)
w = W2r,i and M(w,
in ?3.1. We
s) is the intertwining operator as described
terms
in
summation
has
and
remark that the
converges
only finitely many
(4.5)
series can then be
absolutely
(Corollary 5.2 in [A78]). The truncated Eisenstein
where
rewritten
as
=
AcE(g,s,(P)
^(l9,s,(P)(l-Tc(H(ig)))
J2
1eP{k)\G{k)
-
M^
Y,
*)
W*' s'?))(70)Tc(ff(70))
7eP(k)\G(k)
:= ?1(g)-?2(g).
(4.6)
Let ?sobe a positive real number.
stein series E(g, s, (p) at s = sq:
E(g,
Consider
=
s, (p) ES0(g,
(p)(s
-
the Laurent
s0)e + higher
expansion
of the Eisen
terms
e = ?1, i.e., so is a simple pole of E(g,s,(p),
is the residue
then ESo(g,(p)
When
In general we call ESo(g,(p)
at ?s= so of E(g,s,(p).
the leading term at s = so of
E(g, s, (p). In the following, we assume that so > 0 is a simple pole of E(g, s, (p).
The truncation of the residue ESo(g, (p) is
KcES0(g,(P)
=
ES0(g,(P)-
^
M(w,s)(^,s,(p))So(1g)rc(H(1g))
<yeP(k)\G(k)
(4.7)
:=
Eso(g,0)-?3(g).
consider the period
integral Vzr,r-i(ESQ(',(p),?i(-,(p),(?2r+i)
is replaced by the residue ESQ(-,(p).
but
the
residue
Ei(-,(p)
(4.2),
We
as defined
Using
in
we
(4.7),
ON
THE
OF
NONVANISHING
THE
693
VALUE
CENTRAL
obtain
s, (p) is rapidly decaying,
Since kcE(g,
<?>2r+l)
if2r+l) + P3r,r-Z (Ac?So ( , (p),EX(-,
EX(-, 0),
P3r,r-/(^3,
0),
(p),Ei(-,
V3r,r~l(ESo(',
=
V3r,r-l(AcE(g,
the period
integral
S, (p),Ei(-,
(?>),(f2r+l)
</>),(?>2r+/)
a meromorphic
in the set of
in s with possible poles contained
function
in
hence
series
that
of
the global
Eisenstein
and
of
the
possible poles
E(g,s,(p),
that
It
follows
intertwining operator M(w,s).
defines
s, (p),Ex(-,
Ress=SoP3r?r_/(AcL;(#,
The
4.1.
Proposition
i=
For
1,2,
complex
absolutely
V3r,r-i(kcESo(',
section.
El(',
(p), <P2r+l)
large and have meromorphic
for Re(s)
(p), </?2r+z)
(p),Ex(-,
the periods
V$r,r-l(?i,
converge
=
in the next
will be proved
proposition
following
</>),(p2r+i)
to the whole
continuation
plane.
we
continuation,
meromorphic
By
have
S, (p),EX(-, (p), Lp2r+l)
~
Ei(-, (p), (?>2r+0 p3r,r-/(?2,
V3r,r-l(^cE(g,
=
we
Hence
p3r,r-/(?l,
have
Ress=SoP3r5r_?(AcL;(#,
=
From
EX(-, (p), </?2r+z)
Ei(-,
ReSs=SQV3r,r-l(?l,
(4.6) and
s, (p),Ei(-,
(4.7), one knows
</>),<p2r+i)
(p), (f2r+l)
ReSs=SoP3r,r-z(?2,
E\(-,
(p),^2r+/)
that
=
ReSs=So
we
Therefore,
(4.8)
V3r,r-\
(?2, EX(-,
(p), tp2r+l)
(p),EX(-, 0),
<p2r+l)
V3r,r-\(?3,
EX(-,
=
ReSs==SoV3r,r-l(?l,
In other words, the study of the period V3r,r-i(ESo(-,
(p),E\(-,
4.1 and an explicit calculation
of the integral
Proposition
Ei(-,
V3r,r-l(?\,
We will do this in the next
(p),
EX(-, 0),
<?>2r+z).
(p), (f2r+i) reduces
to
(P2r+l)
section.
4.2. The
proof of Proposition
of
both
large
integrals
p3r,r-z(?l,L?i(-,0),(/?2r+/)
follows as in the proof of Proposition
periods and the possible
calculation
(f2r+l)
obtain
V3r,r-l(ESo(;
of both
(p),
4.1.
First,
the convergence
for the real part of s
and V3r,r-i(?2,E1(',(p),(p2r+l)
continuation
3.1 in [GJR01]. The meromorphic
location of poles will follow from the explicit
of the period integrals.
shall calculate the period integral V3r,r-i(?i,Ei(-,
(p), (p2r+i) for the real part
s
can
of
in a similar
be calculated
large, while the integral V3r^r-\(?2, E\(-, (p), (?>2r+i)
way to that in [GJR01].
We
694
DAVID
Recall
and (p=
first as
(4.9)
DIHUA
GINZBURG,
AND
JIANG,
STEPHEN
RALLIS
that G =
=
P = MU = p2r,z = M2r,iU2r,h Write
(p
Sp4r+2/ and
(p^^{j)?a
Assume
the real part of s is large. The integral can be written
<P<k4(t)?t'
=
7>3r,r-l(5l,^l(-,0),V2r+z)
From
the discussion
above,
large. By (4.6) we have
/./G(fc)\G(A)
it is absolutely
?i(9) =
?1 (fl)^2r+I (#l(-, 0))(<^
convergent
when
the real part
of 5 is
?(l9,s,(P)(l-Tc(H(lg))).
E
yeP(k)\G(k)
Then
(4.9) equals
(4.10)
Recall
/
JP(k)\G(A)
from
^(^^^(l-r^^)))^^^^-,^)^)^.
(4.3) that
J[V2r+l]
where V2r+i = V3r?r+i
we have
as defined
in (2.1) with
i=
3r and
r?
I. From
(2.7),
Recall
as
from
/c2r+z =
/c2r? kl and ? = (&r,&).
subgroup V^+z can be expressed
where
(p2r+i G <S(A2r+?)- We write
that
elements of the unipotent
(2.1)
(n
pi
p2
Pa
hr
0
hi
0
0
:
v(n,pi,p2,p3,z)
=
^(?2r+z(v)50^2r+z(O
^2.+?(?2r+/(^0= E
(4.11)
p
z\
pl
p*2
hr
eSp, 6r
P*
\
1
"7
n G i\Tr_?, the standard maximal
unipotent
subgroup of GLr_?. The pro
to
the
of
G
group #4r+2/+i
jection t2r+i(v)
Heisenberg
V2r+z
v(n,pi,p2,p3,z)
(as
?
defined in (2.2) with p = 3r and i = r
I) can be expressed as (and identified with
elements of V^r+z)
where
//, r-Z-1
\
1 Pi
V2 Vz
z
?2Z
*
*
hr
?2r+z(v(n,pi,p2,i>3,2))
=
hr
e V2r+i
*
1
V
where pi is the last row of pi, and 2; is the
we use the following notation
for ?2r+/(t>):
(4.12)
?2r+l{v(n,Pi,p2,P3,z))
Ir-l-lJ
(r
?
/, l)-th
=t{Pi,P2,Pz,z)
entry of z. For simplicity,
THE
ON
To
carry
(n
0
hr
our
out
P2
0
hi
we
calculation,
P3
0
o
write
V
v(p\)
=
\
v(n,0,p2,P3,z)v(pi)
hr
P?
Ir-lJ
and we write
=
t(Pl,P2iP3iZ)
'[V2r+i]
/ J\v
as
?21
v(Ir-i,p\,0,0,0),
the Fourier-Jacobi
695
VALUE
of V2r+i
*7 v
where
Then
elements
0
hr
CENTRAL
Pl
hr
(h
z\
p5
p5
THE
OF
NONVANISHING
coefficient
the elements
as
^(Pi,p2,p3,^)
?(0>P2>P3>?K(Pi>0>0,0).
can be expressed
T$2r+l(Ei(-,(p))(g)
as
^WO,P2^3^)^l+Pl(6r),O,O,O)50(^2r+/((O,^))
]T
6r
(4.13)
fc2r,^efci
+pi(6r))flf,0)^r-z(w)dv,
x?i(i;(n,0,p2,P3,^(Pi
in Matr_?j2r(/c) with the first r ? / ? 1 rows zero and the
is a matrix
Pi(&r)
last row equal to ?2r. Let
be the subgroup of V2r+i consisting
of elements
V^r+Z
row
zero.
over ?2r
with
the
of
last
the
summation
p\
Collapsing
v(n,pi,p2,P3,z)
in (4.13), we obtain
with integration
where
V^wo,^,^,^)^^^^)^)^^^^,^))
I
V?r+l(k)\V2r+i(A)
?iekl
0,p2,p3,
(4.14) xEx(v(n,
By
the definition
of the Weil
representation
=
u;^(e(0,P2,P3,z)g)ip2r-^i((0,^i))
Hence
the Fourier-Jacobi
coefficient
J2
X
z)v(px)g,
(p)ipr-i(v)dv.
uj^ ,we have
uj,p(e(0,p2,0,z)g)(p2r+i((0,^)).
equals
^2r+l(Ei(-,(p))(g)
?ty(*(0,P2,0, z)?(px,0,0,0)flr)y>2r+i((0,6))
(4.15) xEi(v(n,O,p2,p3,z)?;(pi)^,0)^r-z(^)^.
It follows
that
integral
$c(g,s
,0) [
Y, ^(?(P2,^(Pl)30V2r+z((O,?z))
Jv'
L IP{k)\G{k)
(4.16)
where
(4.10) equals
xE1(v(n,0,p2,P3,z)v(p1)g,(p)ipr-i(v)dvdg,
$c(#,s,0)
=
$(#,
s,(p)(l
-rc(H(g))),
and ?(p2,s)
?(Pi, 0,0,0).
It is easy to check from the definition of the Weil
in [MVW87]) that as a function of g the function
=
e(0,p2,0,z)
representation
J2 ^(^(P2?^(Pl)S0^2r+z((O,?i))
and ffo)
uty (Chapter
2
696
DAVID
DIHUA
GINZBURG,
is left f/2r)/(A)-invariant.
the form
that
Recall
y
hi
RALLIS
radical U =
t/2r,z of P2t%/ has
z'\
*'):=[
In the calculation,
ing the integration
STEPHEN
the unipotent
f ihr
(4.17)
M?/,
AND
JIANG,
?/*j
Sp4r+2i
into the 'middle' of Sp6r. By factor
Sp4r+2Z has been embedded
over
we obtain that integral
from
U2r,i(k)\U2r,i(&)
P(k)\G(A),
(4.16) equals
*c(g,s,
7
/
(4.18)
xEi(v(n,
Y\ ^(^(P2^zY(Pl)9)^2r+l((^^l))
/
Jv'
J[U2r,i]
nU2r,l}JV?r+l(k)\V2r+l(A)ciekl
0,p2,p3,
z)v(p1)ug,
4>)ipr-i(v)dvdpidudg,
where
the integration
in variable g is over
M(k)U(A)\G(A).
over
The integrations
[{/2r,z] and Vr2r_f_/(/c)\V2r+/(A) can be rewritten
We can rewrite v(n, 0,p2,p3, z)v(pi)u
as
(n
pi
hr
p2
y
hi
P3 z\
z' Pi
y* p\
hr
(n
by Vo
Pz
z'
y*
hr
V
the group
consisting
z\
pi
p\
0
v(pi)
n*/
of elements
0
(n
hr
P2
P3
y
z'
pi
y*
hr
p\
0
n*
hi
Note
P2
y
hi
P\
\
Denote
0
hr
as follows.
z\
that
the element of U2r,i sits in the middle
of v?. Hence
in (4.18)
over
can
and
as
rewrite
be
integrations
[U2r,i]
V2/r+z(fc)\V2r+?(A)
Y"
the inner
u;^(?(p2,z)?(p1)g)(p2r-^i((0,^i))Ei(v0v(p1)g,(p)ipr-i(v0^^
//
the integration
where
f dp\ along the variable p\ is in Matr_/?2r with the first
r ? I? 1 rows in
and the last row in A2r.
Matr_z_i52r(fc)\Matr_/_i52r(A)
To calculate the last integral, we define the Weyl element w of SpQn as
hn
/
\
l-n?k
W =
hk
?n?k
hn
\
in Ei(v?v(pi)g,(p)
conjugate
integral equals
(4.19)
We
/ /
V
??e*7
by w
c^Wp2,^?y?)(^^
)
from left to right and obtain
that
the last
ON
that
Note
The
THE
the integration
unipotent
subgroup
OF
NONVANISHING
THE
CENTRAL
VALUE
697
same as the above.
J dpi along the variable pi is the
consists
of elements of the type
V^0)
y
(hr
n
?(0)
P3
p2
z
hi
P\
n*
z'\
p3
V*
V hr)
and
ihr
Pi
Lr-l
v (pi) :=
Pi hr)
V
To continue
a smooth
be
(p
the calculation
automorphic
of (4.19), we formulate a general lemma below. Let
>
form on GL?+J(A)
(i,j
2). We consider integrals of
the type
(4.20)
where the integration
acter of Nj as defined
Matj_i?i(fc)\Matj5i(A);
Lemma
integral
\ipj(n)dndp
P Ij
SJ?(h
in variable n is over
Nj(k)\Nj(A),
ipj(n) is the generic char
in (2.3), and the integration
in variable p is over the quotient
i.e., the last row of p is integrated over the A-points.
Let (p be a smooth
(4.20) is equal to
4.2.
automorphic
Jp J n J q
GL?+j(A)
where
the integration in variable q is
overMaltij(k)\Ma,tij(A)
in variable p is over Mat^(A),
q
of
being zero, the integration
in variable n is the same as in (4.20).
Proof. We define the following
define (C means
'column')
unipotent
:
On
{(* I
> 2).
Then
q^v
with the first column
and the integration
of GL?+J.
subgroups
qeMatij,
(i,j
\dqipj(n)dndp
P h
SIS
on
form
=
0,
For
1 < m
<
j
v^ m
In other words, Cm is the group of all matrices
as above such that all columns
in q are zero except the m-th column. Note that Cm(k)
is isomorphic to kl. For
1 < m < j ? 1 we also define (R means
'row')
t^m
Thus Rm consists
row,
are
zero.
?
\\p
h
of all matrices
p GMat^i,
as above
p^v
=
0,
\i^m
such that all rows in p, except
the m-th
DAVID
698
We
consider
unipotent
the Fourier
first
subgroup
Note that I
?
RALLIS
form (p along
of the automorphic
expansion
*
/
r ) I
k%with
the
'
)
'
=
'q)<Wi(n)dndp.
)
?) f? 7W
*H
is isomorphic to k%,we
Since Rj-i(k)
)
(
the element
we have
(p is automorphic,
0((f /J?>
It is easy
= </>(<?)
to show that
U 0\ (Ii q\ (h
? lj)\0
n)\p
where
is
STEPHEN
Cf
(4.21) /7 ?
Since
AND
JIANG,
this into (4.20), we have
Plugging
identify
DIHUA
GINZBURG,
? q is a j x j-matrix
0
Ij
Ii
q \(
0\ = (li
n
+
+
Ij
\0
?-qjyp n-^
with
all entries
(?'Q)j-i,j
=
zero except
the (j
?
1, j)-th
entry, which
J2&qi?
i=i
Since n_1
is upper
trianglar
we may write
and ? is in Rj-\,
n-1?=P2
+ ?
p2 is in Matj^ with the last two rows zero.
to (4.21) and changing
the above calculation
Applying
where
(4.22)
iff
Jp Jn J
C3
Cjik^CjW
4>{{I%
\ V i)^:
n) \P
to integral
Rj-2),
(4.22) and complete
(Cj-2, Rjs),
we obtain
Ty\dq^(n)dndp
h
where the integration of p is on Mat?_2)?(fc)\Matj}?(A);
are integrated over the A-points.
We apply the same argument above with a sequence
(Cj-i,
variables,
i.e., the last two rows of p
of pairs
>(C2, -Ri)
the proof of this lemma.
D
ON
THE
THE
OF
NONVANISHING
699
VALUE
CENTRAL
In order to apply Lemma 4.2 to the calculation of integral (4.19), we define
?
I define
following unipotent
subgroups of Sp6r. For 1 < m < r
the
q \
[ ihr
h-l
: qG
Mat2r?r_/,
J-21
h-l
=
q{j
0,
j ^ m
<f
hr)
IV
?
of Cm into Sp6r (with i = 2r and j = r
It is clear that Cm is an embedding
I)
as above such that all columns in q are zero
and that it is the group of all matrices
? ? lwe
I
also define
except the ra-th column. For 1 < m <r
(ii
2r
\
:p G
Matr_?52r,
ft?
Pi?
=
i^ m
0,
h-l
P*
IV
hr)
as above such that all rows in p, except the ra-th
1Zm consists of all matrices
row, are zero, and it is an embedding of Rm in Sp6r. We want to apply Lemma 4.2
to integral (4.19). In this case we have to apply the argument
to the series of pairs
Thus
(Cr-i,1Zr-i-i),
in (4.19)
Because
the variable
we
can
apply
4.2 essentially
to the automorphic
4.2, integral (4.19) equals
(4.23)
Lemma
V
/ /
where
V^
to TZr-i and
is related
?(p\)
,7?i,
1Zr-i-i,1Zr-i-2,
subgroups
, (C2,7?i).
(Cr-i-i,1lr-i-2),
the
of
with
the
form Ei(v^v~(pi)wg,(p).
proof
the
of
By Lemma
u?^p^zWpJg^r+i^
1
dieki
in variable p\ is over Matr_jj2r(A).
the integration
consists of elements of the type
(hr
q
y
n
>,(*)
Pi
z
P2
hi
P2
n*
q
is in
Mat2r5r_?
with
the
first
The
unipotent
subgroup
zf\
p3
y*
g*
hr)
V
where
commutes
argument
column
zero.
Finally we consider the Fourier expansion of the residue E\ (g, (p) along the unipo
tent group C\(k)\Ci(A).
The group of characters of C\(k)\C\(A)
is isomorphic to
k2r and the group GL2r(A:) acts on k2r with two orbits. Hence the Fourier expansion
of Ei(g,
(p) along
the unipotent
group C\(k)\C\(A)
is
E1(v(q)g,(P)dv(q)
Ei(g,(P)
/
JC1(fc)\C1(A)
4
(4.24)
+
I
YlEi(v(q)m(-i,hr)g,4>)'ip(q2r,i)dv(q),
jeP1(k)\GL2r(k)'
where m(j,
hr)
G GL2r(/c) xSp2r(fc)
and </2r?i is the (2r, l)-th entry of q in v(q) G Ci.
700
DAVID
GINZBURG.
DIHUA
AND
JIANG,
STEPHEN
RALLIS
Applying
(4.24) to (4.23), integral (4.23) equals a sum of integral J0 and integral
as
defined
follows.
ii,
from the constant term in (4.24), that is,
Integral Iq is obtained
Ei(v(q)g,(p)dv(q).
L /Ci(fc)\Ci(A)
The
of the unipotent
product
'
Vry,
U2r,r
q
(hr
n
subgroups
V^
and C\ can be written
y
Pi
as a product
i.e.,
y
Pi
Z'\
z
P2
hi
q
(hr
pI
y*
n*
q*
n
P3
hi
y*
Ir-i
V hr)
Here Vr? C Sp2r is as defined in (2.1) with
integral h equals
Sp6r as above. Hence
p
p2
z
hi
pi
n*
q*
hr)
\
\
Z'\ (hr
Ir-l
P3
hr)
\
r?
= r and i ?
I and is embedded
in
(4.25)
Yl Ui?,(?(p2,z)e(p1)g)(p2r+i((0,^i))?itp2rr(vv~(p^wg^^r-i^dvdprfg,
/ /
where
the
constant
in variable
integration
term
along
p\
the maximal
is obtained
Integral h
that
is,
(4.24),
is over Matr_/;2r(A),
parabolic
the
from
subgroup
and Ei^p2rr(g,(p)
is the
p2r,r
sum of the nontrivial
Fourier
coefficients
in
^2/E1(v(q)m(j,hr)g^)ip(q2r,i)dv(q).
yePHk)\GL2r(k)Jc^k^Cl^
We now show that integral I\ is zero. Since I\ can be written as a sum of integrals
we will show that each of these summands
parameterized
by 7 G P1(/c)\GL2r(fc),
is zero. As in (4.25) (or in the case of h) the integration over [V^] combined with
that over [C\] is the same as the integration over the product of [t/2r5r] and [Vrj].
The elements of U2r,r are of the form
q
ihr
Pt
y
z'\
Ir
y*
hi
u(q,y,P3,zf)
Ir-i
hr)
V
and the elements
eSP( 6r
q*
of Vr? are of the form
(hr
n
v(n,p2,z)
=
\
The
difference
p2
z
hi
Pi
n*
eSPl
6r
'
hr)
term of
the constant
Iq and I\ is that in Iq it produces
to t/2r)r, but in I\ it produces the nontrivial Fourier coefficient
between
Ei(g,
(p)with respect
character ip(q2r,i)
of Ei(g,
(p)with respect to f72r?r and the nontrivial
The point is that this nontrivial Fourier coefficient of the residue E\
character
respect to U2r,r and the nontrivial
V;(#2r,i) combined with
(g, (p)with
nontrivial
ON
THE
OF
NONVANISHING
THE
CENTRAL
701
VALUE
with respect to Vrj and the nontrivial
character
of E\(g,(p)
a
as
Fourier-Jacobi
coefficient of the residue Ei(g,<p)
produces
ipr-i(v(n,p2,z))
?
orbit with symplectic partition
sociated to the unipotent
((6r
2l)l21). Hence the
of
of
that
summand
the
residue
each
I\
E\
implies
nonvanishing
(g, <p)has a nonzero
orbit with symplectic partition
Fourier-Jacobi
coefficient attached to the unipotent
?
1 of [GRS] and since r > I,
On the other hand, by Proposition
((6r
2l)l21).
nonzero
cannot
have any
the residue E\ (g, (p)
Fourier-Jacobi
coefficient attached to
an
in I\ is identically zero, and so is
such
orbit. This proves that each summand
Fourier
coefficient
h
To carry out the above
we consider
argument,
(4.26)
/
first the Fourier
coefficient
y,p3, z')g, (p)ip(q2r,i)du.
Ex(u(q,
J[U2r,r]
In (4.26), we further consider Fourier expansion
maximal
unipotent
subgroup iV2r of GL2r. Recall
=
p2r,r
Since
the residue E\
(GL2r
(g, (p) has the cuspidal
X
(as a function of g) along
that the parabolic
subgroup
the
is
Sp2r)i72r)r.
support
(GL2r x Sp2r,7ty(r)??),
terms along the subgroups of N2r of integral (4.26) (as a function
the constant
in g) are zero. We end up with the Fourier expansion
similar to the well-known
for a cuspidal automorphic
Whittaker-Fourier
form of GL2r(A)
expansion
([PS71]
and [Shl74]); i.e., (4.26) equals
Y^(4.27) /
SeN2r{k)\GL2r(k)^U2r^
/
^I^r]
E1(nu(q,y,p3,z')g,(p)ip$ir(n)dnip(q2r,i)du,
to 6. For instance, when ? =
ip2r(n) is the generic character corresponding
?
the
the
character
is the generic character
hr,
identity element,
ip2r(n)
$2r(ri)
as defined in (2.3) (with i replaced by 2r). Combining
(4.27) with the unipotent
integration
along Vr? in integral h
(as in I0 in (4.25)), we obtain as an inner
in each summand of h the integral
integration
where
[ff
!\U2r,r]J[N2r)J[VrA
E ^(^(ft.^?Pl^^r+z??O^z))
^tf
(4.28)x?i(vnu(q,y,p3,z,)g,0)ipr-i(v)dvip2r(n)dnip(q2r,i)du.
This
Note that elements S also
integral is parameterized
by ? G P1(/c)\GL2r(/c).
stabilizes the fc-rational orbit of the character
in
(4.24). For automorphic
ip(q2r,i)
of Fourier coefficients depends only on a fc-rational orbit
forms, the nonvanishing
of the characters.
It follows that in order to prove the vanishing of each summand
in h,
it is enough to show the vanishing of (4.28) with ? being the identity.
Note that the product of unipotent
7V2r, C/2r.5r,and Vr,i is Vsr,i (see
subgroups
? 3r and i = 3r ?
for
the
definition
with
p
(2.1)
I) of Sp6r and the product
of characters
is
and
the
generic character of iV3r_/ (see
ipr-i(v),
ip2r(n),
ip(q2r,i)
is
Hence
the
Fourier-Jacobi
of the residue Ei(g,
coefficient
integral (4.28)
(2.3)).
<j>)
?
to the unipotent
attached
orbit with partition
((6r
2/)l2Z). They are all zero by
1 in [GRS]. This proves that h is identically zero.
Proposition
702
DAVID
DIHUA
GINZBURG,
JIANG,
AND
STEPHEN
RALLIS
from the sum of h and I\, there remains one integral in (4.25), which
Therefore,
nonzero.
In other words,
be
from (4.18), (4.19), and (4.25), we obtain the
may
following proposition.
4.3.
Proposition
tegral (4.25),
The period V3r,r-i(?i,Ei(-,
in (4.9) equals in
(p), <??2r+z)defined
that is,
&(9,S,4>)t
/ [ E ^('(P2.^(Pl)s)^2r+z((0,6))
J
JMatr^,2r(^)J[yr,i]?ieki
xEhp2rr(vv~(p1)wg,^)ipr-i(v)dvdp1dg,
in variable g is over M(k)U(A)\G(A),
term of the residue Ei(g,<p)
along the maximal
the integration
where
constant
In the following we
shall simplify
?^2r+a(?(p2,mPl)9)
parabolic
in Proposition
the integral
~
(4.29)
and Ei^p2rr(g,
(p) is the
subgroup p2r,r
4.3. First we write
^^P2^(Pl)^2r+l((^l))
E
tieki
constant
The
term Ei^p2rr(g,
(p) of the residue Ei(g,
(p) along the standard maximal
is the residue at s = 1 of the
which
parabolic
subgroup P2r.5r equals Mi(<&)(g),
defined in ?3.2. By the definition
in ?3.2,
operator A4(w2r%r,s)($)(#)
intertwining
sections
maps
M(ii;2r,r5s)
$ = $>(g,S,0Wv,(f)?r)e 7(5, 7ty(?)? f)
to those
in I(?s,W2r,r(flV>(^)
Proposition
4.3) equals
Hence
? t)).
the integral
in
(or the integral
[
?c(g,s,(P)
?
j e^a(?(p2,z)?(Pi)9)
?/Matr_z)2r(A)
J{Vr,i)
JM(k)U(A)\G(A)
(4.30) xMi($)(vv~
By
in (4.25)
the Iwasawa
G (A)
decomposition
=
(pi)wg)ipr-i(v)dvdpidg.
P(A)K
(P ='P2r,z)? we have
M(k)U(A)\G(A)= [M(k)\M(A)]K.
Recall
=
that M
GL2r x Sp2Z and write
m = m(a,
If we write
g
=
umk, we have,
b)
=
m(a)m(b).
in (4.30),
=
=
$c(g, s,ft
$(g, s, 4>){1 Tc(H(g))) $(mk, s, </>)(l Tc(H(m)))
and
^2r+iA?(P2^y(Pi)Ma^b)]?)
factor.
7^ is the Weil
of
integral (4.30)
[Vr,i]
where
//
=
Thus
Ideta|^^(deta)oV2r+z>i(?(p2,^)?(pr^)m(fc)k),
the
inner
integrations
6^2r+l^i(?(p2,z)?(p1)m(a,b)k)Mi(^)(vv~(p1)wm^
over Matr_??2r(A)
and
THE
ON
OF
NONVANISHING
THE
CENTRAL
VALUE
703
equals
/
Idetal^+^deta)
/
JMatr_?)2r(A)
(4.31)
J[Vra]
^ar+l,z(?(p2^)^(Pi)^(&)k)
xMi($)(m(a)vv~(pi)wm(b)k)ipr-i(v)dvdpi,
G GL2r(A) x Sp2r(A).
where m(a) = rh(a, 72r),ra(fr) = m(/2r,6)
From the definition of $(/i, s,
and $(</, s,
<?>^{j)?a)
^(f)?f)
as given
in ?3, we
have
=
$(m(a, b)k, s,
^(r)?*)
Idet a|s+r+/+i ^(f)?a(^i(a,
=
|deta|s+2r+i
$(m(a,6)k,s,0^(f)0f)
Hence,
sition,
7~1(deta)
taking into account the Jacobian which results
the integral in (4.30) can be expressed as
/ /
JK
x /
^[Sp2il
</>^(f)^a(m(a,&)k);F^^
b)v~(pi)wk) is given by
"<P2r+ly
^2r+i,/(?(p2,z)?(p1)m(&)k)^(t'm(a,6)v"(pi)^k)^r-z(^)^
//
for v = v(n,p2,
from the Iwasawa decompo
\deta\s-Hl-Tc(H(a)))
/
where the function
^((^(f)?^)^^
(4.33)
<^(f)0f(ra(a,&)k).
J[GL2r]
JMatr_z,2r(A)
(4.32)
6)k)
=
z) and (p
We
(^(f)??-
consider
the Langlands
decomposition
for
GL2r(A)
=
GL2r(A)
and we set PM2r,/
the number
=
GL2r(A)1.i4+,
(ZGL2r(A)GL2r(/c)\GL2r(A))x (Sp2Z(fe)\Sp2Z(A)).Let d be
of the real archimedean
places
of the number
field k. Then
integral
(4.32) equals
(4.34)
X(s)
where
the function
X(s)
(PnAma(mk)J^(^)(mv-(Pl)wk)dmdPldk
[
/KxMatr_i)2r(A)xPM2r-,z
as
is defined
A(s):=vo1(A7A:x)
/
\t\2rd^-i\l-Tc(t))dtx.
It is easy to see that
i
JR+
which
has a simple pole at s =
JK x Matr_z;2r
which
\t\2rd(*-*\i
is holomorphic
\.
(4.35) /
(A) x PM2r,?
in s. Hence
-
r2rd{s-\)
cr
TC(t))de
2rd(s-\)
In (4.34) we are left with
(r)?* (mk)^ (4>)(mv~ (pi)wk)dmdpidk
(pnip
we obtain
the main
identity.
704
DAVID
4.4.
Theorem
When
s0 ^
\,
so =
zero. When
JIANG,
AND
\,
RALLIS
^2r+z)
07r^(f)?f)?
the period
(', ?^(t)?^
V3r,r-l(E?
STEPHEN
the period
07rVF(f)?a)?E\(-,
V3r,r-l(Es^(',
25 identically
DIHUA
GINZBURG,
Ei(-,
07rv>(f)?f)J V?2r+z)
is equal to
ex /
0^(f)0a(mk)^(0^(f)0f)(m?;"(pi)^k)dmG?7?i?/k
?/KxMatr_?)2r(A)xPM2r)?
where PM2r,?= (ZcL2r(A)GL2r(A:)\GL2r(A))x (Sp2Z(fc)\Sp2Z(A))and the constant
c is equal to the residue
Remark
4.5. From
at s =
formulas
(4.34) and
V3r,r-l(?l,Ei(-,
has
continuation
meromorphic
This
to
for the period
completes,
the
is
which
| of \(s),
(4.35),
vo
2J??*
the period
V?2r+z)
07ty(f)?f)j
whole
complex
plane.
Vzr,r-i(?\,Ei(-,
07rv,(f)?f)?
<^2r+z)> the proof
4.1. The proof for the period P3r,r-z(^25 Ei(-,
Proposition
^2r+z)
</>7rV;(f)?f)?
from the same argument
(as in [GJR01]). Hence the proof for Proposition
now
of
follows
4.1
is
completed.
4.4 for the case of / > r. In this case
Remark 4.6. There is an analog of Theorem
one has that 4r + 21 > 6r. One has to consider the Fourier-Jacobi
coefficient of the
on
is
residue
Note
that
this
coefficient
Fourier-Jacobi
Sp4r+2Z(A).
Ei(g,(p7Tip^)?a)
an automorphic
form on Sp6r (A). The period will be the integral over Sp6r of this
on Sp6r(A). Again the
coefficient against the residue E\(-,
Fourier-Jacobi
?^(f)??)
no
two
of
is
forms
product
genuine automorphic
longer genuine, so the integration
can
be
taken
the details
over
here
the
linear
group
Sp6r.
In order
to
avoid
extra
notation,
we
omit
for / > r.
5. A
SUFFICIENT CONDITION FOR
L(\,1ti
X 7T2)^ 0
In this section we prove Theorem
for
5.1, which gives a sufficient condition
x 7T2) to be nonzero.
L(|,7Ti
of
Let
7Ti be an irreducible
representation
unitary
cuspidal
automorphic
a
L-function
has
such
that
the
square
symmetric
pole
)
L(s,7Ti,Sym
GL2/+i(A)
at s = 1, and let a be an irreducible unitary generic cuspidal automorphic
repre
to 7Ti. Let 7r2be an irreducible unitary
of Sp2Z (A) which lifts functorially
that the exte
of
representation
cuspidal automorphic
GL2r(A) with the properties
= 1 and the standard L-function
a
s
at
has
rior square L-function
7r2,
pole
L(s,
A2)
=
Take f to be an irreducible unitary generic
L(s,7T2) does not vanish at s
\.
of Sp2r (A) which has the ^-transfer
7r2. Note
representation
cuspidal automorphic
descent method
that the existence
of a and r is established
by the automorphic
sentation
([GRS99c], [GRS01],and [S02]).
Theorem
5.1.
Let (pa G VG and <p?G Vf.
If the period
integral
Vr,r-l(<t>a,4>r,Vi) (r > 1) Or Vl,l-r(4>r, <Pv,<fr) (r < I)
ON
THE
NONVANISHING
OF
THE
CENTRAL
as defined in (2.12) and (2.13),
attached to (a,r,ip),
value of the Rankin-Selberg
product L-function
L(|,7Ti
VALUE
705
is nonzero,
then the central
x 7r2) is nonzero.
Remark 5.2. If the irreducible unitary cuspidal automorphic
7r2 of
representation
has the properties
that the exterior square L-function
has
GL2r(A)
L(s,7r2,A2)
=
a pole at s = 1 and the standard L-function
s
at
vanishes
the
L(s,7r2)
|,
and we hope to deal with this case in a future
lifting theory is more complicated
publication.
(2.12) and
By
(5.1)
are defined
the periods
(2.13)
Vi,i-r(fc,<l><T,<Pr)= i
as follows.
If / > r, we have
M9)^r(M(9)dg,
JSp2r(k)\Sp2r(A)
function on Ar (i.e.
ipr is a Bruhat-Schwartz
of two genuine functions
is no longer genuine,
<
over
is taken
If I r, we have
Sp2r(fc)\Sp2r(A).
where
(pa G Va, (pr G Vf, and
<S(Ar)). Note that the product
hence
the integration
(5.2)
Pr,r-z(^,0f,^)=
(pr G
and
M9)^t?r)(9)dg
/
JSp2l(k)\Sp2l(A)
where
(pa G VG, (p? ? Vf, and <piG S(Al).
We prove this theorem in detail for the case where / < r using Theorem 4.4. The
other case where
I > r can be proved by the same idea and the same argument.
We omit the details here.
as in (5.2). Denote
When
I < r, the period is Vr^-i((pa,(pr^i)
by tt^(t) the
of
the
our
of
from
to
f
at the
image
^-transfer
By
assumption
Sp2r(A)
GL2r(A).
=
we
use
of
this
have
we
To
Theorem
consider
7r2
section,
beginning
4.4,
tt^(t).
the generic cuspidal data (P2r,i^t?)(j) 0 cr) of
and
? t) of
Sp4r+2/(A)
(P2r,r^^(j)
Sp6r(A),
assume
We may
respectively.
(5-3)
that
=
07ty(f)?<7
^(f)?^,
=
(5-4) 4>*i,{t)?t
^(f)^^f,
where
Recall from (3.2), (3.3) that the
<pn (?) is the complex conjugate of (^(f)function
is given by
^(g,s,(pn^{r)?a)
=
(5.5)
&(g,s,(pn^r)?<T)
(t>^{r)??9)^v(sJrpP2r^Hp2rl(g))
s is normalized
as in (3.3) (i.e., s is identified with
where the parameter
so;2r), and
the function
$(#,
(5.6)
*(^S,07rv,(f)?f)
s, 4>^{j)?t)
is given by
=
^(r)?r(g)
following (3.13) and (3.14).
We need the following proposition
Proposition
5.3.
exp(s + pp2rr, Hp2rr(g))
to finish
the proof of Theorem
5.1.
// the period
Vr,r-l((p(j,(pT,?l)
does not vanish for some given
(4.35), which equals
^KxMatr_;,2r(A)xPM2r)Z
<paG Va, fc
G Vf, and
^(f)
G
K>(?)>
then integral
/ 07r^(f)?ff(^k)^(^(f)0f)(mi;"(pi)iyk)dmdpidk,
706
DAVID
does not
vanish
GINZBURG,
Recall
from (5.2),
AND
STEPHEN
RALLIS
in (5.5)
and
(5.6),
(2.11), and (2.13) that the period Vr^r-i(?a,
where
the
(p?, <pi)equals
(Pa(g)
f ^(^(v^Mvg^r-i^dvdg.
/
It defines
JIANG,
the corresponding
data defined
is defined as in (4.33).
for
function ^,(07TV;(f)?f)
Proof.
DIHUA
a continuous
functional
on the space of
is the space generated
by the theta functions 0^ with
ipi G S (A1) and
Qf
coefficients
of auto
Vf )yirtr-i js the space generated by the Fourier-Jacobi
(Of <S>
forms in r.
morphic
where
Recall
from
that the function
(4.33)
^r^(07rv,(f)?f)(^(?,
b)v~(pi)wk)
equals
(5.7) / ^2r+l?i(?(p2,?)?(p^
J[Vr,l)
It is clear that S(A2r+l) = S(A2r)?S(Al).
of variables),
then we have
=
0v2r+MP2,mPi)9)
for g G Sp2?(A).
functions
For any fixed
=
</>2r+Z
If we
take
=
<?>2r+z ?>2r? y>i (separation
?>2r(?(Pi)) Ol{i{p2,z)g)
</?2rG <S(A2r), we
consider
all Bruhat-Schwartz
G <S(A2r_M),
? <?>Z
<?>2r
with
(fi G S (A1). It follows that the space generated
by 0<p2r+l,i(?(P2,z)g)
(with a
fixed if2r ? <S(A2r) and all (pi G S (A1)) is the same as the space
(generated by all
?f
as
of
the
Jacobi
automorphic
representations
group Sp2Z(A) x Hi(A),
0^t (?(p2iz)g))
where Hi is the Heisenberg
all
In
the following we may
group generated by
?(p2,z).
assume that ip2r is supported
in a small neighborhood
of zero. This is needed for the
in (5.10) below, in particular
of X(\?)
for the integration
in variable
nonvanishing
of the period Vr,r-i((pa,
P\. It follows that the nonvanishing
(pri^Pi) is equivalent to
the nonvanishing
of the integral
/
^[SP2Z]
It is clear that
(pa(b)/
0^2r+l^i(?(p2,z)b)^r(vb)ipr-i(v)dvdb.
J[Vr,l)
the integral
JZGL2r(A)GL2r(fc)\GL2r(A)
/ (t>^{T)(o)(t>^{r)(o)da
is not zero for some choice of
and
Using
(5.3) and (5.4), the nonvan
(p<K^(r)
^^(r)two
of
of
the
the
combination
is
above
ishing
integrals
equivalent to the nonvanishing
of the integral
/ (5.8)^(f)??rM^/,(07r^(f)?f)M?/m,
JPM2rJ
ON
THE
THE
OF
NONVANISHING
CENTRAL
VALUE
707
x (Sp2i(fc)\Sp2,(A)) as in (4.34), and
where PM2r,z= (ZGL2r(A)GL2r(A:)\GL2r(A))
for m = m(a,b),
=
^(0Mf)??)M
/
J[Vr,l]
Hence
(5.8) gives
^2r+i,z(^fe^)^(&))^(f)?f(^(?)^^(^)^r-z(^)^.
rise to a nonzero
(5-9)V^)0a
continuous
linear functional
J
on the space
? [V^(f) ? (6? ? t)^"']
It remains to show that there are some choices of data, so that the integration
over the maximal
and over the variable p\ of
compact subgroup K of Sp4r+2/(A)
in (5.8) will be nonzero. To do so, we have to realize the relevant
the functional
in different models
of functions.
For example,
for the integration
representations
over K, one has to realize the induced representation
in the compact model, while
for the integration over variable p\, one has to realize the parabolic
induced rep
in the model of functions over the opposite unipotent
resentation
from P = MN
radical N~.
The argument has been used in [JR92], [Jng98a], [GJR01],
We
sketch the main ideas following
[GJR].
[GJR01].
As in the proof of Theorem
3.2 in [GJR01], we consider the function
[GJR03]
and
*
with
:K x
Matr_,j2r(A)
-+
V^(m<r
left quasi-invariance
tt(pk)
for p G ?*2r,/(A) nK.
Here p(p) denotes
!(*):=/"
=
?
[V^
?
p(p)9(k)
the representation
(5.9). Define
J(*(pi,k))d|?idk.
=
for a given (p <g>
J^Q)
?^(f)?* ? -^(^(f)??)
p, there is a smooth function 9 such that
representation
=
in the space
^KxMatr_i)2r(A)
Then
(5.10)1(9)
(6f ? t)v"?'*-']
in the space
(5.9) of the
[ J(*(Pl,k))dPldk
?KxMatr_/
xMatr_2;2r 2r(A)
(p(mk)F^ ((p)(mv~ (pi)wk)dmdpidk.
L /KxMatr_z,2r(A)xPM2r.,?
the proof of this proposition,
it suffices to show that one can choose the
9 with restricted support so that the integral T(&) is nonzero.
Recall
that [/2r,r is the unipotent
radical of the maximal
parabolic
subgroup
the
in
of
Since
the
is in a subgroup of f/2~ r
variable v~(p\)
integration
Sp6r.
P2r?r
To finish
function
of ?72r,r), we can choose a section 4> with the property
that it is
(the opposite
as
a
modulo
function
and
compactly
supported
compatible with
ofv~(pi)
P2r?r(A)
the support of </?2ras remarked before. Hence the integration
in pi of (5.10) is
nonzero.
Since the variable k is independent
of the variable p\, the nonvanishing
in variable k is proved the same way as in the proof of Theorem
of the integration
3.2 in [GJR01]. D
now return to the proof of Theorem
5.1. By Proposition
if
the
not
does
vanish
for some given
4.4,
period Pr?r_/(0cr, 0f, tpi)
and
then the period
<t>^{j) K-^f),
We
V3r,r-l(Ei
(',
^(f)?*^
Ei(-, ^(f)??)>
T?2r+l)
5.3 and Theorem
(pa G Va,
</>f? Vf,
708
DAVID
GINZBURG,
DIHUA
JIANG,
AND
STEPHEN
RALLIS
not vanish
In particular,
for the corresponding
data.
this implies that the
not
Hence
does
vanish
for
the
data.
from Proposition
given
(g, <P>k^{j)?<j)
3.2 the product of L-functions
does
residue E\
L(s,t?^(t)
has a simple pole at s =
|.
Since
x
g)L(2s,t?^(t),K2)
the exterior
square L-function
L(s,7t7P(t),A2)
x a) cannot vanish at s =
has a simple pole at s = 1, the L-function
L(s,tt^(t)
^.
of r from Sp2r and i?\ is the image of the
Since 7T2= 7T^(r) is the ^-transfer
lift of a from Sp2/, we obtain that the tensor product L
functorial
Langlands
This completes
the proof of
function L(s,7Ti x 7r2) does not vanish at s =
\.
Theorem
5.1.
6. A NECESSARY CONDITION FOR
L(|,7Ti
X 7T2)^ 0
In this section we prove the second part of the Main Theorem.
an assumption
on the nonvanishing
of certain Fourier
will make
In this proof, we
coefficients of the
See Assumption
(FC) below.
Ei(g,(p7r^^)^(7)'
we
recall
In order to state Assumption
briefly from [GRS03] the notion
(FC),
to unipotent
orbits. The
of Fourier coefficients
of automorphic
forms associated
orbits was
twisted Jacquet modules with unipotent
p-adic version, which associates
orbits in Sp2n(C)
well studied in [MW87]. Recall
that the set U of all unipotent
residue
ordered.
The partial
(or in any complex reductive group) is finite and partially
relation. Let ?\ and ?2 be two
ordering in U is given by the 'included-in-closure'
in U. One defines ?\ < ?2 if ?\
is included in the closure ?2 of ?2.
It
members
is also known that unipotent
of
orbits in Sp2n(C) are parameterized
by partitions
type (see [CM93] for example).
symplectic
Let (p be an automorphic
form on Sp2n(A). Following
[GRS03], we let ?((p) be
that ? G 0((p)
orbits
with
the
the subset of U consisting
of unipotent
property
a
nonzero
to
orbit
if (p has
Fourier coefficient associated
? and for any unipotent
?' G ZYbigger than ?, i.e. ?' > ?, (pdoes not have any nonzero Fourier coefficient
to ?'.
associated
Let a be an irreducible automorphic
of Sp2n(A)
representation
in the discrete spectrum. We let O (a) be 0((p) for some nonzero (p G a.
occurring
is independent
of the choices
Since a is irreducible, one can check easily that 0((p)
of nonzero
a.
(p in
of
Let a be an irreducible unitary generic cuspidal automorphic
representation
r
an
and
let
be
irreducible
represen
automorphic
unitary
generic
cuspidal
Sp2/(A),
tation of Sp2r (A). Assume
of r to GL2r (A)
that the image ir^ (r) of the ^-transfer
is cuspidal. We make the following assumption.
is nonzero, then it has a nonzero
If the residue Ei (g,
Assumption
(FC).
(p^^^^)
to the unipotent
orbit
Fourier coefficient associated
by
parameterized
?((2r+2J)i2r)
the symplectic
+
partition
((2r
2/)l2r).
to the unipo
Recall from (2.1) and (2.4) that the unipotent
subgroup attached
tent orbit
?f Sp4r+2? can be expressed as (with p replaced by 2r + I
?((2r+2/)i2r)
THE
ON
and i by r +
NONVANISHING
=
n G Nr+i
character
<v
and x
=
is an
attached
For an automorphic
(r +
to
=
ip(nlj2
+
+
form (p of Sp4r+2? (A), the ipQ
Sp4r+2?
rcr+i_ijr+?
+
2r -Fourier
coefficient
The
a2r+?}i).
of (p
. L
^)^((2r+2i)l2r)(^.
the /c-rational orbits
a
in kx. Assumption
of
of the residue Ei(g,
(p^^r^a)
a.
we
conjecture.
fact,
make
6.1.
Conjecture
that
a
stronger
If the residue E\
are parameterized
by the
0((2r+2i)i2r)
2r -Fourier
(FC) requires that the ipQ
of
is nonzero
(g, ^^(f)?^)
=
0(Ei(-,(p7rij^)^a))
is easy
a
ipQ
to check
that
2r -Fourier
Conjecture
coefficient
form on Sp2r (A), and a ipQ
(g, 07r^(f)?ir) produces aWhittaker-Fourier
ular unipotent
of the ibfn
orbit)
automorphic
Ei
Ei(g,(pTrlp{j)^cr)the residue Ei(g,
6.1
of
for at least one square class of
i*snonzero,
then it has the property
?((2r+2/)(2r))
implies Assumption
the residue Ei(g,<pn
In fact,
*s an
(?)?cr)
(FC).
t 2r -Fourier coefficient of the residue
to the reg
coefficient (i.e., associated
-Fourier
coefficient
of the residue
the nonvanishing
of a ipQ 2r 2i ^ -Fourier coefficient of
of a ipQ
implies the nonvanishing
2r -Fourier co
^^(f)?^)
Hence
of Ei
(g,
^(f)?^).
as in Proposition
1 in [GRS], we
using the same argument
which supports the conjecture.
following proposition,
By
Proposition
is greater
Ei(-,(pn
row.
?((2r+2i)l2r)J
that
square classes
efficient
G
x
coefficient
It
x*
by
remark
first
hr
(2r)-matrix with zero bottom
1Sgiven (see (2.5)) by
0((2r+2?)i2r)
I)
ipr+i(n)ip(azr+hi)
1
In
I
v(n,x,z)
=
^ol(n ((2r +
xon,2r,(v)
2t)l
)
We
=
709
VALUE
CENTRAL
J
^
ipQ
is defined
THE
I)
V0{{2r+2l)12r)
where
OF
can prove
the
6.2. Let ? be a unipotent orbit
If the unipotent orbit O
o/Sp4r+2Z.
than or not related to the unipotent orbit
^hen the residue
?((2r+2i)(2r))>
no nonzero Fourier coefficient associated
to the unipotent orbit
has
(t)?(t)
o2.
=
be any unipotent
is bigger than the orbit
orbit which
<9(ni...nq)
an<^ sucn that the residue Ei (g, (p^
bas a nonzero Fourier co
?((2r+2Z)(2r))
(?)&(?)
as in Lemmas 2.4 and 2.6
to ?. Then n\ > 2r + 21. Arguing
efficient associated
in [GRS03], we deduce that there is a number p which is larger than 2r + 21 such
that the residue E\ (g,
bas a nonzero Fourier coefficient associated
to the
(^^(f)?^)
orbit
in
Lemma
3
and
Lemma
6.8
unipotent
However, using
[GRS]
?((2p)i4r+2z-2P).
which we will prove in ?6.1, it follows that E\(g,
bas no nonzero Fourier
^^(f)?^)
to the unipotent
coefficients
associated
orbit
Thus we derive a
0((2p)i4r+2z-2P).
contradiction.
D
Proof.
Let ?
710
DAVID
DIHUA
GINZBURG,
JIANG,
AND
STEPHEN
RALLIS
below. Let 7Ti be an irreducible
of
with the property that
unitary cuspidal automorphic
representation
GL2z+i(A)
=
a
the symmetric
L-function
has
square
1, and let a
pole at s
L(s,7Ti, Sym2)
be an irreducible unitary generic cuspidal automorphic
of Sp2/(A)
representation
to tti . Let 7r2be an irreducible unitary cuspidal automorphic
which lifts functorially
of GL2r(A) with the properties
that the exterior square L-function
representation
= 1 and the standard L-function
a
s
at
has
pole
L(s, 7T2,A2)
L(s, 7r2)does not vanish
r
an
at s =
and let
be
irreducible unitary genuine generic cuspidal automorphic
|,
to 7r2.
of
which
has the ^-transfer
representation
Sp2r (A)
The main
result of this section
is the theorem
6.3.
Theorem
Assume
that Assumption
(FC) holds for the pair (r,o~). If the
x 7t2) is nonzero,
product L-function
of the Rankin-Selberg
L(|,7Ti
then there exists an irreducible cuspidal automorphic
a' o/Sp2/(A),
representation
which is nearly equivalent to a, and there exists an irreducible cuspidal automorphic
central
value
is nearly
which
r' o/Sp2r(A),
representation
equivalent
to r,
such
that the period
integral
(r > I) Or Vl:l-r(^r^(pa'^r)
Vr,r~l(<Pa'^T'^l)
is nonzero
to (a',Tf,ip)
attached
for some
choices
(r < I)
of the data.
> / = 1 in ?7.
shall verify Assumption
(FC) for the case r
=
1.
for the case r > I
Hence Theorem
6.3 holds unconditionally
x 7r2) ^ 0. Since 7r2=
To prove Theorem
6.3, we assume that L(|,7Ti
7ty(r)
x a) ^ 0 since t?\ = n(a)
is the image of the
and 7Ti= 7r(cr), we have
L(|,7r^(r)
functorial
lifting of a. Hence the product of L-functions
Langlands
We
remark
that we
L(s,7T1p(r)
has a simple pole at s =
x
a)L(2s,n^(r),K2)
3.2 implies
\. Proposition
that the residue
E^g^n^i?)?*)
does
not
vanish.
(FC), we know
By Assumption
Fourier coefficient
corresponding
that
the residue
to the unipotent
Ei(g,(p7r^^^)a)
orbit
O((2r+2?)i2r-).
has a nonzero
That
is, the
integral
/
(6-1)E\ (?0>
K(m^o{{2r+2l)l2n(v)dv
the square classes a G kx associated
is nonzero for some a G kx. In order to match
to the Fourier coefficients which will be discussed below, we have to twist the residue
by a similitude element in GSp4r+2?(&).
#i(0,07ty(f)?(7)
Let
rahr+i
o
d(o)-<
be a diagonal
Sp4r+2/(A),
by
0
i2r+l
similitude element in GSp4r+2Z(A:). Since the element d(a) normalizes
II of
one may define the twist by d(a) of a representation
Sp4r+2/(A)
nd^(g)
=
U(d(a)9d(arl).
on twists of repre
4 in [MVW87] for more discussions
See Section II of Chapter
sentations.
Then one can check that the twist by d(a) of the residual representa
is the residual representation
tion generated
generated by the
by Ei(g,
(pn^(f)?^)
ON
THE
NONVANISHING
OF
THE
CENTRAL
711
VALUE
Here o~a is the twist of a by the similar element d(a) in
Ei(g,(p7r^^0(Ja).
It is easy to check that aa is generic (with respect to the twisted generic
GSp2?(fc).
to a. This implies in particular
that the residue
character) and nearly equivalent
to
Now the
is nonzero and is nearly equivalent
??i(^,0^(f)0iTa)
Ei(g,(pni}^)?a)residue Ei(g,(pn
has the nonzero Fourier coefficient
^^aa)
residue
Ei / (vg,
(6-2)
^(f)?(Tfl)^((2p+al)iap)
J[V?((2r
where
en,,
^((2r+2Z)l2r-)
It follows
N~
=
(v)dv
+ 2Z)l2r)]
Y
U
ip}n
from Lemma
n
.
((2r+2l)l^)
1.1 in [GRS03] that the nonvanishing
to the nonvanishing
of the integral
of the Fourier
in (6.2) is equivalent
coefficient
(6.3) ^J'(Ei)(g)=
i
J[V2r+l,r]
?&MirivWEiivg^^J??^dv.
It is clear that T?f>l~fl(Ei)(g)
is an automorphic
form on Sp2r(A). Similar automor
descent map
phic forms were studied in [GRS99c] in the context of the automorphic
the residue representation
from GL to classical groups. However,
used in [GRS99c]
was
different.
__
Let ra be the automorphic
of Sp2r(A) generated
representation
defined in (6.3). We claim that
phic forms Fif>2?l(Ei)(]j)
by all automor
ra is cuspidal;
(1) the representation
to r.
(2) any irreducible summand of ra is nearly equivalent
To prove the cuspidality of ra, it is enough to show that for any standard maximal
?
< j <
terms of the
parabolic
subgroup Pj,r-j
(1
r), the constant
^j,r-jUj^r-j
forms ?F^>2rl(Ei)(g)
radical f7j,r-j are all zero;
automorphic
along the unipotent
i.e.,
^(Ei)(ug)du
/ (6.4)
= 0.
This is done as in [GRS99c, pages 844-847]. The idea is to take Fourier expansions
of the constant
term in (6.4) along some unipotent
In such Fourier
subgroups.
are
there
two
of
Fourier coefficients occurring
in the summands.
expansions,
types
One type is the Fourier coefficient of the residue
to
associated
Ei(g,(pnip^)<S)aa)
the unipotent
orbit
for p > r + /. It follows from Proposition
6.2
?((2p)i4r+2i-2P)
that the residue has no nonzero Fourier coefficients
to such unipotent
associated
orbits.
Another
of
type is the Fourier coefficients which contain an integration
the constant
terms of the residue
the standard maximal
along
Ei(g,(plx^^^(Ja)
parabolic
subgroups other than P2r%/. These are zero due to the cuspidal support
of the residue Ei (g,
This proves the cuspidality
of ra.
^^(f)?^)Remark
6.4. The
second assertion above is proved by studying the structure of the
of ra and r. This can be done in the same way as
components
that of Proposition
5 in [GRS]. We will give some details at the end of ?6.1, after
certain preparation
of local results.
local unramified
It follows
ra is cuspidal.
By assertion
(1) as proved above, the representation
can
that r0
be written as a direct sum of irreducible cuspidal automorphic
repre
sentations.
Let t' be one of the irreducible summands of r0. By assertion
(2), rf is
712
DAVID
nearly
to r. From
equivalent
(6-5)
4>r'(g)/
J[V2r
/
J[SP2r]
does not vanish
DIHUA
GINZBURG,
AND
JIANG,
STEPHEN
RALLIS
(6.3), we have
+ l,r]
e^2r(?2r(v)g)Ei(vg,(P{^^aa)ipr+i(v)dvdg
for some choices of data. We
note here that the irreducible
r' of Sp2r(A)
is nearly
automorphic
representation
ducible generic cuspidal automorphic
representation
cuspidal
to
and
the irre
r,
equivalent
aa of Sp2?(A) is nearly equiv
to a.
alent
6.5. Let us mention
Remark
that
In integral
E(g,
(6.5), we replace
s, <^(f)?<ra) and obtain
(6.6)
the
the residue E\
4>r'(g) i
/
6.1 holds,
then r1 will be generic.
(-, (pn^^<S)(Ta)by the Eisenstein
o*2r(?2r(v)^S(^,s,^
J[V2r+l,r]
^[Sp2r]
if Conjecture
series
(^)0aa)^r+z(i;)d^.
inner
in integral
an automorphic
form on
integration
(6.6) produces
we
is
know
that
for
</>f/ cuspidal,
Sp2r(A)
integral (6.6) converges absolutely
the real part of s large and has meromorphic
to the whole C-plane.
continuation
Since integral (6.5) is the residue of integral (6.6) at s =
one concludes
that
|,
not
some
does
vanish
for
choice
of
data
when
the real part of s is
integral (6.6)
large, because
integral (6.5) is not identically zero.
Since
and
Hence
the proof
of Theorem
6.3 will
be completed
by means
of Proposition
6.6
below.
6.6.
Proposition
// the period
Vr,r-i((pa,(pr,<fi)
vanishes
identically,
integral
(r >
I)
then the integral
or
Vifl_r(<i>T,<t>a,<Pr) (r
in (6.6) vanishes
identically
<
I)
for all choices
of
data.
Proof. The idea is to unfold integral (6.6), for the real part of s large, and to obtain
the above period as inner integration.
For the real part of s large, we unfold the Eisenstein
series in (6.6) and we obtain
(6.7)
where
cosets
VJ f fc(g)6$2r(e2r(v)g)$(wvg,s,</>)?^v)dvdg
the
in
summation
on w
is taken
from
the
set of representatives
of double
and
the
is over
integration
L2r^(A:)\Sp4r+2?(A:)/Sp27.(A:)V2r+/5r(A:),
D
Since the double coset de
(w~1P2r,i(k)w
Sp2r(/c)V2r+;5r(fc))\Sp2r(A)V2r+/?r(A).
is of generalized Bruhat type, the representatives
w can be chosen from
composition
the Weyl
in
elements
Sp4r+2?.
We shall first show that all the terms in (6.7), except one, vanish by means of the
=
argument of double cosets. We write w
admissibility
(wij) where Wij is the (i,j)
th entry of w. We may assume that Wi? take values in {0,
If v e V2r+i,r such
?1}.
that wvw'1
G U2r,i and ipr+i(v) ^ 1, then the summand
in (6.7) corresponding
to
this w is zero, because of the left and right quasi-invariance
the
of
property
integral.
In this case, we say that w is not admissible.
For 1 < i < 2r + I, let o?i denote the z-th simple root of
Sp4r+2?. Let xa.(c)
the one parameter
denote
of
to the
unipotent
subgroup
Sp4r+2/ corresponding
=
we
root
In
matrices
have
Here ei ?+1 =
ol{.
simple
hr+2i + cea+ixa.(c)
ON
e?,?+i
THE
=
denote
xr+i(c)
Sp4r+2?
+
hr+21
to
attached
/-
if ? < 2r +
e4r+2?-z,4r+2/-?+i
the
CENTRAL
1 and
eM+1
is
which
cer+/53r+?+i,
root
positive
THE
OF
NONVANISHING
the
+
2ar+i
+
=
VALUE
713
if i =
eiji+i
2r +
one-parameter
of
subgroup
+
2a2r+/_i
I. Also
cv2r+i-
Clearly,
G V2r+i,r for 1 < ? < r + ! - 1. It follows from the definition
of
xoti(c),xrjri(c)
= 2r +1 and i = r +
case
n
in
for
the
and
the
left
quasi-invariant
Z)
ipr+i,a (as
(2.5)
< ?< r + / ?
1
property of 0^2r that if one of wxai(c)w~1
(for
l)or wxr+i(c)w~l
in (6.7) corresponding
to w is zero; i.e., w is
belongs to ?/2r?/, then the summand
not admissible.
Let w = (w?j) be one of the representatives.
We first assume w^i = 1 with
a
1 < i < 2r. In this case, by multiplying
by Weyl element of P2r%/ on the left, we
=
= 1. Next we consider
a
new
w
obtain
may
representative
Wi?
(w?j) with w\?
for i > 1. If i > 2r, we will have wxai(c)w~x
G U2r,i and hence this representative
w is not admissible.
We obtain that Wi? = 1 for some i < 2r.
In this case,
a
we
on
the
left
element
of
suitable
may obtain a new
by multiplying
by
LW,/?
=
w
with
representative
(w??)
=
Wl,l
Continuing
admissible
=
^2,2
with this argument, we deduce that the representative
or can be chosen to be in the form
have
p depends
p
=
r +
/.
But
in
j
w
If r >
/ as follows.
we
case
this
is either not
w'
\
on r and
w
\
(h
w =
where
I
have
I, then 2r >
G
wxrjri(c)w~1
r+
I and we would
w
i-e-'
U2r,i'i
ls no^
= 2r. In this case we have
I, then 2r < r + 1and we would have p
cases
w such that wi? = 1
G
in
Thus
both
any representative
wxa2r(c)w~l
t/2r,/.
with 1 < i < 2r is not admissible.
Note that we may start with Wi^ = ?1 with
1 < i < 2r, but the positive or negative
is not essential
sign of the representatives
to the admissibility
of the representatives.
Next assume w^\ = 1 where 2r + 1 < i < Ar. In this case, by multiplying
a
If r <
admissible.
suitable
element
=
in
P2r,i
on
the
we
left,
obtain
may
a new
w
representative
=
(w?j)
=
1- Suppose Wi?
1- If i > 2r + 21, then wx^^w-1
G C/2r??.This
the representative
is not admissible.
If i < 2r, we can obtain a new
= 1
w =
a suitable element in
representative
by multiplying
(w?j) with wi?
P2rj
on the left. Arguing
as above, we deduce that the representative
w is either not
admissible or has the form
with W2r+i,i
implies that
*
0
0\
0
*
0
0
0
0
*
0
1
0
0*00
0 * Ip 0/
/0
w~
_
Ip
0
0*00
1
\0
'
If r > /, then p = r + Zand t?;xr+/(c)i<;~1 G [/2r%/, and hence i?; is not admissible.
If r < I, then p = 2r - 1 and i?;xa2r(c)i?;~1 G ?72r,/, and hence w is not admissible.
- I
we deduce
that if Wi? ?
Therefore,
1, then 2r + 2 < i < 2r + 21
(since
?
^2r+i,i
that
i=
!)
Multiplying
2r + 2. Arguing
a
suitable
as above,
element
this
in
P2r,i
time with
on
Wi?
the
=
left,
we
1 and
may
assume
so on, we
can
714
DAVID
continue
this process
DIHUA
GINZBURG,
AND
JIANG,
STEPHEN
RALLIS
that
and deduce
* * 0\
/0
0
0 0 I
iz
0 0 0 7/1*
* *
0/
\0
=
now, if w?,z+i
1, then either z > 2r + 2/ or i < 2r. Hence,
deduce that this w is not admissible.
this process
Thus we may assume that W2r+2Z+i,i ? 1- Repeating
2r columns, we are finally left with a unique element which is
But
in (6.7).
summand
element represents an admissible
This Weyl
this summand follows from the vanishing of the periods.
First assume that r <l. For wq we have
H (Sp2rVr2r+Z,r) = N2r
^o"lp2r,/^0
where
refers to ?2. Thus,
the notation
X
with
The
we
the first
vanishing
of
(Sp2rV/,r)
(6.6) equals
integral
as above,
in this case
0T! (W^2r(?2r(v)g)^(wvg, s, (p)ipr+i(v)dvdg
I
x
the
Factoring
integrate over N2r(k)
(Sp2rV??r)(fc)\Sp2r(A)V2r+/jr(A).
as
an
inner
with respect to Sp2rV/5r, we obtain the period Vij~r((p?,
(pa? </v)
integral in the above integral.
in P2r>? on the left and by
Next assume that r > I. Multiplying
by elements
in Sp2r on the right, we may assume that
elements
where
we
measure
=
wo
In this case we obtain
H
WQ1P2r,lW0
where Pr-i,i
Sp2? and
is the standard
N2r,r-l
=
parabolic
=
|
N2r,r-l
(Sp2rV2r+Z,r)
r1
subgroup
j2
j
x
Pr-l,l
of Sp2r whose
Levi part
is GLr_/
x
x hi GV2r+w
as in the proof of Theorem
v? G Nr+i and V2 G Mat(n+fc)X(n_fc).
Continuing
5.1 in [GRS98], we once again obtain the period Vr,r-i(<Pcn<l>Ti<Pi) as an inner
in the above integral.
integration
the above integral is zero for all choices of data. This
Hence, by assumption,
6.6. D
the proof of Proposition
completes
Here
ON
THE
NONVANISHING
THE
OF
VALUE
CENTRAL
715
In this subsection we study the structure of the local un
the
residue E\ (g,
of
and prove the second assertion
components
?^(t)??^)
stated above Remark 6.4.
Let F be a nonarchimedean
to the number field k. Let a be
local field associated
an irreducible admissible generic unramified
of Sp2? (F) and let f be
representation
6.1. Local
structures.
ramified
an irreducible admissible
representation
generic unramified
a
we
can
as
and
f
induced
express
[GRS99b],
(normalized)
=
<T
(6.8)
where
respectively,
is a character
p
?=
In4P2'(M);
Indfa-(xo7*),
I unramified
of
consisting
of Sp2r(F).
Following
representations:
characters
characters xi, X2-, * * ,Xr of Fx,
pi of Fx, x consists of r unramified
to the Weil
Weil
factor attached
representation
uty. By assumption
transfer can be defined as ([GRS99b])
p\,
,
/x2,
and j^ is the
the local ip
(6-9)
7r^)=Ind^-L2r)(x')
where
x'
1San unramified
character
,tr,tr+i,
x'(diag(?i,?2,
of GL^r defined
:=
,t2r-iMr))
by
Xi(r^)X2(?^-).
t2r
t2r-l
-X?t1?)
tr+1
Let Qi^r denote the standard parabolic
subgroup of Sp4r+2? whose Levi part is
x
on
definition
/
unramified
characters
?i
\i{ of Fx and %
By
depends
GLX
GL2.
r
on
an
unramified
characters
of
Fx.
We
define
character
unramified
depends
Xj
(li, x) on GL^ x GL2 as
r
/
(6.10)
(frX)(a>w-
for all ai G GLi(F)
of
,ai,hi,--
,hr)
and hj G GL2(F).
=
We
JJ/x?(ai) JJ Xj(det hj)
extend
(?i, x) canonically
to a character
Qi,r
First we have
Lemma
6.7.
the unique
The
local unramified
unramified
quotient
component
of the residue Ei(g,(pni){j^(Ta)
induced representation
of the (normalized)
is
Ind^Vx)
Proof. The
the details.
proof
D
is the same as the proof of Proposition
1 in [GRS99b]. We
omit
Next we proceed as in Sections 2 and 3 in [GRS]. We recall from (2.1) that for
1 < P < 2r + /, V2r+i-p
is the unipotent
radical of the standard parabolic subgroup
of Sp4r+2Z whose Levi part is GL^ x
As in (2.4), V2r+i-P has a normal
Sp4r+2/_2p.
For
subgroup
V0{{2p)l4r+2l_2py
rest of this section.
Up
=
{v
=
More
precisely,
we
we have
denote
V0{{2p)l4r+2l_2p)
by
Up
in
the
: vPij = 0 for all p + 1 < j < 4r + 21 p}.
=
are
and
in
defined
ipp
(2.3) and (2.5).
ipjjp
rip%2p)i4r+2l-2p)
(e,Xe) of Sp4r+2?(F), we let Cp,a be a linear functional on
(v?j) G V2r+i-p
For a G Fx,
characters
Given a representation
the space Xe
simplicity
satisfying
CPja(ux)
for all u e Up and x GXe.
=
1p^jp(u)Cp^a(x)
DAVID
716
As
Lemma
2 in [GRS] we
in Lemma
6.8.
DIHUA
GINZBURG,
JIANG,
AND
STEPHEN
RALLIS
state
Let Xn
component
of the residue
^)(Sx7aiv be the local unramified
=
a
v
a^
local
Then
representation
fin^e
place
for all a G
(g, ^^(f)?^)
(F
kv).
=
no
nonzero
r +1 +1.
Fx
the representation Xn ^)?aa,v
has
functional CPja for p
Ei
as in the proof of Lemma
2 in [GRS]. We sketch the idea
of the proof. By Lemma 6.7 it is enough to prove Lemma 6.8 for
Indg 4r+2l(fi, x).
This is done by the Mackey
theory. In other words we need to consider the space
and show that for every representative
7 of the double cosets,
Qz,r\Sp4r+2?/?7r+?+i
u
1. This is done
there exists
G ?7r_M+i such that 7i?7_1 G Qi,r and
ip?jr t x(u) ^
as in the proof of Lemma 2 of [GRS]. We omit the details here. D
Proof.
This
is done exactly
the proof of the second assertion above Remark 6.4. If r' is an
of ra, then arguing as in [GRS], we show that the unramified
of f' is given by an induced representation
Finally, we sketch
irreducible summand
local component
Indf(^(X,o7,J
element. Recall
and that 7^ = 7^ and Xv is equal to x up to a twist by a Weyl
means
are
in
defined
that the local unramified
that 7-0 and x
(6.8) and (6.9). This
to the corresponding
local unramified
of
of r' is equivalent
component
component
can be found in
t. This proves the second assertion.
The detail of the argument
the proof of Proposition
5 in [GRS].
7. On Assumption
(FC)
=
In this section we shall prove Assumption
1, while
(FC) for the case 1
I= 0, it is proved in [GRS99a]. In other words, we shall prove
for the
case
the Eisen
be the residue at s =
\ of
Ei(g,(p^^{j)^a)
on
as defined in (3.10).
the
residue
If
Sp4r+2(A)
E(g,s,(p7rip^^(T)
to
i>snonzero,
associated
then it has a nonzero Fourier
coefficient
Ei(g,(p^lP(T)^a)
class parameterized
the unipotent
by the symplectic partition
((2r + 2)l2r).
7.1.
Proposition
stein series
Let
Proof. Let U denote the unipotent
group of Sp4r+2 whose Levi part
affine space
Au=
(7.1)
where Mat?Xj
?0Mat2x2
=
where Wj is the longest Weyl
elements of Au by
(7.2)
zi G Mat2x2,
a character
ipu(u)
=
eMat2x3eMat3X3
{X e Matjxj
element
u =
define
j
the space of all i x j-matrices,
denotes
Mat^xj
where
radical subgroup of the standard parabolic sub
with the
is GL2_1 x GL3. We identify U/[U,U]
x =
of GLj.
,zr-2,
u(z1,
(x?j) GMat2x3,
on U by
ipjj
^(tr(zi)
+
=
:
Xwj}
WjX1
To define
x, y)
and y
+ tr(zr_2)
and
=
characters
on U, we denote
G Au,
(yij)
GMat3x3.
For a G kx, we
+ x1A + x2,3 + yhl + ay2?)>
ON
Define
THE
OF
NONVANISHING
THE
CENTRAL
717
VALUE
the function
f(g)=
(7.3)
Ei(ug,(P
/
JU(F)\U(A)
{r)?*)^u(u)du
in [GRS03], this is a Fourier coefficient corresponding
As explained
orbit
0((2r)22).
Let ? denote the Weyl element of GL2r+i defined by
to the unipotent
'0 0 V
(7.4)
/?=( (?
07
,1 i),
and let ?o = m(?) =
Let
'
t(b), for b G Ax,
=I
(7.5) t(b)
Then m(t(b))
?
'
-,(?
0;
V1 ?),(0
a
^
)
denote
G Sp4r+2. It is easy to check that f(?og) = /(#)
the torus element
in GL2_1
x GL3 defined
by
,I 1 _i gGLr1xGL3.
(b b_,),---,(b b_^
jJ
is a diagonal
element
in
idea of the proof is to show
Fourier coefficients associated
hold.
following two statements
The
nonzero
(1) The
1 0) IGGL^xGLs
o o>
function
f(g)
defined
Sp4r+2(A).
that
It is clear that
if the residue
Ei(g,(pn^(f)?tr)
orbit
0^2r-\-2)i2r)i
to the unipotent
in (7.3)
is not
=
(2) For all b GAx, f(m(t(b))g)
\b\Af(g).
identically
bas no
then the
zero.
the other hand, statements
Indeed,
(1) and (2) lead to a contradiction.
left
of
and
the
invariance
f(g) by ?o, it is easy to show the identity
(2)
On
=
= f(0om(t(b))g) = f
\b\Af(g) f(m(t(b))g)
{m^1))
Since
for all b G Ax and all g G Sp4r+2(A).
must be identically zero, which contradicts
using
?0g) =^?{g)
b and g are arbitrary, the function f(g)
statement
(1). This proves Proposition
7.1.
It remains
to show that
associated
Ei(g,(pir^{T)?a)
(1) and (2) above.
We define a Weyl
the vanishing of the Fourier coefficients
of the residue
to the unipotent
orbit
implies statements
0((2r+2)i2r)
w =
we can
In term of matrices
(w??) of Sp4r+2.
to the set {0,1,-1}.
choose Weyl
elements of Sp4r+2 to have entries belonging
a Weyl
to determine
in
this convention,
With
element
it is
Sp4r+2 completely,
enough to specify its entries in the first 2r + 1 rows. The Weyl element we need to
= 1 and for
define is as follows. For 1 < i < r we set Wi^i-i = 1. Also wr+i?r+2
=
=
2 < i < r we set uv+?)2r+2?+i
1. Finally we set W2r+i,2r
1- All other entries in
element
the first 2r +1 rows of w are zeroes. One can check that this Weyl element
one that conjugates
the character ipjj in (7.2) to the character ipa stated
we conjugate
In (7.3), since the residue Ei (g,
is automorphic,
^^(f)?^)
element w from the left to the right across the integration and obtain
w is the
in (7.7).
the Weyl
that f(g)
718
DAVID
DIHUA
GINZBURG,
JIANG,
AND
STEPHEN
RALLIS
hr
P
h
I) \ Q P* hr)
Here
v\ G AT2r(fc)\A^2r(A)
(N2r
]Wg,(pnxlj{r)^a)lpa((vi,V2))d(...).
is the standard
maximal
unipotent
of
subgroup
GL2r),t;2GAr2(fc)VV2(A). Let
=
Zi
: Zij = 0 for all i >
j}.
{z E Mat2rx2r
in Z\. The
y and q over matrices
such that x^j = 0 if i > r and also
is denoted by Z3. The variable p is integrated
subgroup
2 x 2r such that pij = 0 if j < r and P2,r+i = 0- We shall
is integrated over the quotient
Z2. Each of these variables
We
integrate variables
over all 2r x 2 matrices
respectively.
Finally,
(7.7)
variable x is integrated
= 0. This matrix
xr_i,i
over all matrices
of size
denote
this subgroup by
?= 1,2,3,
Zi(k)\Zi(A),
we have
=
l?a((vi,V2))
^((Vl)l,2
+
*'
+
+ ?(^2)1,2).
(Vl)2r-l,2r
1 in [GRS99a], we consider
the proof of Theorem
Following
in
for
the
Let
pansions
integral
(7.6).
L=U
where
?\ GMat2rx2
such that
h=
that L is abelian.
Note
[ihr
(??)ij
{ (q ?)
t\ GMat2x2r
t2
We
=
h\
?\\
Fourier
ex
GSp4r+2
at the (r ? 1, l)-th
= 0
except
entry and
:(hkj= 0forall*< j\ .
In (7.6) we expand
the integral
h
|
t\ hr/
T={\h
where
h
h
certain
along L(k)\L(A).
Let
Sp4r+2
= 0
except at the (l,r) position and
(t\)%j
=
: qij
0 for all 1 < i < r - 1; r + 1 < j < 2r}.
such that
{q G Z1
denote
T1xT2:=
| eT
[h h
j(*i,t2) y| \tl t*2 hr)
that 7i x 7^ is a subgroup of Z\ x Z2. Conjugating
from left to
in
summation
the
and
the
with
matrices
right by
collapsing
integration, we
T(k)
deduce that f(g) equals
Then
we know
(7.8)
/ /Vi
I
Eil
J
VV
X y\
V2 x*
(hr
h
\\p
V\) \q
\
(hr
ti
p* hr) \t2
h
\
t\ hr)
\
\wg,(p)iPa((v1,V2))d(...)
)
=
The integrations over v\ and i>2 are the same as in (7.6). The
(p ^^(f)?^are
variables
t\ and t2
integrated over T? (A) x 7^ (A) and p and q are integrated
x T2)(A)\(Zi
x Z2)(A).
over (Z\ x Z2)(k)(Ti
The variable x is integrated over all
where
ON
THE
NONVANISHING
OF
THE
CENTRAL
of size 2r x 2 such that Xij = 0 for i > r +
of the form
matrices
matrices
{u)2ryt
=
>
yw2r
yi,j
= 0if?>r
+
VALUE
719
1 and y is integrated
over all
?
l and j <r
1}.
in x and y are over the quotient
precisely, the integrations
over
the
fc-rational
points
points of the corresponding
algebraic
as described above, respectively.
of all matrices
More
of the A-rational
groups
consisting
In the following we consider the Fourier expansion of the above integral along the
additive subgroup Xa(') attached
to a positive
one-parameter
long root a, which
consists of all matrices
of the form {/4r+2 + cer+i53r+2} where c G k and e?j is
the (Ar + 2) x (Ar + 2) matrix with one at the (i, j)-th entry and zero elsewhere.
The nontrivial
Fourier
is zero.
coefficient
of the function defined in (7.8) along the quotient
the
integrations
along variables v\, x, y and Xa in
Xa(&)\Xa(A)
the integral in (7.8) (against the nontrivial
character ipa(vi)ip(bc) for some b G kx)
attached to the unipotent
yield the Fourier coefficient of the residue Ei (g, ^^(f)?^)
our
orbit
this
Fourier
is zero. This means
coefficient
By
assumption
?((2r+2)i2r)that in the Fourier expansion of (7.8) along Xa(h)\Xa(&)
the
trivial character
onry
Indeed,
(or the constant term) contributes.
this process as in the proof
Continuing
obtain that the function f(g)
is equal to
(7.9)
of Theorem
p h
?E^1lm(v1,V2)?
2
J(Z1xZ2)(A)J/
V
\ ?
P* hr)
1 in [GRS99a], we finally
)wg,AiPa(v1,v2))d(...).
)
Here
radical of the parabolic subgroup P2r,i of
C/2r,i is the unipotent
Sp4r+2 whose
Levi part is GL2r x Sp2 and Ei2r,1(g,
is the constant term of the residue
?^(f)?^)
same
In
the
1 and 2
way as in the proofs of Lemmas
along P2r,iEi(g^(p7r^(T)?a)
on pages 895-896
in [GRS99a], we deduce that if f(g)
is zero for all choices of the
data, then the following integral
(7.10)
/
EU^^\m(vi,v2)g,(p^^r)?cj)^a(vi,V2)dv1dv2
is zero for all choices of the data. In (7.10), the integration along the variable v\ is
the same as taking the Whittaker-Fourier
coefficient of
and the integration
(pn^(r)
coefficient of (pa.
along the variable v2 is the same as taking the Whittaker-Fourier
If we choose a G kx such that a has a nonzero
ipa-Whittaker-Fourier
coefficient,
then the above integral represents a nonzero Whittaker-Fourier
coefficient of the
cusp forms in the generic cuspidal datum (P2r,i, n^?r) <&a). Thus the function f(g)
defined in (7.3) is not identically zero and statement
(1) is proved.
we
To prove statement
first
notice
that
(2),
=
,b~x) G Sp4r+2
is in the center of the Levi subgroup GL2r x GLi. Then we replace g by
m(t(b))g
in integral (7.9) and conjugate
it to the left. By changing variables
in Zi(A)
and
r~
T
we
obtain
the
factor
the
a
fact
that
has
trivial
Z2(A),
Using
|6|^
7r^(f)
central character, we obtain a factor of
from
the
left
quasi-invariance
|fr|A
wm(t(b))w~1
properties
of Ek2r,1(g,
diag(fr,
^^(f)?^)-
f(m(t(b))g)
=
,b, 1,1, b~l,
Thus we obtain
=
|&i:(2r2+2r-1)+2^+1)/(<7) |6|A/(fl).
720
This
DAVID
proves
GINZBURG.
statement
DIHUA
(2) and hence
AND
JIANG,
STEPHEN
the proposition
RALLIS
D
follows.
7.2. (1) The way we proved the nonvanishing
of the function f(g) relies on
the vanishing
of the Fourier coefficient of the residue Ei (p,
assumption
^^(f)?^)
to the unipotent
attached
orbit
However, we point out that the non
?((2r+2)i2r)this assumption.
vanishing of f(g) may be proved without
can
This
7.1
of
be
proof
Proposition
Indeed, if r > I, we can
generalized.
(2)
show that the residue Ei (g,
bas a nonzero Fourier coefficient attached to
^^(t)?^)
the unipotent
if I > 1, this is not enough to verify
orbit
Clearly,
?((2r+2)i2(r+i-1))-
Remark
Assumption
(FC).
Acknowledgment
on
We would
and suggestions
like to thank M. Harris for his useful comments
the earlier version of this paper, and we thank the referee for his/her very helpful
suggestions
and
remarks.
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G(Q).
[A80]
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