REFINED GLOBAL GAN–GROSS–PRASAD CONJECTURE FOR
FOURIER–JACOBI PERIODS ON SYMPLECTIC GROUPS
HANG XUE
Abstract. In this paper, we propose a conjectural identity between the Fourier–Jacobi periods
on symplectic groups and the central value of certain Rankin–Selberg L-functions. This identity
can be viewed as a refinement to the global Gan–Gross–Prasad conjecture for Sp(2n)×Mp(2m).
To support this conjectural identity, we show that when n = m and n = m±1, it can be deduced
from the Ichino–Ikeda’s conjecture in some cases via theta correspondences. As a corollary, the
conjectural identity holds when n = m = 1 or when n = 2, m = 1 and the automorphic
representation on the bigger group is endoscopic.
Contents
1.
Introduction
Part 1.
1
Conjectures
8
2.
Conjectures for the Fourier–Jacobi periods
3.
Convergence and positivity
12
4.
Unramified computations
25
Part 2.
Compatibility with Ichino–Ikeda’s conjecture
8
36
5.
Some assumptions and remarks
36
6.
Ichino–Ikeda’s conjecture for the full orthogonal group
40
7.
Compactibility with Ichino–Ikeda’s conjecture: Sp(2n) × Mp(2n)
46
8.
Compactibility with Ichino–Ikeda’s conjecture: Sp(2n + 2) × Mp(2n)
50
References
62
1. Introduction
In this paper, we propose a conjectural identity between the Fourier–Jacobi periods on symplectic groups and the central value of certain Rankin–Selberg L-functions. This identity can be
viewed as a refinement to the (global) Gan–Gross–Prasad conjecture [7] for Sp(2n) × Mp(2m).
Date: May 13, 2015.
1
The Gan–Gross–Prasad conjecture predicts that the nonvanishing of certain periods is equivalent to the nonvanishing of the central value of certain L-functions. There are two types of
periods: Bessel periods and Fourier–Jacobi periods. Bessel periods are periods of automorphic
forms on orthogonal groups or hermitian unitary groups. A lot of work has been devoted to the
study of Bessel periods, starting from the pioneering work of Waldspurger [43]. In their seminal work [22], based on an extensive study of the known low rank examples, Ichino and Ikeda
proposed a precise formula relating the Bessel periods on SO(n + 1) × SO(n) and the central
value of some Rankin–Selberg L-functions. The analogous formula for Bessel periods on the
hermitian unitary groups U(n + 1) × U(n) has been worked out by N. Harris in his thesis [16] .
W. Zhang [46, 49] then proved a large part of the conjectural formula for U(n + 1) × U(n), using
the relative trace formulae proposed by Jacquet–Rallis [23]. Recently, Liu [34] proposed a conjectural formula for Bessel periods in general, i.e. the Bessel periods on SO(n + 2r + 1) × SO(n)
or U(n + 2r + 1) × U(n). Some low rank cases has also been considered in [34].
There is a parallel theory for the Fourier–Jacobi periods. They are the periods of automorphic
forms on Mp(2n + 2r) × Sp(2n) or U(n + 2r) × U(n). The case of Fourier–Jacobi periods on
U(n) × U(n) has been considered in [44, 45]. We proposed a conjectural formula relating the
Fourier–Jacobi periods on U(n)×U(n) and the central value of some L-functions. We proved this
conjectural formula in some cases, using the relative trace formula proposed by Liu [33]. In the
other extreme case, where one of the groups is trivial, the Fourier–Jacobi periods is simply the
Whittaker–Fourier coefficients. In this situation, Lapid–Mao [27] proposed a formula computing
the norm of the Whittaker–Fourier coefficients. In a series of papers [28–30], they proved the
formula for Whittaker–Fourier coefficients on Mp(2n), under some simplifying conditions at the
archimedean places.
The goal of this paper is to formulate a conjectural identity between the Fourier–Jacobi
periods and the central value of some Rankin–Selberg L-functions for symplectic groups. We
also verify that this conjecture is compatible with Ichino–Ikeda’s conjecture in some cases. As
a corollary, the conjectural identity holds in some low rank cases. We now describe our results
in more detail.
For simplicity, in the introduction, we consider only the Fourier–Jacobi periods on Sp(2n +
2r) × Mp(2n) (r ≥ 0). The case r < 0 will be explained in the main context of the paper. Let F
be a number field and ψ : F \AF → C× be a nontrivial additive character. Let (W2 , q2 ) be the
symplectic space over F with an orthogonal decomposition W0 + R + R∗ where R and R∗ are
isotropic subspaces and R + R∗ is the direct sum of r − 1 hyperbolic planes. We fix a complete
filtration of R and let Nr−1 be the unipotent radical of the parabolic subgroup of G2 fixing the
complete filtration.
2
f0 = Mp(W0 ) (the metaplectic double cover). Let π2
Let G2 = Sp(W2 ), G0 = Sp(W0 ) and G
(resp. π0 ) be an irreducible cuspidal tempered (resp. genuine) automorphic representation of
f0 (AF )). Let ϕ2 ∈ π2 and ϕ0 ∈ π0 . Let H = W0 n F be the Heisenberg group
G2 (AF ) (resp. G
f0 (AF ) which is realized on the
attached to W0 and ωψ be the Weil representation of H(AF ) o G
Schwartz space S(AnF ). Let φ ∈ S(AnF ) be a Schwartz function and θψ (·, φ) be a theta series on
f0 (AF ). Let ψr−1 be an automorphic generic character of Nr−1 (AF ) which is stable
H(AF ) o G
under the conjugation action of H(AF ) o G0 (AF ). The Fourier–Jacobi period of (ϕ2 , ϕ0 , φ) is
the following integral
(1.0.1)
FJ ψ (ϕ2 , ϕ0 , φ)
Z
Z
=
G0 (F )\G0 (AF )
Z
ϕ2 (uhg0 )ϕ0 (g0 )ψr−1 (u)θψ (hg0 , φ)dudhdg0 .
H(F )\H(AF )
Nr−1 (F )\Nr−1 (AF )
This integral is absolutely convergent since ϕ2 and ϕ0 are both cuspidal. It defines an element
in
HomNr−1 (AF )o(H(AF )oG0 (AF )) (π2 ⊗ π0 ⊗ ωψ ⊗ ψr−1 , C).
This space is at most one dimensional [35, 42].
The Gan–Gross–Prasad conjecture predicts [7, Conjecture 26.1] that if the above Hom-space
is not zero, then the integral (1.0.1) does not vanish identically if and only if LSψ ( 12 , π2 × π0 ) is
nonvanishing, where S is a sufficiently large finite set of places of F and LSψ (s, π2 × π0 ) is the
tensor product L-function of π2 and π0 (note that this L-function depends on ψ).
The conjectural identity that we propose is
(1.0.2)
|FJ ψ (ϕ2 , ϕ0 , φ)|2 =
Y
LSψ ( 12 , π2 × π0 )
∆SG2
×
αv (ϕ2,v , ϕ0,v , φv ),
S
|Sπ2 ||Sπ0 | LS (1, π2 , Ad)Lψ (1, π0 , Ad)
v∈S
where
– ϕ2 = ⊗ϕ2,v , ϕ0,v = ⊗ϕ0,v , φ = ⊗φv .
n+r
Q S
– ∆SG2 =
ζF (2i);
i=1
– LSψ (s, π2 × π0 ) is the tensor product L-function and LS (s, π2 , Ad), LSψ (1, π0 , Ad) are
adjoint L-functions;
– αv is a local linear form defined by integration of matrix coefficients (see Section 2.2 for
the definition). It is expected that αv 6= 0 if and only if HomNr−1 (Fv )o(H(Fv )oG0 (Fv )) (π2,v ⊗
π0,v ⊗ ψr−1,v ⊗ ωψv , C) 6= 0.
– dg0 in the definition of FJ ψ is the Tamagawa measure on G0 (AF ), du and dh are the
self-dual measures on Nr−1 (AF ) and H(AF ) respecively;
– Sπ2 and Sπ0 are centralizers of the L-parameters of π2 and π0 respectively. They are
abelian 2-groups (see Section 2.3 for a discussion).
3
This conjectural identity can be viewed as a refinement to the Gan–Gross–Prasad conjecture.
It is motivated by the existing conjectural identities of this type [16, 22, 34, 45]. The conjectural
identity claims that we should expect the same for both the Bessel periods and the Fourier–
Jacobi periods. In the first part of this paper, we show that the conjectural identity (1.0.2) is
well-defined, i.e. the local linear form αv is well-defined and the right hand side of (1.0.2) is
independent of the set S. In the definition of the local linear form αv , we introduce a new way to
regularize a divergent oscillating integral over a unipotent group. This gives the same results as
the existing regularizations [27, 34], but has the advantage of being elementary, purely function
theoretic and uniform for both archimedean and non-archimedean places.
One might be asking what happens for the Fourier–Jacobi periods on skew-hermitian unitary
groups. An identity similar to 1.0.2 should also hold. We exclude that in the present paper
for two reasons. First, sticking to the symplectic groups greatly simplifies the notation. More
importantly, in showing that the right hand side of (1.0.2) is independent of S, we make use of
some results in [13]. The analogue results for unitary groups have not appeared in print yet.
D. Jiang has informed the author that X. Shen and L. Zhang are working on a more general version of the results in [13], which should cover Fourier–Jacobi periods for both symplectic groups
and skew-hermitian unitary groups. Once such results are available, one can then formulate the
refined Gan–Gross–Prasad conjecture in the context of skew-hermitian unitary groups.
To support our conjecture, in the second part of this paper, we show, under some hypothesis
on the local and global Langlands correspondences which we will state in Section 5, that our
conjecture is compactible with Ichino–Ikeda’s conjecture in some cases. Thus (1.0.2) holds in
some low rank cases when the Ichino–Ikeda’s conjecture is known. We have the following cases.
(1) If n = 1 and r = 0, then (1.0.2) has been proved in [37, Theorem 4.5].
(2) If r = 0 and π2 is a theta lift of some irreducible cuspidal tempered automorphic representation of O(2n), then (1.0.2) can be deduced from Ichino–Ikeda’s conjecture for
SO(2n + 1) × SO(2n). In this case, if π0 is not a theta lift from any O(2n + 1), then both
sides of (1.0.2) vanish.
(3) If r = 1 and π2 is a theta lift of some irreducible cuspidal tempered automorphic representation of O(2n + 2), then (1.0.2) can be deduced from Ichino–Ikeda’s conjecture for
SO(2n + 2) × SO(2n + 1). In this case, if π0 is not a theta lift from O(2n + 1), then both
sides of (1.0.2) vanish. In particular, when n = 1, (1.0.2) holds for Sp(4) × Mp(2), if the
automorphic representation on Sp(4) is a theta lift from O(4).
See Theorem 7.1.1 and 8.1.1 for the precise statments. See also Theorem 8.6.1 for an analogous
statement in the case r = −1. In the course of proving these results, we derive a variant for
4
the Ichino–Ikeda’s conjecture for the full orthogonal group, c.f. Conjecture 6.3.1 and Proposition 6.3.2. I hope that this variant is of some independent interest. See [9] for the case of the
triple product formula on GO(4).
Ichino informed the author that there is some minor inaccuracies in the original formulation
of Ichino–Ikeda’s conjecture [22, Conjecture 2.1] when the automorphic representation on the
even orthogonal group appears with multiplicity two in the discrete automorphic spectrum. In
this case, one needs to specify an automorphic realization. Moreover, the size of the centralizer
of the Arthur parameter needs to be replaced by half of it.
It is expected that our conjecture is compatible with the refined Gan–Gross–Prasad conjecture
for SO(2n + 2r + 1) × SO(2n) proposed by Liu [34]. To keep this paper within a reasonable
length, we postpone to check this more general compatibility in a future paper.
This paper is organized as follows. The first part of the paper consists of Sections 2, 3 and 4.
In Section 2, we first define the Fourier–Jacobi periods and the local linear form αv . Then
we state the conjectural formula for the Fourier–Jacobi periods. In Section 3, we show that
the local linear form αv is well-defined, i.e. its defining integral is absolutely convergent. We
also prove a positivity result for αv . In Section 4, we compute αv when all the data involved
are unramified. The argument is mostly adapted from [34]. The second part of this paper
consists of Sections 5, 6, 7 and 8. In Section 5, we state some working hypotheses on the local
and global Langlands correspondences and make some remarks on the theta correspondences.
For orthogonal groups and symplectic groups, these hypotheses should follow from the work
of Arthur [2]. For metaplectic groups, they should eventually follow from the on-going work
of Wen-Wei Li (e.g. [32]). In section 6, we review the Ichino–Ikeda’s conjecture and derive a
variant of it for the full orthogonal group. In Section 7, we study the conjecture in the case
Mp(2n) × Sp(2n) via a seesaw argument. This type of argument has also been used in [4, 10, 45].
In Section 8, we study the conjecture in the case Sp(2n + 2) × Mp(2n). This section contains
some of the main innovation of this paper. For the convenience of the readers, we remark that
Section 3, 4 and the second part of the paper are logically independent. Section 7 and 8 are also
logically independent. They can be read in any order.
Notation. The following notation will be used throughout this paper. Let F be a number
field, oF the ring of integers and AF the ring of adeles. For any finite place v, let oF,v be the
ring of integers of Fv and $v a uniformizer. Let qv = |oF,v /$v | be the number of elements in the
residue field of v. We fix a nontrivial additive character ψ = ⊗ψv : F \AF → C× . We assume
that ψ is unitary, thus ψ −1 = ψ. For any a ∈ F × , we define an additive character ψa of F \AF
by ψa (x) = ψ(ax). Let (·, ·)F be the Hilbert symbol of F and γψ the Weil index, which is an
eighth root of unity. We put χψ (a) = γ(ψa )/γ(ψ). Then χψ (a)2 = (a, −1)F .
5
Suppose that V is a vector space and v1 , · · · , vr ∈ V . Then we denote by hv1 , · · · , vr i the
subspace of V generated by v1 , · · · , vr . We write S(V ) for the space of Schwartz functions on
V.
Let (V, qV ) be a quadratic space of dimension n over F where V is the underline vector
space and qV is the quadratic form. We can choose a basis of V so that its quadratic form is
represented by a diagonal matrix with entries a1 , · · · , an . We define the discriminant disc V of
V by
disc V = (−1)
n(n−1)
2
a1 · · · an ∈ F × /F ×,2 .
Define a quadratic character χV : F × \A×
F → {±1} by χV (x) = (x, disc V )F .
Let (W, qW ) be a symplectic space of dimension 2n over F where W is the underline vector
space and qW is the symplectic form. Then we denote by Sp(W ) or Sp(2n) the symplectic group
attached to W and Mp(W ) or Mp(2n) the metaplectic double cover. By definition, if v is a
place of F , then Mp(W )(Fv ) = Sp(W )(Fv ) n {±1} and the multiplication is given by
(g1 , 1 )(g2 , 2 ) = (g1 g2 , 1 2 c(g1 , g2 )),
where c(g1 , g2 ) is some 2-cocycle on Sp(W ) valued in {±1} [39]. Moreover
Y
Y
0
Mp(W )(AF ) =
Mp(W )(Fv )/{(1, v )v |
v = 1}.
v
v
If g ∈ Sp(W )(AF ) (resp. Sp(W )(Fv )), then we define ι(g) = (g, 1) ∈ Mp(W )(AF ) (resp.
Mp(Fv )). Note that g 7→ ι(g) is not a group homomorphism.
By a genuine function on Mp(W )(Fv ), we mean a function on Mp(W )(Fv ) which is not the
pullback of a function on Sp(W )(Fv ). We always identify a function on Sp(W )(F ) with a nongenuine function on Mp(W )(F ). Suppose that f1 , · · · , fr are genuine functions on Mp(W )(Fv )
and h1 , · · · hs are functions on Sp(W )(Fv ) such that the product f1 · · · fr is not genuine. Then
we write
Z
Z
f1 (g) · · · fr (g)h1 (g) · · · hs (g)dg =
Sp(W )(Fv )
f1 (ι(g)) · · · fr (ι(g))h1 (g) · · · hs (g)dg.
Sp(W )(Fv )
An irreducible representation of Mp(W )(Fv ) is said to be genuine if the element (1, ) acts by
. We always identify an irreducible representation of Sp(W )(Fv ) with a non-genuine representation of Mp(W )(Fv ). We make similar definitions for genuine functions and representations of
Mp(W )(AF ).
Suppose v is a non-archimedean place of F whose residue characteristic is not two. Suppose
that the conductor of ψv is oF,v . By an unramified principal series representation of Mp(2n)(Fv ),
Mp(2n)(F )
v
we mean the induced representation I(χ) = Ind e
γψv χ, where B = T U is a Borel
B
e
e
subgroup of Sp(2n), B = T U is the inverse image of B in Mp(2n)(Fv ) and χ is a character
6
of T ' Fvn defined by χ(t1 , · · · , tn ) = |t1 |α1 · · · |tn |αn , α1 , · · · , αn ∈ C and γψ χ is a genuine
character of Te defined by
γψv χ(t1 , · · · , tn ; ) = γψv (t1 · · · tn )χ(t1 , · · · , tn ).
The convention of parabolic inductions of the metaplectic group is as in [12]. If πv is an unramified representation of Mp(2n)(Fv ), then we can find an unramified character χ of T as above
and πv ⊂ I(χ). The complex numbers (α1 , · · · , αn ) are called the Satake parameters of πv . Note
that the Satake parameters of πv depend also on ψv .
We write 1r for the r×r identity matrix. We recursively define w1 = {1} and wr =
wr−1
!
.
1
Suppose a = (a1 , · · · , ar ) ∈ (F × )r . We let diag[a1 , · · · , ar ] be the diagonal matrix with diagonal
entries a1 , · · · , ar .
Suppose that G is a unimodular locally compact topological group and dg a Haar measure.
Suppose that π is a representation of G, realized on some space V . Let f be a continuous
function on G. Then we put (whenever it makes sense, e.g. f is compactly supported and
locally constant)
Z
π(f )v =
f (g)π(g).vdg.
G
Let G be a connected reductive group over F . Let M be the motive associated to G as in [15]
and M ∨ its dual. Let v be a place of F , then we denote by ∆G,v = Lv (1, M ∨ ) and for any finite
Q
set of places S of F , we define ∆SG,v = v6∈S ∆G,v . If G = Mp(2n), we define ∆SG = ∆SSp(2n) .
If T is a maximal torus in Sp(2n) and Te is the inverse image of T in Mp(2n), then we define
∆Te = ∆T . If G = O(V ), then we define ∆SG = ∆SSO(V ) . More concretely, if G = Sp(2n), then
∆SG =
n
Y
ζFS (2i).
i=1
If G = SO(V ) for some n dimensional orthogonal space V , then
∆SG =
 S
 ζF (2)ζFS (4) · · · ζFS (n − 1),
if n is odd
 ζ S (2)ζ S (4) · · · ζ S (n − 2)LS ( n , χV ), if n is even,
F
F
F
2
Acknowledgement. I thank Atsushi Ichino for pointing to me the inaccuracy in [22] and some
inaccuracies in an early draft of this paper. I thank Raphael Beuzart-Plessis, Wee-Teck Gan,
Yifeng Liu, Wei Zhang and Shou-Wu Zhang for many helpful discussions. This material is based
upon work supported by the National Science Foundation under agreement No. DMS-1128115.
7
Part 1. Conjectures
2. Conjectures for the Fourier–Jacobi periods
2.1. Global Fourier–Jacobi periods. Let (W2 , q2 ) be a 2m dimensional symplectic space over
F . We choose a basis {e∗m , · · · , e∗1 , e1 , · · · , em } of W2 so that q2 (e∗i , ej ) = δij . For 1 ≤ i ≤ m,
let Ri = hem−i+1 , · · · , em i and Ri∗ = he∗m , · · · , e∗m−i+1 i be isotropic subspaces of W2 . Put
R0 = R0∗ = {0}. Let 0 ≤ r ≤ m be an integer and put n = m − r and (W0 , q0 ) the orthogonal
complement of Rr + Rr∗ . We define (W1 , q1 ) = W0 + hen+1 , e∗n+1 i. Let Gi = Sp(Wi ) and
fi = Mp(Wi ).
G
Let 0 ≤ i ≤ n be an integer. Let Pi be the parabolic subgroup of G2 stabilizing the flag
0 = R0 ⊂ R1 ⊂ · · · ⊂ Ri ,
with the Levi decomposition Pi = Mi Ni . Here and below in this article, the notation P = M N
signifies that M is the Levi subgroup and N is the unipotent radical of P . We denote by W i
the orthogonal complement of Ri + Ri∗ and Gi = Sp(W i ). Then Mi = Gi × GLi1 . Let ψm be the
character of Nm defined by

m−1
X
ψi (n) = ψ 

q2 (ne∗m−j+1 , em−j ) + q2 (ne∗1 , e∗1 ) .
j=1
Let ψi be the restriction of ψm to Ni .
Let H = H(W0 ) be the Heisenberg group attached to the symplectic space W0 . By definition,
H = W0 n F and the group law is given by
(w1 , t1 )(w2 , t2 ) = (w1 + w2 , t1 + t2 + q0 (w1 , w2 )).
Let L = he1 , · · · , en i and L∗ = he∗n , · · · , e∗1 i. Then W0 = L + L∗ is a complete polarization. We
sometimes write an element h ∈ H as h(l + l∗ , t) where l ∈ L, l∗ ∈ L∗ and t ∈ F . Let v be a
place of F and ωψv be the Weil representation of H(Fv ) which is realized on S(L∗ (Fv )). It is
defined by
ωψv (h(y + x, t))f (l∗ ) = ψ(t + q2 (2x + l∗ , y))f (l∗ + x), f ∈ S(L∗ (Fv )), l∗ , x ∈ L∗ (Fv ), y ∈ L(Fv ).
This is the unique irreducible infinite dimensional representation of H(Fv ) whose central charf0 (Fv ) on S(L∗ (Fv )). We denote the joint action of
acter is ψv . It induces an action of G
f0 (Fv ) on S(L∗ (Fv )) again by ωψ . We take the convention that if W0 = {0}, then
H(Fv ) o G
v
ωψv = ψv .
Taking restricted tensor product of the Weil representations ωψv , we obtain a global Weil
f0 (AF ) which is realized on S(L∗ (AF )). We define the theta
representation ωψ of H(AF ) o G
8
series
θψ (hg0 , φ) =
X
ωψ (hg0 )φ(l∗ ),
f0 (AF ).
φ ∈ S(L∗ (AF )), h ∈ H(AF ), g0 ∈ G
l∗ ∈L∗ (F )
We now talk about automorphic representations. There are two cases.
Case Mp: Let π2 = ⊗π2,v be an irreducible cuspidal genuine automorphic representation of
f2 (AF ) and π0 = ⊗π0,v be an irreducible cuspidal automorphic representation of G0 (AF ).
G
Case Sp: Let π2 = ⊗π2,v be an irreducible cuspidal automorphic representation of G2 (AF )
f0 (AF ).
and π0 = ⊗π0,v be an irreducible cuspidal genuine automorphic representation of G
Let S be a sufficiently large finite set of places of F containing all archimedean places and
finite places whose residue characteristic is two, such that π2,v and π0,v are both unramified and
the conductor of ψv is oF,v if v 6∈ S. Let (α1,v , · · · , αm,v ) and (β1,v , · · · , βn,v ) be the Satake
parameters of π2,v and π0,v respectively. Put
A2 =

−1
−1
 diag[α1,v , · · · , αm,v , αm,v
, · · · , α1,v
]
Case Mp,
 diag[α1,v , · · · , αm,v , 1, α−1 , · · · , α−1 ] Case Sp,
m,v
1,v
and
A0 =

−1
−1
 diag[β1,v , · · · , βn,v , 1, βn,v
] Case Mp,
, · · · , β1,v
 diag[β1,v , · · · , βn,v , β −1 , · · · , β −1 ]
n,v
1,v
Case Sp.
We then define the tensor product L-function
Lψv (s, π2,v × π0,v ) = det(1 − A2 ⊗ A0 · qv−s )−1 ,
LSψ (s, π2 × π0 ) =
Y
Lψv (s, π2,v × π0,v ).
v6∈S
The partial L-function is convergent for <s 0. We denote by Lψv (s, πi,v , Ad) and LSψ (s, πi , Ad) =
Q
v6∈S Lψv (s, πi,v , Ad) the (local and partial) adjoint L-functions of πi . If πi is an automorphic
representation of the metaplectic group (resp. symplectic group), then they depend (resp. do
not depend) on ψ. We include the subscript ψ in both cases to unify notation. We assume that
these L-functions can be meromorphically continued to the whole complex plane.
Let ϕ2 ∈ π2 , ϕ0 ∈ π0 and φ ∈ S(L∗ (AF )). Define
FJ ψ (ϕ2 , ϕ0 , φ)
Z
Z
=
G0 (F )\G0 (AF )
H(F )\H(AF )
Z
ϕ2 (uhg0 )ϕ0 (g0 )ψr−1 (u)θψ (hg0 , φ)dudhdg0 .
Nr−1 (F )\Nr−1 (AF )
The measures du and dh are the self-dual measures on Nr−1 (AF ) and H(AF ) respectively. The
measure dg0 is the Tamagawa measures on G0 (AF ).
9
2.2. Local Fourier–Jacobi periods. We fix a Haar measure dg0,v on G0 (Fv ) for each v such
R
Q
that G0 (oF,v ) dg0,v = 1 for almost all v. Then there is a constant C0 such that dg0 = C0 v dg0,v .
Following [22], we call C0 the measure constant.
Let Bπi (i = 0, 2) be the canonical bilinear pairing between πi and πi∨ defined by
Z
∨
ϕ(g)ϕ∨ (g)dg, ϕ ∈ πi , ϕ∨ ∈ πi∨ .
Bπ2 (ϕ, ϕ ) =
G2 (F )\G2 (AF )
∨ for each place v such that B
We fix a bilinear pairing Bπi,v between πi,v and πi,v
πi =
Put Φϕi,v ,ϕ∨i,v (g) =
Bπi,v (πi,v (g)ϕi,v , ϕ∨
i,v )
if ϕi,v ∈ πi,v and
ϕ∨
i,v
∈
Q
v
Bπi,v .
∨ .
πi,v
The contragredient representation of ωψ is ωψ−1 (again realized on S(L∗ (AF )) and there is a
canonical pairing between ωψ and ωψ−1 given by
Z
Bωψ (φ, φ∨ ) =
φ(l∗ )φ∨ (l∗ )dl∗ ,
L∗ (AF )
φ, φ∨ ∈ S(L∗ (AF )),
where the measure dl∗ is the self-dual measure on L∗ (AF ). Similarly, for any place v, there is a
canonical pairing between ωψv and ωψv−1 given by
Z
∨
∗
∗
Bωψv (φv , φv ) =
φv (l∗ )φ∨
v (l )dl ,
L∗ (Fv )
∗
φv , φ∨
v ∈ S(L (Fv )),
where the measure dl∗ is the self-dual measure on L∗ (Fv ). Then Bωψ =
Q
v
Bωψv . Put Φφv ,φ∨v (g) =
Bωψv (ωψv (g)φv , φ∨
v ).
We now fix a place v of F . Recall that the group Pm of G2 is a minimal parabolic subgroup
which is contained in Pr−1 . For any real number γ or γ = −∞, define
Nm,γ = {u ∈ Nm (Fv ) | |q2 (ue∗1 , e∗1 )| ≤ eγ , |q2 (ue∗i+1 , ei )| ≤ eγ , 1 ≤ i ≤ m − 1}.
For any γ ≥ −∞, we define Ni,γ = Ni (Fv ) ∩ Nm,γ . Define
Z
Fψv Φϕ2,v ,ϕ∨2,v (hg0 ) = lim
Φϕ2,v ,ϕ∨2,v (hg0 u)ψr−1,v (u)du,
γ→∞ N
r−1,γ (Fv )
∨
ϕ2,v ∈ π2,v , ϕ∨
2,v ∈ π2,v
f0 (Fv ) in the case Mp). Define
where h ∈ H(Fv ) and g0 ∈ G0 (Fv ) in the case Sp (resp. g0 ∈ G
Z
Z
∨
∨
αv (ϕ2,v , ϕ∨
,
ϕ
,
ϕ
,
φ
,
φ
)
=
Fψ Φϕ2,v ,ϕ∨2,v (hg0 )Φϕ0,v ,ϕ∨0,v (g0 )Φφv ,φ∨v (hg0 )dhdg0 ,
0,v
v
2,v
0,v
v
G0 (Fv )
H(Fv )
∨
∨
∗
for ϕi,v ∈ πi,v , ϕ∨
i,v ∈ πi,v , φv , φv ∈ S(L (Fv )). If r ≤ 1, then it is to be understood that
Fψ Φϕ2,v ,ϕ∨2,v = Φϕ2,v ,ϕ∨2,v . Moreover, if r = 0, then it is to be understood that the integral over
H(Fv ) is void.
Proposition 2.2.1. Assume that π2,v and π0,v are both tempered. Then the limit in the definition of Fψv Φϕ2,v ,ϕ∨2,v exists. Moreover, the defining integral of αv is absolutely convergent.
10
∨ with π . We then define
If πi,v is unitary, then we may identify πi,v
i,v
αv (ϕ2,v , ϕ0,v , φv ) = αv (ϕ2,v , ϕ2,v , ϕ0,v , ϕ0,v , φv , φv ).
Proposition 2.2.2. Assume that π2,v and π0,v are unitary and tempered. Then αv (ϕ2,v , ϕ0,v , φv ) ≥
0 for all smooth vectors ϕ2,v ∈ π2,v , ϕ0,v ∈ π0,v and φv ∈ S(L∗ (Fv )).
These two propositions will be proved in Section 3.
We now consider the unramified situation. Note first that the symplectic spaces Wi ’s, the
isotropic subspaces Ri ’s and hence the groups Gi ’s are naturally defined over oF . Let S be
a sufficiently large finite set of places of F containing all archimedean places and finite places
whose residue characteristic is two, such that if v 6∈ S, then the following conditions hold.
(1) The conductor of ψv is oF,v .
(2) φv = φ∨
v = 1L∗ (oF,v )
∨
(3) For i = 0, 2, ϕi,v and ϕ∨
i,v are fixed by Gi (oF,v ) and satisfies Bπi,v (ϕi,v , ϕi,v ) = 1. In
∨ are unramified.
particular, the representations πi,v and πi,v
R
(4) G0 (oF,v ) dg0,v = 1.
Proposition 2.2.3. If v 6∈ S and the defining integral of αv is convergent, then
∨
∨
αv (ϕ2,v , ϕ∨
2,v , ϕ0,v , ϕ0,v , φv , φv ) = ∆G2 ,v
Lψv ( 21 , π2,v × π0,v )
.
Lψv (1, π0,v , Ad)Lψv (1, π2,v , Ad)
We will prove this proposition in Section 4. Note that in this proposition, we do not assume
that the representations π2,v and π0,v are tempered.
2.3. Conjectures. Following [22] and [34], we say that the representations π2 and π0 are almost
locally generic if for almost all places v of F , the local components π2,v and π0,v are generic.
Suppose that we are in the case of Mp. As explained in [22], the automorphic representations
π2 and π0 should come from some elliptic Arthur parameters
c
f2 = Sp(2m, C),
Ψ2 : LF × SL2 (C) → G
c0 = SO(2n + 1, C)
Ψ0 : LF × SL2 (C) → G
where LF is the (hypothetical) Langlands group of F . If πi is tempered, then Ψi is trivial on
SL2 (C). It is believed (Ramanujan conjecture) that almost locally generic representations are
c
f2 (resp. G
c0 ).
tempered. We define Sπ (resp. Sπ ) to be the centralizer of the image of Ψ2 in G
2
0
They are finite abelian 2-groups. In the case Sp, we have the same discussion, except that we
f2 by G2 and replace G0 by G
f0 .
replace G
Conjecture 2.3.1. Assume that π2 and π0 are irreducible cuspidal automorphic representations
that are almost locally generic. Then the following statements hold.
11
∨
∨
(1) The defining integral of αv (ϕ2,v , ϕ∨
2,v , ϕ0,v , ϕ0,v , φv , φv ) is convergent for any Ki -finite
∨
vectors ϕi,v , ϕ∨
i,v and K0 -finite Schwartz functions φv , φv , where Ki is a maximal compact
subgroup of Gi (Fv ), i = 0, 2;
(2) αv (ϕ2,v , ϕ0,v , φv ) ≥ 0 for any Ki -finite vectors ϕi,v and K0 -finite Schwartz function φv .
Moreover, αv (ϕ2,v , ϕ0,v , φv ) = 0 for all Ki -finite ϕi,v and K0 -finite φv precisely when
HomNr−1 (Fv )o(H(Fv )oG0 (Fv )) (π2,v ⊗ π0,v ⊗ ψr−1,v ⊗ ωψv , C) = 0;
(3) Assume that ϕi = ⊗v ϕi,v ∈ πi (i = 0, 2) and φ = ⊗v φv ∈ S(L∗ (AF )) are factorizable,
then
(2.3.1)
|FJ ψ (ϕ2 , ϕ0 , φ)|2 =
Y
LSψ (s, π2 × π0 )
C0 ∆SG2
αv (ϕ2,v , ϕ0,v , φv ).
1×
1
1
S
S
|Sπ2 ||Sπ0 | Lψ (s + 2 , π2 , Ad)Lψ (s + 2 , π0 , Ad) s= 2
v∈S
Remark 2.3.2. It follows from the Proposition 2.2.3 that the right hand side of (2.3.1) does not
depend on the finite set S.
Remark 2.3.3. Assume that π2 and π0 are both tempered. It is then believed that LSψ (s, π2 × π0 )
and LSψ (s, πi , Ad) should be holomorphic for <s > 0. Moreover, LSψ (1, πi , Ad) 6= 0.
Remark 2.3.4. Without the assumption of almost local genericity of π2 and π0 , we expect that
local linear forms αv can be “analytically continued” in some way so that it is defined for all
representations π2,v and π0,v . This is indeed the case if v 6∈ S. Thus αv is well-defined for all v
if π2 and π0 satisfy the property that π2,v and π0,v are both tempered if v ∈ S. Moreover, we
expect that the identity 2.3.1 holds with the quantity |Sπ2 ||Sπ0 | replaced by some 2−β where β
is an integer. The nature of β, however, remains mysterious at this moment.
3. Convergence and positivity
For the rest of Part I of this paper, we fix a place v of F and suppress it from all notation.
Thus F is a local field of characterstic zero. To shorten notation, for any algebraic group G or
G = Mp(2n) over F , we denote by G instead of G(F ) for its group of F -points. We have fixed a
basis {e∗m , · · · , e∗1 , e1 , · · · , em } of W2 . We thus realize the group G2 and its various subgroups as
groups of matrices. We also identify Wi , L, L∗ as spaces of row vectors. We put Ki = Gi (oF ).
This is a maximal compact subgroup of Gi . The group Pm consists of upper triangular matrices.
The group Pm ∩ Gi is a minimal parabolic subgroup of Gi .
Suppose a = (a1 , · · · , an ) ∈ (F × )n . Then we let d(a) ∈ G0 so that d(a)e∗i = ai e∗i for any
1 ≤ i ≤ n. We also put a = diag[an , · · · , a1 ] ∈ GLn .
12
3.1. Preliminaries. We recall some basic estimates in this subsection. We follow [22, Section 4]
rather closely.
Let G be a reductive group over F . Let AG be a maximal split subtorus of G, M0 the
centralizer of AG in G. We fix a minimal parabolic subgroup P0 of G with the Levi decomposition
P0 = M0 N0 . Let ∆ be the set of simple roots of (P0 , AG ). Let δP0 be the modulus character of
P0 . Let
A+
G = {a ∈ A0 | |α(a)| ≤ 1 for all α ∈ ∆}.
We fix a special maximal compact subgroup K of G. Then we have a Cartan decomposition
G = KA+
G K. We also have the Iwasawa decomposition
G = M0 N0 K,
g = m0 (g)n0 (g)k0 (g).
Let f and f 0 be two nonnegative functions on G. We say that f f 0 if there is a constant
C such that f (g) ≤ Cf 0 (g) for all g ∈ G. We say that f ∼ f 0 if f f 0 and f 0 f . In this case
we say that f and f 0 are equivalent.
For any function f ∈ L1 (G),
Z
Z
(3.1.1)
f (g)dg =
ZZ
ν(m)
A+
G
G
f (k1 mk2 )dk1 dk2 dm,
K×K
where ν(m) is a positive function on A+
G such that
ν(m) ∼ δP0 (m)−1 .
(3.1.2)
1
Let 1 be the trivial representation of M0 and let e(g) = δP0 (m0 (g)) 2 be an element in IndG
P0 1.
Let dk be the measure on K such that vol K = 1. We define the Harish–Chandra function
Z
Z
1
e(kg)dk =
δP0 (m0 (kg)) 2 dk.
Ξ(g) =
K
K
This function is bi-K-invariant. This function depends on the choice of K. However, different
choices of K give equivalent functions on G. So this choice will not affect our estimates.
We define a height function on G. We fix an embedding τ : G → GLn . Write τ (g) = (aij )
and τ (g −1 ) = (bij ). Define
(3.1.3)
ς(g) = sup{1, log|aij |, log|bij | | 1 ≤ i, j ≤ n}.
There is a positive real number d such that
(3.1.4)
1
1
δ0 (a) 2 Ξ(a) δ0 (a) 2 ς(a)d ,
a ∈ A+
0.
Now let π be an irreducible admissible tempered representation of G. Let Φ be a smooth
matrix coefficient of G. Then there is a constant B such that
(3.1.5)
|Φ(g)| Ξ(g)ς(g)B .
13
This is classical and is called the weak inequality when Φ is K-finite and due to [41] when Φ is
smooth.
We finally assume that G = Mp(2n). This is not an algebraic group, but it behaves in
many ways like an algebraic group. In particular, we have a Cartan decomposition for G, i.e.
G = KA+
G K where K is the inverse image of a special maximal compact subgroup of Sp(2n)
(e.g. Sp(2n)(oF ) if F is nonarchimedean and U(n) is F is archimedean) and A+
G is the inverse
image of A+
Sp(2n) in G. We define ΞG = ΞSp(2n) ◦ p where p : G → Sp(2n) is the canonical
projection. Then the weak inequality holds for tempered representations of G.
3.2. Some estimates.
Lemma 3.2.1. Fix a real number D. Then there exists some β > 0, such that
Z
ΞGi (um)ς(u)D du γ β ς(m)β ΞGi+1 (m), m ∈ Gi+1 .
Ni+1,γ ∩Gi
Proof. We fix some real number b to be determined later. We denote the left hand side of the
inequality by I. Then, I = I<b + I≥b with
Z
I<b =
1ς<b (u)ΞGi (um)ς(u)D du
Ni+1,γ ∩Gi
Z
I≥b =
Ni+1,γ ∩Gi
1ς≥b (u)ΞGi (um)ς(u)D du,
where 1ς<b is the characteristic function of {u ∈ Ni+1 ∩ Gi | ς(u) < b} and 1ς≥b is the characteristic function of {u ∈ Ni+1 ∩ Gi | ς(u) ≥ b}.
We have
d
Z
I<b b
Ni+1,γ
∩Gi
1ς<b (u)ΞGi (um)ς(u)D−d du
d
b ς(m)β1 ΞGi+1 (m),
where β1 is a positive real number and d is a positive real number so that the integral
Z
ΞGi (um)ς(u)D−d du
Ni+1 ∩Gi
is convergent.
Let λ : Ni+1 ∩ Gi → F be a character defined by λ(n) = q2 (ne∗m−i , em−i−1 ). Then by [5,
Corollary B.3.1], there is an > 0, such that the integral
Z
ΞGi (u)eς(u) ς(u)D (1 + |λ(u)|)−1 du
Ni+1 ∩Gi
14
is convergent. We have ΞGi (um) eας(m) ΞGi (u) for some α > 0. It follows that
Z
ας(m)
1ς≥b (u)ΞGi (u)ς(u)D eς(u) (1 + |λ(u)|)−1 e−ς(u) (1 + |λ(u)|)du
I≥b e
Ni+1,γ ∩Gi
ας(m)−b
e
γ
Z
(1 + e )
Ni+1,γ
eας(m)−b (1 + eγ )
Z
Ni+1
ας(m)−b
e
∩Gi
∩Gi
1ς≥b (u)ΞGi (u)ς(u)D eς(u) (1 + |λ(u)|)−1 du
ΞGi (u)ς(u)D eς(u) (1 + |λ(u)|)−1 du
γ
(1 + e ).
There is a constant c > 0, such that ΞGi+1 (m) e−cς(m) , then we have
I bd ΞGi+1 (m)ς(m)β1 + e(α+c)ς(m)−b (1 + eγ )ΞGi+1 (m).
We may thus choose b = −1 (log(1 + eγ ) + (α + c)ς(m)) and get
I (−d ς(m)β1 (γ + (α + c)ς(m))d + 1)ΞGi+1 (m).
Note that α, β1 , d and c are constants which are independent of γ or m. We therefore conclude
that there is some β > 0, such that
I γ β ς(m)β ΞGi+1 (m).
This proves the lemma.
Lemma 3.2.2. Fix a real number D. Then
Z
ΞGi (um)ς(u)D du γ β ς(m)β ΞGi+1 (m),
m ∈ Gi+1 .
Ni+1,−∞ ∩Gi
Proof. Choose a subgroup N † of Ni+1 ∩Gi so that the multiplication map N † ×(Ni+1,−∞ ∩Gi ) →
Ni+1 ∩ Gi is an isomorphism. It follows from the Diximier–Milliavin theorem [6] that ΞGi is a
finite linear combination of the functions of the form
Z
g 7→
f (n)Φ(ng)dn,
N†
where f (n) is a compactly supported function on N † and Φ is a smooth matrix coefficient of a
tempered representation of Gi . The lemma then follows from Lemma 3.2.1.
Lemma 3.2.3. Let f be a nonnegative function on L∗ such that p(x)f (x) is bounded for any
polynomial function on L∗ (e.g. f is compactly supported). Let p : H oG0 → L∗ be the projection
given by
hg0 7→
n
X
q2 (hg0 e∗n+1 , ei )e∗i .
i=1
15
Then there is a real number B such that
Z
ΞG1 (hg0 )f (p(hg0 ))dh ΞG0 (g0 )ς(g0 )B ,
g0 ∈ G0 .
H
Proof. By the Cartan decomposition of G0 , we may assume that g = d(a) where a = (a1 , · · · , an ) ∈
(F × )n , |an | ≤ · · · ≤ |a1 | ≤ 1. Then p(h(l + l∗ , t)d(a)) = l∗ a where a = diag[an , · · · , a1 ].
We fix some γ which will be detemined later. Let Hγ = H ∩ Nm,γ and H γ be the complement
of Hγ in H. Then
Z
Z
∗
ΞG1 (hd(a))f (l a)dh =
H
Z
∗
ΞG1 (hd(a))f (l a)dh +
Hγ
Hγ
ΞG1 (hd(a))f (l∗ a)dh.
By Lemma 3.2.1, the first integral is bounded by
γ β ΞG0 (d(a))ς(d(a))B .
Write l∗ = (l1∗ , · · · , ln∗ ) and l0∗ = (0, l2∗ , · · · , ln∗ ) ∈ F n . Then
Z
Z
∗
ΞG1 (h(l + l0∗ , t)d(a)h(l1∗ an , 0, · · · , 0))f (l∗ a)dh.
ΞG1 (hg0 )f (l a)dh =
Hγ
Hγ
There is some positive constant α such that
∗
ΞG1 (h(l + l0∗ , t)d(a)h(l1∗ an , 0)) ΞG1 (h(l + l0∗ , t)d(a))eα log max{|l1 an |,1} .
Therefore
Z
Z
ΞG1 (hd(a))f (l∗ a)dh Hγ
Z
ΞG1 (hd(a))dh ×
H−∞
|l1∗ an |≥eγ
∗
eα log max{|l1 an |,1} f1 (l1∗ an )dl1∗ ,
where f1 is a function on F such that f1 (x)p(x) is bounded for any polynomial function p on
F . It follows from Lemma 3.2.2 that there is a positive real number D such that
Z
ΞG1 (hd(a))dh ΞG0 (d(a))ς(d(a))D .
H−∞
Since f1 (x)p(x) is bounded for any polynomial function p on F , we have
Z
∗
eα log max{|l1 an |,1} f1 (l1∗ an )dl1∗ |an |−1 e−γ ,
|l1∗ an |≥eγ
where the implicit constant in does not depend on an or γ. We may choose γ with γ >
− log|an |. Then
Z
∗
|l1∗ an |≥eγ
eα log max{|l1 an |,1} f1 (l1∗ an )dl1∗ 1.
Therefore
Z
Hγ
ΞG1 (hd(a))f (la)dh ΞG0 (d(a))ς(d(a))D .
The desired estimate then follows.
16
Lemma 3.2.4. Let Φ be a smooth matrix coefficient of a tempered representation of G2 . Then
the limit
Z
lim
γ→∞ N
r−1,γ
Φ(ng)ψr−1 (n)dn,
g ∈ G2
exists and defines a continuous function in ψr−1 (for a fixed g). If F is non-archimedean, then
the integral is in fact a constant for sufficiently large γ. Moreover if g ∈ G1 , then
Z
Φ(ng)ψr−1 (n)dn ΞG1 (g)ς(g)D .
lim
γ→∞
Nr−1,γ
Proof. If F is non-archimedean, the fact that the integral is a constant for sufficiently large γ is
well-known. The bound of the limit follows from Lemma 3.2.1. So from now on we assume that
F is archimedean.
To simplify notation, we put
Z
I(γ, g, Φ) =
Φ(ng)ψr−1 (n)dn,
g ∈ G1 .
Nr−1,γ
Note that to prove the limit exists, we may even assume that g = 1. Recall that Mr−1 '
×G1 is the Levi factor of Pr−1 . Put T = GLr−1
GLr−1
1 . By the Diximier–Malliavin theorem, it
1
is enough to prove the lemma for limγ→∞ I(γ, g, f ∗ Φ) where f ∈ Cc∞ (T ) and
Z
f ∗ Φ(g) =
f (t)Φ(t−1 gt)dt
T
is a function on G2 . When there is no confusion, we write I(γ) = I(γ, g, f ∗ Φ) for short.
Let (x1 , · · · , xr−1 ) ∈ F r−1 and n(x1 , · · · , xr−1 ) ∈ N so that n(x1 , · · · , nr−1 )e∗n+i = e∗n+i +
xi−1 e∗n+i−1 for i = 2, · · · , r. Let N† = {n(x1 , · · · , xr−1 ) | (x1 , · · · , xr−1 ) ∈ F r−1 }. It is a
subgroup of Nr−1 which is stable under the conjugation by T and the multiplication map N† ×
c† the group
Nr−1,−∞ → Nr−1 is an isomorphism. Let N†,γ = N† ∩ Nr−1,γ . We denote by N
reg
c†
of additive characters of N† and by N
the open subset consisting of generic characters.
c† reg . Let ψ t be the character of N† defined by ψ t (n) = ψr−1 (tnt−1 ). The map
Then ψr−1 ∈ N
c† reg . A compactly supported function on T is
t 7→ ψ t defines a homeomorphism from T to N
c† reg . We may thus talk about the
then identified with a compactly supported function on N
Fourier transform of f , which is a Schwartz function on N† . Let t1 , · · · , tr−1 ∈ F × so that
tn(x1 , · · · , xr−1 )t−1 = n(t1 x1 , · · · , tr−1 xr−1 ). The measure |t1 · · · tr−1 |dt is, up to a positive
b to N
b reg under this homeomorphism. We
constant, the restriction of the self-dual measure of N
may assume that the constant is one.
17
We have
Z
Z
f (t)Φ(t−1 ntg)ψr−1 (n)dtdn
I(γ) =
Nr−1,γ
(3.2.1)
T
Z
Z
=
N†
Z
−1
t
f (t) 1N†,γ (tnt )ψ (n)dt
T
!
Φ(nn0 g)dn0
dn.
Nr−1,−∞
Since f ∈ Cc∞ (T ), there are positive constants c1 and c2 such that c1 ≤ |ti | ≤ c2 for any i. We
put c = c−1
2 γ. Let Ω ⊂ {1, · · · , r − 1} be a subset and
N†Ω = {(x1 , · · · , xr−1 ) | |xi | ≤ c if i ∈ Ω; |xi | > c if i 6∈ Ω}.
P R
Then I(γ) = Ω N Ω ∗ where Ω runs over all subsets of {1, · · · , r − 1} and ∗ is the integrand
†
of (3.2.1).
We now fix an Ω and n ∈ N†Ω . We claim that there is a constant C, depending only on f , so
that
Z
Y
f (t) 1N (tnt−1 )ψ t (n)dt ≤ C ×
|xi |−1 .
†,γ
T
Let f1 (t) = f (t)|t1 · · · tr−1
|−1
i6∈Ω
∈
Cc∞ (T ).
If i ∈ Ω, then the integral over ti is the Fourier transform
of f1 with respect to the variable ti . If i 6∈ Ω, then we integrate over ti by integration by parts.
The anti-derivative of
1{x∈F ||x|≤γ} (tx)ψ(tx)
(as a function of t ∈ F ) is |x|−1 Xγ (tx), where Xγ is a function on F which is bounded by a
constant that is independent of γ. Our claim then follows.
We now prove that
Z
(3.2.2)
N†Ω
Z
Y
|xi |−1 ΞG2 (nn0 g)dn0 dn
Nr−1,−∞ i6∈Ω
is convergent. We write n = n(0, · · · , 0, xr−1 )n(x1 , · · · , xr−2 , 0) and n0 = n01 n02 where n01 ∈ N1,−∞
and n02 ∈ Nr−1,−∞ ∩ G1 . We integrate over xr−1 and n01 first. If r − 1 ∈ Ω, then it follows from
Lemma 3.2.1 that the integral is bounded by
Y
c3 ×
|xi |−1 ΞG1 (n(x1 , · · · , xr−2 , 0)n02 g)ς(n(x1 , · · · , xr−2 , 0)n02 g)D ,
i6∈Ω
where c3 and D are absolute constants. If r − 1 6∈ Ω, then we need to estimate
Z
Z
|xr−1 |−1 ΞG2 (n(0, · · · , 0, xr−1 )n01 n(x1 , · · · , xr−2 , 0)n02 g)dn01 dxr−1 .
|xr−1 |≥c
N1,−∞
e r−1 ) be the anti-derivative of
Let Ξ(x
Z
ΞG2 (n(0, · · · , 0, xr−1 )n01 n(x1 , · · · , xr−2 , 0)n02 g)dn01
N1,−∞
18
as a function of xr−1 , with x1 , · · · , xr−2 , n0 , g fixed. It follows from Lemma 3.2.1 that
e r−1 )| (log|xr−1 |)β ΞG1 (n(x1 , · · · , xr−2 , 0)n02 g)ς(n(x1 , · · · , xr−2 , 0)n02 g)D .
|Ξ(x
We integrate over xr−1 by integration by parts to get
(3.2.2)
Y
|xi |−1 ×
c−1 (log c)β ΞG1 (n(x1 , · · · , xr−2 , 0)n02 g)ς(n(x1 , · · · , xr−2 , 0)n02 g)D
i6∈Ω∪{r−1}
!
Z
−2
|xr−1 |
+
β
(log|xr−1 |) dxr−1 × ΞG1 (n(x1 , · · ·
|xr−1 |≥c
Y
, xr−2 , 0)n02 g)ς(n(x1 , · · ·
, xr−2 , 0)n02 g)D
|xi |−1 × ΞG1 (n(x1 , · · · , xr−2 , 0)n02 g)ς(n(x1 , · · · , xr−2 , 0)n02 g)D .
i6∈Ω∪{r−1}
Therefore in any case, the convergence of (3.2.2) is reduced to an analogous problem for G1 .
Then we can repeat this process and prove by induction that (3.2.2) is convergent and
(3.2.2) ΞG1 (g)ς(g)D
for some constant D.
By the Lebesgue dominated convergence theorem, we have
Z
Z Z
−1
t
lim I(γ) =
lim f (t) 1N†,γ (tnt )ψ (n)dt
γ→∞
T γ→∞
N†
Z
Z
=
N†
!
Φ(nn0 g)dn0
dn
Nr−1,−∞
fb1 (n)Φ(nn0 g)dn0 dn.
Nr−1,−∞
The rest of the assertions of the lemma follow easily from this expression.
3.3. Proof of Proposition 2.2.1. The case r = 0 is rather straight forward. Indeed, in this
case G0 = G1 = G2 . By the weak inequality, we only need to prove that
Z
ΞG0 (g)2 |Φφ,φ∨ (g)|dg
G0
is absolutely convergent. By the Cartan decomposition and the estimates (3.1.2) and (3.1.4),
the convergence is reduced to the convergence of
!D
Z
n
X
1
log|ai |
da1 · · · dan .
|a1 · · · an | 2 −
|an |≤···≤|a1 |≤1
i=1
This is clear. Proposition 2.2.1 is thus proved when r = 0.
The case r ≥ 2 follows from the case r = 1 by Lemma 3.2.4.
We now treat the case r = 1. In this case G2 = G1 . The defining integral of α reduces to
Z Z
∨
∨
α(ϕ2 , ϕ∨
,
ϕ
,
ϕ
,
φ,
φ
)
=
Φϕ,ϕ∨ (hg0 )Φϕ0 ,ϕ∨0 (g0 )Φφ,φ∨ (hg0 )dhdg0 ,
0
2
0
G0
H
19
Since π2 and π0 are both tempered, we need to prove that
Z
Z
ΞG1 (hg0 )ΞG0 (g0 )|Φφ,φ∨ (hg0 )|dhdg0
G0
H
is convergent.
Let g0 = k1 d(a)k2 be the Cartan decomposition of g0 where a = (a1 , · · · , an ) ∈ (F × )n with
|as | ≤ · · · ≤ |a1 | ≤ 1. We first estimate |Φφ,φ∨ (hd(a))|. We claim that there is a function f on
L∗ so that f (l∗ )p(l∗ ) is bounded for any polynomial function p on L∗ , such that
1
|Φφ,φ∨ (h(l + l∗ , t)d(a))| |det a| 2 f (l∗ a).
(3.3.1)
Indeed
1
|Φφ,φ∨ (h(l + l∗ , t)d(a))| ≤ |det a| 2
Z
|φ(xa + l∗ a)φ∨ (x)|dx.
L∗
Thus to prove (3.3.1), it is enough to prove that for any polynomial function p on L∗ ,
Z
sup |p(y)|
y∈L∗
|φ(xa + y)φ∨ (x)|dx < ∞.
L∗
We have
Z
sup |p(y)|
y∈L∗
|φ(xa + y)φ∨ (x)|dx ≤
Z
L∗
L∗
!
sup |p(y)φ(xa + y)| |φ∨ (x)|dx.
y∈L∗
Since p is a polynomial function, we may choose a sufficiently large N , such that
sup |p(y)φ(xa + y)| (1 + |x1 an | + · · · |xn a1 |)N (1 + |x1 | + · · · |xn |)N ,
y∈L∗
where x = (x1 , · · · , xn ) ∈ L∗ . We have the second inequality because |ai | ≤ 1 for all i. Then
Z
∨
Z
( sup |p(y)φ(xa + y)|)|φ (x)|dx L∗ y∈L∗
L∗
(1 + |x1 | + · · · |xn |)N |φ∨ (x)|dx < ∞.
We have thus proved (3.3.1).
By (3.1.1), to prove the convergence of the defining integral of α, it is enough to show the
convergence of
Z
A+
G0
Z
H
1
ΞG1 (h(l + l∗ , t)d(a))ΞG0 (d(a))δP−1
(d(a))|det a| 2 f (l∗ a)dhda.
m ∩G0
Then Lemma 3.2.3 reduces the convergence of this integral to the case r = 0.
20
3.4. Proof of Proposition 2.2.2. We are going to use the notation in the proof of Lemma 3.2.4,
one paragraph before (3.2.1). To simplify notation, we write Φϕi = Φϕi ,ϕi , i = 0, 2 and Φφ =
Φφ,φ . In order to prove that α(ϕ2 , ϕ0 , φ) ≥ 0, it is enough to show that for any function
f ∈ Cc∞ (T ), we have
Z Z
Z
!
Z
lim
T
G0
H
γ→∞ N
r−1,γ
Φϕ2 (nhg0 )ψr−1 (tnt−1 )dn Φϕ0 (g0 )Φφ (hg0 )f (t)f (t)dhdg0 dt ≥ 0.
We denote this expression by I. Since f (t) is compactly supported, by Fubini’s theorem, we
have
Z
Z
!
Z
Z
lim
I=
G0
H
T γ→∞ Nr−1,γ
Φϕ2 (nhg0 )f (t)f (t)ψr−1 (tnt−1 )dndt Φϕ0 (g0 )Φφ (hg0 )dhdg0 .
We denote the integral in the parentheses by II. It follows from Lemma 3.2.4 that
Z
lim
Φϕ2 (nhg0 )ψr−1 (tnt−1 )dn
γ→∞ N
r−1,γ
is bounded by a constant which depends continuously on ψr−1 . Since f is compactly supported
on T , we can choose this constant to be independent of t (but depends on hg0 ). Then by the
Lebesgue dominated convergence theorem, we have
Z Z
II = lim
Φϕ2 (nhg0 )f (t)f (t)ψr−1 (tnt−1 )dndt.
γ→∞ T
Nr−1,γ
Moreover, the double integral on the right hand side is absolutely convergent. We can thus
interchange the order of integration. Finally we conclude that
Z
Z
II = lim
Φϕ2 (nhg0 )f (t)f (t)ψr−1 (tnt−1 )dtdn.
γ→∞ N
r−1,γ
T
Let f1 (t) = f (t)|t1 · · · tr−1 |−1 ∈ Cc∞ (T ). We have
Z
Z
Z
II = lim
Φϕ2 (n1 n0 hg0 )fb1 (n1 n2 )fb1 (n2 )dn2 dn0 dn1
γ→∞ N
†,γ
Z
Z
Nr−1,−∞
Z
=
N†
N†
Z
=
Nr−1,−∞
where π2 (fb1 )ϕ2 =
R
N†
Nr−1,−∞
N†
b
0
0
b
Φϕ2 (n1 n−1
2 n hg0 )f1 (n1 )f1 (n2 )dn2 dn dn1
Φπ2 (fb1 )ϕ2 (n0 hg0 )dn0 ,
fb1 (n)π2 (n)ϕ2 dn. This expression makes sense since fb1 is a Schwartz
function on N† . Thus to show that I ≥ 0, it remains to show that
Z Z Z
Φπ2 (fb1 )ϕ2 (n0 hg0 )Φϕ0 (g0 )Φφ (hg0 )dn0 dhdg0 ≥ 0.
G0
H
Nr−1,−∞
21
Actually, we will show that
Z Z Z
(3.4.1)
G0
H
Φϕ2 (n0 hg0 )Φϕ0 (g0 )Φφ (hg0 )dn0 dhdg0 ≥ 0,
Nr−1,−∞
for all smooth vectors ϕ2 ∈ π2 and ϕ0 ∈ π0 . Unlike the proof of [22, Proposition 1.1] and [34,
Theorem 2.1(2)], we cannot apply [18, Theorem 2.1] directly, as G2 × HG0 is not reductive.
However, we are going to mimic the proof of [18, Theorem 2.1] to prove (3.4.1).
We first claim that it is enough to prove (3.4.1) for a K2 -finite (resp. K0 -finite) vector ϕ2 ∈ π2
(resp. ϕ0 ∈ π0 ). This is only an issue when F is archimedean. So we assume temporarily that
F is archimedean. Since K2 -finite vectors are dense in the space of smooth vectors in π2 , we
(i)
may choose a sequence of K2 -finite vectors ϕ2 which is convergent to ϕ2 . It follows that Φϕ2
is approximated pointwisely by Φϕ(i) . Moreover by [41], there exists an element X in the Lie
2
algebra of G2 , which depends on K2 only, such that
(i)
(i)
(i)
Φϕ(i) (g2 ) ≤ Bπ2 π2 (X)ϕ2 , π2 (X)ϕ2 ΞG2 (g2 ) = |π2 (X)ϕ2 |2 ΞG2 (g2 ).
2
(i)
(i)
Since ϕ2 is convergent to ϕ2 , we see that |π2 (X)ϕ2 |2 is convergent to |π2 (X)ϕ2 |2 . In particular,
(i)
it is bounded by some constant which is independent of ϕ2 . Similarly we choose a sequence
(i)
ϕ0 of K0 -finite vectors in π0 which approximate ϕ0 . Since
Z Z Z
ΞG2 (n0 hg0 )ΞG0 (g0 )Φφ (hg0 )dn0 dhdg0
G0
H
Nr−1,−∞
is absolutely convergent, by the Lebesgue dominated convergence theorem
Z Z Z
Φϕ2 (n0 hg0 )Φϕ0 (g0 )Φφ (hg0 )dn0 dhdg0
G0
H
Z
Nr−1,−∞
Z Z
= lim
i→∞ G0
H
Nr−1,−∞
Φϕ(i) (n0 hg0 )Φϕ(i) (g0 )Φφ (hg0 )dn0 dhdg0 .
2
0
So the positivity in (3.4.1) for smooth vectors follows from the positivity for K-finite vectors.
From now on, we assume that ϕ2 and ϕ0 in (3.4.1) are K2 -finite and K0 -finite respectively.
We come back to the situation F being an arbitrary local field of characteristic zero.
Since π2 is tempered, by (the proof of) [18, Theorem 2.1] (which is also valid when F is
(i)
non-archimedean), one can find a sequence of compactly supported continuous functions f2,j on
P i (i)
(i)
G2 and a sequence of positive real numbers aj , j = 1, · · · , si , such that sj=1
aj = 1 and the
functions
g20 7→ A(i) (g20 ) =
si
X
(i)
Z
aj
G2
j=1
(i)
(i)
f2,j (g2 g20 )f2,j (g2 )dg2
approximate Φϕ2 ,ϕ2 (g20 ) pointwisely. Moreover, there is a constant C2 , such that
|A(i) (g20 )| ≤ C2 ΞG2 (g20 ).
22
(i)
Similarly, we can find a sequence of compactly supported continuous functions f0,j on G0 and a
P i (i)
(i)
sequence of positive real numbers bj , j = 1, · · · , ki , such that kj=1
bj = 1 and the functions
g00
(i)
7→ B
(g00 )
=
ki
X
(i)
bj
Z
j=1
(i)
(i)
f0,j (g0 g00 )f0,j (g0 )dg0
G0
approximate Φϕ0 ,ϕ0 (g00 ) pointwisely. Moreover, there is a constant C0 , such that
|B (i) (g00 )| ≤ C0 ΞG0 (g00 ).
Since the integral
Z
Z Z
G0
H
ΞG2 (n0 hg0 )ΞG0 (g0 )Φφ (hg0 )dn0 dhdg0
Nr−1,−∞
is absolutely convergent, by the Lebesgue dominated convergence theorem, to prove (3.4.1), it
is enough to prove that for any i, j,
Z
Z Z Z
(3.4.2)
G0
H
Nr−1,−∞
G2
Z
G0
(i)
(i)
f2,j (g2 n0 hg00 )f2,j (g2 )dg2
(i)
(i)
f0,j (g0 g00 )f0,j (g0 )dg0
Φφ (hg00 )dn0 dhdg00 ≥ 0.
We denote the left hand by Q. Note that this integral is absolutely convergent. To simplify
(i)
(i)
notation, we write f2 = f2,j and f0 = f0,j .
We can write the inner product on S(L∗ ) as
Z
0
Bωψ (φ, φ ) =
ωψ (h0 )φ(0)ωψ (h0 )φ0 (0)dh0 .
L+F \H
Using this expression of the inner product, we have
Z
Z
Z Z Z
0
0
Q=
f2 (g2 n hg0 )f2 (g2 )dg2
G0
H
Nr−1,−∞
G2
f0 (g0 g00 )f0 (g0 )dg0
G0
!
Z
L+F \H
ωψ (h0 hg00 )φ(0)ωψ (h0 )φ0 (0)dh0
dn0 dhdg00 .
We make the following change of variables
g00 7→ g0−1 g00 ,
h 7→ g0−1 hg0 ,
h 7→ h0−1 h,
g2 7→ g2 hg0 .
Then
Z
Z
ZZ
ZZ
Q=
G2
Nr−1,−∞
(L+F )\H×H
G0 ×G0
f2 (g2 n0 h0 g00 )f1 (g2 hg0 )
f0 (g0 )f0 (g0 )ωψ (hg00 )φ(0)ωψ (h0 g0 )φ(0)dg0 dg00 dhdh0 dn0 dg2 ,
where L + F embeds in H × H diagonally.
23
Finally we decompose the integral over G2 as
Z
Z
Z
.
G2 /(Nr−1,−∞ o(L+F ))
Nr−1,−∞
L+F
We conclude that
Z
Q=
Z
2
Z Z
f2 (g2 nhg0 )f0 (g0 )ωψ,µ (hg0 )φ(0)dg0 dhdn dg2 ≥ 0.
G2 /(Nr−1,−∞ o(L+F ))
Nr−1,−∞ H G0
We have thus proved (3.4.1) and hence Proposition 2.2.2.
3.5. Regularization via stable unipotent integral. In this subsection, we give an alternative but equivalent way to define the linear functional α when F is non-archimedean following [27,34]. This definition is better for the unramified computation and is valid for nontempered
representations. In this subsection, F is always assumed to be non-archimedean.
Let N be a unipotent group over F and f a smooth function on N . We say that f is compactly
supported on average if there are compact subsets U and U 0 of N , such that L(δU 0 )R(δU )f is
compactly supported. Here δU stands for the Dirac measure on U , i.e. δU = (vol U )−1 1U , and
Z Z
L(δU 0 )R(δU )f (n) =
δU 0 (u0 )δU (u)f (u0 nu)du0 du.
N
N
If f is compactly supported on average, we then define
Z st
Z
f (n)dn :=
L(δU 0 )R(δU )f (n)dn.
N
N
This is called the stable integral of f on N . The definition is independent of the choice of U
and U 0 .
We denote temporarily by G a reductive group over F . Let Pmin = Mmin Nmin be a fixed
minimal parabolic subgroup of G. Let P = M N ⊃ Pmin be a parabolic subgroup of G. Let Ψ
be a generic character of N , i.e. the stabilizer of Ψ in Mmin is the center of Mmin . Let π be an
irreducible admissible representation of G and Φ a matrix coefficient of π. Then
Proposition 3.5.1 ([34, Proposition 3.3]). The function Φ|NP · Ψ is compactly supported on
average.
Now let G = Mp(2n). Then Proposition 3.5.1 still holds. The same proof as in [34, Proposition 3.3] goes through as it uses only the Bruhat decomposition and Jacquet’s subrepresentation
theorem, which are valid for G.
f2 ).
Now we retain the notation G0 , G1 , G2 etc. Let Φ be a matrix coefficient on G2 (resp. G
Define
Fψst Φ(g) =
Z
st
Φ(gn)ψr−1 (n)dn,
Nr−1
f2 ). This definition makes sense because of Proposition 3.5.1.
which is a function on G2 (resp. G
24
Lemma 3.5.2. Assume that Φ is a matrix coefficient of a tempered representation of G2 (resp.
f2 ). Then
G
Fψst Φ(hg0 ) = Fψ Φ(hg0 ),
f0 ).
h ∈ H, g0 ∈ G0 , (resp. g0 ∈ G
Proof. By definition,
Fψst Φ2 (hg0 )
Z
(vol U )
=
−1
Φ(unhg0 )ψr−1 (un)du dn,
Z
U
Nr−1
where U is an open compact set of Nr−1 . The inner integral, as a function of n, is compactly
supported. Therefore we may take a sufficiently large γ, such that Nr−1,γ contains U and the
support of the inner integral (as a function of n) and that
Z
Fψ Φ2 (hg0 ) =
Φ2 (nhg0 )ψr−1 (n)dn.
Nr−1,γ
It follows that
Fψst Φ2 (hg0 )
Z
=
(vol U )
−1
Nr−1,γ
Z
Φ(unhg0 )ψr−1 (un)dudn,
U
Z
−1
Z
Φ(nhg0 )ψr−1 (n)dn × (vol U )
=
du
U
Nr−1,γ
= Fψ Φ2 (hg0 ),
where in the second equality, we have made a change of variable n 7→ u−1 n.
Thanks to Lemma 3.5.2, if F is nonarchimedean, then we may use Fψst instead of Fψ in the
definition of the local linear form α. We will not distinguish Fψst and Fψ from now on and write
just Fψ .
4. Unramified computations
In this section, we assume the conditions prior to Proposition 2.2.3. In particular, F is a
non-archimedean local field of residue characteristic different from two. The argument is mostly
adapted from [34], except that at the end we use a different trick, which avoids the use of
the explicit formulae of the Whittkaer–Shintani functions as in [34, Appendix]. Some of the
arguments which are identical to [34] are only sketched.
4.1. Reduction steps. For i = 0, 1, 2, let Bi = Pm ∩ Gi = Ti Ui be the upper triangular Borel
subgroup of Gi where Ti is the diagonal maximal torus of Gi . We have a hyperspecial subgroup
fi → Gi splits uniquely over Ki . We can
Ki = Sp(Wi )(oF ) of Sp(Wi ). Recall the two fold cover G
fi . Let Ξ (resp. ξ) be an unramified character of T2 (resp. T0 ).
thus view Ki as a subgroup of G
f0 .
In the case Sp, we consider the unramified principal seires π2 = I(Ξ) of G2 and π0 = I(ξ) of G
f2 and π0 = I(ξ) of G0 .
In the case Mp, we consider the unramified principal seires π2 = I(Ξ) of G
Note that the unramified principal series representation of the metaplectic group depends on the
25
additive character ψ, even though this is not reflected in the notation. We frequently identify
Ξ with an element in Cm which we also denote by Ξ = (Ξ1 , · · · , Ξm ), the correspondence being
given by
−1
Ξ1
Ξ(diag[am , · · · , a1 , a−1
· · · |am |Ξm
1 , · · · , am ]) = |a1 |
Similarly we identify ξ with an element in Cn . The contragredient of π2 (resp. π0 ) is I(Ξ−1 )
(resp. I(ξ −1 )). Let fΞ ∈ I(Ξ), fΞ−1 ∈ I(Ξ−1 ) (resp. fξ ∈ I(ξ), fξ−1 ∈ I(ξ −1 )) be the K2 -fixed
(resp. K0 -fixed) elements with fΞ (1) = fΞ−1 (1) = 1 (resp. fξ (1) = fξ−1 (1) = 1). Let
Z
Z
Z
ωψ (hg0 ) 1L∗ (oF ) (x)dx.
fξ (k0 g0 )dk0 , Φφ (hg0 ) =
fΞ (k2 g2 )dk2 , Φξ (g0 ) =
ΦΞ (g2 ) =
L(oF )
K0
K2
and
Z
Z
I(g2 , Ξ, ξ, ψ) =
G0
H
Fψ ΦΞ (g2−1 hg0 )Φξ (g0 )Φφ (hg0 )dhdg0 .
Then α(fΞ , fΞ−1 , fξ , fξ−1 , φ, φ) = I(1, Ξ, ξ, ψ).
f0 . We define the Borel subgroup BJ (resp. B e) of J (resp.
Let J = H o G0 and Je = H o G
J
e as a subgroup of J (resp. J)
e consisting of elements of the form hb0 where b0 ∈ B0 (resp. B
f0 ,
J)
f0 ) and h ∈ H is of the form h(l, t), l ∈ L. We define the unramified
the inverse image of B0 in G
e as
principal series representation of J (resp. J)
1
1
I J (ξ, ψ) = {f ∈ C ∞ (J) | f (h(l, t)b0 hg0 ) = δB2 J (b0 ) 2 ξ(b0 )ψ(t)f (hg0 )},
resp.
1
e
e | f (h(l, t)b0 hg0 ) = δ 2 (b0 )χψ (b0 )ξ(b0 )ψ(t)f (hg0 )},
I J (ξ, ψ) = {f ∈ C ∞ (J)
BJ
−1
where χψ (b0 ) = γψ (t1 · · · tn )χ(t1 · · · tn ) and tn , · · · , t1 , t−1
1 , · · · , tn are diagonal entries of b0 .
e acts on I J (ξ, ψ) (resp. I Je(ξ, ψ)) via the right translation. Let
The group J (resp. J)
e
KJ = J ∩ K1 . There is a canonical J (resp. J)-invariant
pairing given by
Z Z
BJ (f, f ∨ ) =
f (h(l∗ , 0)k0 )f ∨ (h(l∗ , 0)k0 )dk0 dl∗ ,
L∗
K0
where f ∈ I J (ξ, ψ), f ∨ ∈ I J (ξ −1 , ψ) (resp. f ∈ I J (ξ, ψ), f ∨ ∈ I J (ξ −1 , ψ)).
e
e
In the case Mp, there is a canonical inner product preserving isomorphism
ωψ ⊗ I(ξ) → I J (ξ, ψ),
e
φ ⊗ fξ → fξ,ψ ,
f0 . In the case Sp, there is a canonical
where fξ,ψ (hg0 ) = ωψ (hg0 )φ(0)fξ (g0 ), h ∈ H and g0 ∈ G
inner product preserving isomorphism
ωψ ⊗ I(ξ) → I J (ξ, ψ),
φ ⊗ fξ → fξ,ψ ,
where fξ,ψ (hg0 ) = ωψ (hι(g0 ))φ(0)fξ (ι(g0 )).
For the ease of the exposition, we slightly modify our notation in the case Mp for the rest of
this section. For i = 0, 1, 2, we put Gi = Mp(Wi ) and Bi = Ti Ni the standard Borel subgroup
26
of Gi . Denote by J = H o Mp(W0 ), which is a subgroup of G1 , and BJ its Borel subgroup. We
denote by Ki = Sp(Wi )(oF ) a hyperspecial maximal subgroup of Sp(Wi ). The metaplectic cover
Mp(Wi ) → Sp(Wi ) splits canonically over Ki , so we view Ki as a compact (but not maximal)
subgroup of Gi and an element in Ki is naturally viewed as an element in Gi . Let KJ = K1 ∩ J.
The subgroup Pi = Mi Ni is a parabolic subgroup of Sp(Wi ) as before. The metaplectic double
cover splits canonically over Ni , so we consider Ni as subgroups of G2 . By the Weyl group of
Mp(Wi ), we mean the Weyl group of Sp(Wi ). We let
w2,long =
wm
!
,
−wm
w1,long =
wn+1
!
−wn+1
,
w0,long =
wn
!
−wn
be representatives of the longest elements in the Weyl groups of WG2 , WG1 and WG0 respectively.
They are viewed as elements in G2 , G1 and G0 respectively.
For (Ξ1 , · · · , Ξm ) ∈ Cm and (ξ1 , · · · , ξn ) ∈ Cn , we denote by Ξ and ξ the genuine character
of B2 and B0 respectively, defined by
−1
Ξ((diag[tm , · · · , t1 , t−1
1 , · · · , tm ], )) = · (χψ Ξ1 )(t1 ) · · · (χψ Ξm )(tm ),
−1
ξ((diag[tn , · · · , t1 , t−1
1 , · · · , tn ], )) = · (χψ ξ1 )(t1 ) · · · (χψ ξn )(tn ).
We have the unramified principal series representation I(Ξ) of G2 and I(ξ, ψ) of J. We let
fξ,ψ be the KJ fixed element in I(ξ, ψ) such that fξ,ψ (1) = 1. We will need to integrate over
Mp(W0 ). For this, we pick a measure dx on Mp(W0 ), such that for any f ∈ Cc∞ (Sp(W0 )), we
R
R
have Sp(W0 ) f (g)dg = Mp(W0 ) f (x)dx. When integrating over Ki ’s or KJ , we always use the
measure so that the volume of the domain of the integration is one.
With this modification of notation, the integral I(g2 , Ξ, ξ, ψ) in both cases Mp and Sp can be
written as
Z Z
I(g2 , Ξ, ξ, ψ) =
J
KJ
Fψ ΦΞ (g2−1 gJ )fξ,ψ (kJ gJ )dkJ dgJ .
We distinguish two cases: r = 0 and r ≥ 1. We treat the case r ≥ 1 first.
−1
Let ẇ = w1,long
w2,long be a representative of the longest element in WG1 \WG2 .
Lemma 4.1.1. If g2 ∈ G2 and gJ ∈ J, then
Fψ ΦΞ (g2−1 gJ )
=w
−1
Z
K1
Z
st
Fψ (π2 (gJ )fΞ )(k1 ẇn)(π2∨ (g2 )fΞ−1 )(k1 ẇn)dk1 dn,
Nr−1
where
Z
∆T2
w=
fΞ (ẇn)fΞ−1 (ẇn)dn =
∆G2
Nr−1
27
∆T1
∆G1
−1
.
Proof. By definition,
Fψ ΦΞ (g2−1 gJ )
Z
st
=
Nr−1
Z
st
Bπ2 (π2 (g2−1 gJ u)fΞ , fΞ−1 )ψ(u)−1 du
Bπ2 (π2 (gJ u)fΞ , π2∨ (g2 )fΞ−1 )ψ(u)−1 du.
=
Nr−1
By [34, Lemma 3.2] (it is valid also for metaplectic groups since the Bruhat decomposition
is valid for metaplectic groups), there is an open compact subgroup U of Nr−1 , such that
(π2∨ (g2 )fΞ−1 )◦ = R(δU ψ)(π2∨ (g2 )fΞ−1 ) and (π2 (g2 )fΞ )◦ = R(δU ψ)(π2 (g2 )fΞ ) are supported in
B2 ẇPr−1 . Then
Z
Fψ ΦΞ (g2−1 gJ ) =
Bπ2 (π2 (u)(π2 (gJ )fΞ )◦ , (π2∨ (g2 )fΞ−1 )◦ )ψ(u)−1 du.
Nr−1
We use the following realization of Bπ2 :
Z Z
∨
−1
Bπ2 (ϕ, ϕ ) = w
K1
st
ϕ(k1 ẇn)ϕ∨ (k1 ẇn)dndk1 ,
Nr−1
where
Z
∆T2
w=
fΞ (ẇn)fΞ−1 (ẇn)dn =
∆G2
Nr−1
∆T1
∆G1
−1
.
Then
Fψ ΦΞ (g2−1 gJ )
=w
−1
Z
Z
Nr−1
Z
K1
(π2 (gJ )fΞ )◦ (k1 ẇnu)(π2∨ (g2 )fΞ−1 )◦ (k1 ẇn)ψ(u)−1 dndk1 du,
Nr−1
where the integrand is compactly supported. It equals
Z Z
−1
w
Fψ (π2 (gJ )fΞ )(k1 ẇn)(π2∨ (g2 )fΞ−1 )◦ (k1 ẇn)dndk1
K1
= w−1
Z
Nr−1
Z
K1
st
Fψ (π2 (gJ )fΞ )(k1 ẇn)(π2∨ (g2 )fΞ−1 )(k1 ẇn)dndk1 .
Nr−1
By Lemma 4.1.1, we have
Z Z Z
−1
I(g2 , Ξ, ξ, ψ) = w
J
K1
st
Nr−1
Z
KJ
Fψ fΞ (k1 ẇngJ )π2∨ (g2 )fΞ−1 (k1 ẇn)fξ,ψ (kJ gJ )dkJ dndk1 dgJ .
Let
l0∗ = (1, · · · , 1) ∈ L∗ ,
η1 = w1,long h(l0∗ , 0) ∈ G1 ,
η = ẇη1 ∈ G2 .
Lemma 4.1.2. The double coset B2 η(Nr−1 o BJ ) is open dense in G2 .
Proof. This is straightforward to check.
Thanks to this lemma, we can define a function YΞ,ξ,ψ on G2 with the following properties.
28
−1/2
1/2
(1) YΞ,ξ,ψ (b2 g2 h(l, t)b0 u) = (Ξ−1 δB2 )(b2 )(ξδBJ )(b0 )ψ(t)ψr−1 (u)YΞ,ξ,ψ (g2 ) for any b2 ∈ B2 ,
b0 ∈ B0 , l ∈ L and u ∈ Nr−1 .
(2) The support of YΞ,ξ,ψ is B2 η(Nr−1 o BJ ).
(3) YΞ,ξ,ψ (η) = 1.
The space of functions that satisfy the first two conditions is one dimensional by Lemma 4.1.2.
We have
1/2
−1/2
YΞ,ξ,ψ (b2 ηh(l, t)b0 u) = (Ξ−1 δ2 )(b2 )(ξδBJ )(b0 )ψ(t)ψr−1 (u),
for b2 ∈ B2 , b0 ∈ B0 , l ∈ L and u ∈ Nr−1 . We define a function TΞ,ξ,ψ on G2 as
Z


Fψ fΞ (g2 gJ )fξ,ψ (gJ )dgJ , g2 ∈ B2 η(Nr−1 o BJ ),
J
TΞ,ξ,ψ (g2 ) =

 0,
otherwise.
If the defining integral of TΞ,ξ,ψ is convergent, then we have
TΞ,ξ,ψ (g2 ) = TΞ,ξ,ψ (η)YΞ−1 ,ξ−1 ,ψ−1 (g2 ),
g2 ∈ G2 .
Assuming that the defining integrals of TΞ,ξ,ψ is convergent, which will be proved later, it
follows that
I(g2 , Ξ, ξ, ψ)
Z Z st
−1
=w
K1
=w
−1
Z
TΞ,ξ,ψ (k1 ẇnkJ )π2∨ (g2 )fΞ−1 (k1 ẇn)dkJ dndk1
Nr−1
KJ
Z
Z
st
Z
YΞ−1 ,ξ−1 ,ψ−1 (k1 ẇnkJ )π2∨ (g2 )fΞ−1 (k1 ẇn)dkJ dndk1 .
TΞ,ξ,ψ (η)
K1
Nr−1
KJ
Define
(4.1.1)
SΞ0 −1 ,ξ−1 ,ψ−1 (g2 )
=w
−1
Z
Z
K1
st
Z
Nr−1
YΞ−1 ,ξ−1 ,ψ−1 (k1 ẇnkJ )(π2∨ (g2 )fΞ−1 )(k1 ẇn)dkJ dndk1 .
KJ
Then we have
I(g2 , Ξ, ξ, ψ) = TΞ,ξ,ψ (η)SΞ0 −1 ,ξ−1 ,ψ−1 (g2 ).
We now treat the case r = 0.
The integral we need to compute is
Z Z
I(gJ , Ξ, ξ, ψ) =
G0
KJ
Z
K0
fΞ (k0 g)fξ,ψ (kJ gJ−1 g)dk0 dkJ dg.
We define
l0 = (1, · · · , 1) ∈ L,
η = w0,long h(l0 , 0) ∈ J.
Similar to Lemma 4.1.2, it is straightforward to prove the following lemma.
Lemma 4.1.3. The double coset BJ ηB0 is open dense in J.
29
We define a function YΞ,ξ,ψ on J which is supported on BJ ηB0 by
1
1
YΞ,ξ,ψ (h(l, t)b00 ηb0 ) = (ξ −1 δJ2 )(b00 )(Ξδ02 )(b0 )ψ(t),
b0 , b00 ∈ B0 , l ∈ L.
We define the function TΞ,ξ,ψ on J by
Z


fΞ (g)fξ,ψ (gJ g)dg, gJ ∈ BJ ηB0 ,
G0
TΞ,ξ,ψ (gJ ) =

 0,
otherwise.
and the function SΞ,ξ,ψ by
Z
Z
SΞ,ξ,ψ (gJ ) =
KJ
K0
YΞ,ξ,ψ (kJ gJ−1 k0 )dk0 dkJ .
It follows that
I(gJ , Ξ, ξ, ψ) = TΞ,ξ,ψ (η)SΞ−1 ,ξ−1 ,ψ−1 (gJ ).
We now prove the convergence of the defining integral of TΞ,ξ,ψ and SΞ−1 ,ξ−1 ,ψ−1 . Assume
that r ≥ 0.
Lemma 4.1.4. The defining integrals for TΞ,ξ,ψ and SΞ−1 ,ξ−1 ,ψ−1 are absolutely convergent if Ξ0
and ξ are sufficiently close to the unitary axis, where Ξ0 is the restriction of Ξ to T1 .
Proof. If r = 0, then it follows from Proposition 2.2.1 (or its proof, applied to |Ξ| and |ξ|) that
I(gJ , Ξ, ξ, ψ) is convergent if Ξ and ξ are sufficiently close to the unitary axis. It then follows that
for a fixed gJ ∈ J, the defining integral of TΞ,ξ,ψ (kJ gJ k0 ) is convergent for almost all kJ ∈ KJ
and k0 ∈ K0 such that kJ gJ−1 k0 ∈ BJ ηB0 . By the definition of TΞ,ξ,ψ , its defining integral is
convergent for some gJ ∈ BJ ηB0 is and only if it is convergent for all gJ ∈ BJ ηB0 . Therefore
the defining integral of TΞ,ξ,ψ (η) is convergent. This then implies that the defining integral of
SΞ−1 ,ξ−1 ,ψ−1 is convergent.
The convergence in the case of r = 1 can be proved similarly. We only need to change the
notation at several places.
Now assume that r ≥ 2. By [34, Lemma 3.3], there is an open compact subgroup U of Nr−1 ,
such that for all gJ ∈ J,
Z
Fψ fΞ (ηgJ ) =
fΞ (ηgJ u)ψr−1 (u)du.
U
Therefore there is a constant C, such that
|Fψ fΞ (ηgJ )| ≤ C × f|Ξ0 | (η1 gJ ).
The Lemma in the case r ≥ 2 then follows from the case r = 1.
30
4.2. Proof of Proposition 2.2.3. Assume that r ≥ 1. Let Ξ0 = (Ξ1 , · · · , Ξn ) ∈ Cn . Let σ be
the unramified principal series representation of G0 defined by Ξ0 . We let τ be the unramified
principal series representation of GLr defined by the unramified characters (Ξn+1 , · · · , Ξm ).
Following the notation of [22] and [34], we shall denote TΞ,ξ,ψ (η) by ζ(Ξ, ξ, ψ).
Lemma 4.2.1. We have

Y
Lψ ( 21 , π0 × τ )

1


ζ(Ξ0 , ξ, ψ),
Case Sp,

2

)
 L(1, σ × τ )L(1, τ, ∧ ) 1≤i<j≤r L(1, Ξn+i Ξ−1
n+j
ζ(Ξ, ξ, ψ) =
Y

L( 12 , π0 × τ )
1

0



−1 ζ(Ξ , ξ, ψ), Case Mp.
2
 Lψ (1, σ × τ )L(1, τ, Sym )
L(1, Ξn+i Ξn+j )
1≤i<j≤r
Proof. By definition,
Z
(4.2.1)
Z Z
ζ(Ξ, ξ, ψ) =
G0
H
fΞ (w2,long h(l0∗ , 0)uhg0 )fξ (g0 )ψr−1 (u)ωψ (hg0 )φ(0)dudhdg0 .
Nr−1
We combine the integral over J and Nr−1 to get an integral over Nr and get
Z
Z
fΞ (w2,long h(l0∗ , 0)g0 v)fξ (g0 )ψr−1 (v)ωψ (g0 `(v))φ(0)dvdg0 ,
ζ(Ξ, ξ, ψ) =
G0
Nr
where ` : Nr → H is the natural projection whose kernel is Nr−1 . We make a change of variable
v 7→ h(l0∗ , 0)−1 v and get
Z
Z
ζ(Ξ, ξ, ψ) =
G0
Nr
fΞ (w2,long vg0 )fξ (g0 )ψr−1 (v)ωψ (h(l0∗ , 0)−1 `(v)g0 )φ(0)dvdg0 ,
Let NR be the unipotent radical of the upper triangular Borel subgroup of GLr and
Z
fWτ ,Ξ0 (g) =
fΞ (wr ng)ψr (n)dn,
g ∈ G2 .
NR
Then by the Casselman–Shalika formula, we have
fWτ ,Ξ0 (1) =
Y
1
1≤i<j≤r
L(1, Ξn+i Ξ−1
n+j )
.
We can then write the integral (4.2.1) as
Y
1
L(1, Ξn+i Ξ−1
n+j )
1≤i<j≤r
Z
Z
×
NR \Nr
G0
fWτ ,Ξ0 (w0,long g0 ẅv)fξ (g0 )ω(h(l0∗ , 0)−1 g0 `(v))φ(0)dvdg,
31

1r

where ẅ = 
−1
w0,long
vw0,long .


−1
. We make a change of variable g 7→ w0,long gw0,long and v 7→
12n
−1r
Then since w0,long ∈ K0 , we conclude that
1
Y
ζ(Ξ, ξ, ψ) =
1≤i<j≤r
Z
×
L(1, Ξn+i Ξ−1
n+j )
Z
fWτ ,Ξ0 (g ẅv)fξ (w0,long g)ω(w0,long h(l0∗ , 0)g`(v))φ(0)dvdg.
NR \Nr
G0
By definition,
0
Z
ζ(Ξ , ξ, ψ) =
G0
fΞ0 (g)fξ (w0,long g0 )ωψ (w0,long h(l0∗ , 0)g)φ(0)dg.
We then apply [13, Theorem 4.3] and [13, End of Section 4, (43)] to get the lemma. (In the
notation of [13], we apply this to the case r = 0 and bν (fΞ0 , fξ , φ) = ζ(Ξ0 , ξ, ψ)).
We now compute SΞ0 −1 ,ξ−1 ,ψ−1 (1). Define the projection pr2 : Cc∞ (G2 ) → I(Ξ) by
Z
1/2
F2 (b2 g2 )(Ξ−1 δ2 )(b2 )db2 ,
pr2 (F2 )(g2 ) =
B2
where the measure db2 is the left invariant measure on B2 so that pr2 (1K2 ) = fΞ . Then we
define
lΞ,ξ,ψ ∈ HomNr−1 oJ (I(Ξ), I J (ξ −1 , ψ) ⊗ ψr−1 )
by
Z
lΞ,ξ,ψ (f2 )(gJ ) =
f20 (g2 gJ )YΞ,ξ,ψ (g2 )dg2 ,
G2
where f20 is any element in Cc∞ (G2 ) with pr2 (f20 ) = f2 . It is not hard to check that lΞ,ξ,ψ is
independent of the choice of f20 . We define
SΞ,ξ,ψ (g2 ) = BI J (ξ,ψ) (fξ,ψ , lΞ,ξ,ψ (π2 (g2 )fΞ )).
The defining integral of lΞ,ξ,ψ is convergent if YΞ,ξ,ψ is continuous. By [40, Section 3], whose
method is valid for both cases Mp and Sp, YΞ,ξ,ψ is continuous if (Ξ, ξ) lie in some (nonempty)
open subset of Cr+s × Cs . We refer the readers to [40, Section 3] for a precise description of this
open subset.
0
Lemma 4.2.2. SΞ,ξ,ψ
= SΞ,ξ,ψ .
0
Proof. We check that SΞ,ξ,ψ and SΞ,ξ,ψ
agree when YΞ,ξ,ψ is continuous.
32
Let Ξ1 = (Ξ1 , · · · , Ξn+1 ) and I(Ξ1 ) be the unramified principal series representation of G1
defined by the character Ξ1 . Let Fψ0 (f2 )(g2 ) := Fψ (f2 )(g2 ẇ). Then Fψ0 (f2 )|G1 ∈ I(Ξ1 ). Define
the projection pr1 : Cc∞ (G1 ) → I(Ξ1 ) by
Z
pr1 (F )(g1 ) =
1/2
B1
F (b1 g1 )((Ξ1 )−1 δ1 )(b1 )db1 ,
where the left invariant measure db1 is the one so that pr1 (1K1 ) = fΞ1 . Note that pr1 is surjective
and for any element f ∈ I(Ξ1 ), one can choose F whose support lies in K1 such that pr1 (F ) = f .
0
Define the intertwining operator lΞ,ξ,ψ
∈ HomNr−1 oJ (I(Ξ), I J (ξ −1 , ψ) ⊗ ψr−1 ) by
Z
0
lΞ,ξ,ψ
(f2 )(gJ ) =
f200 (g1 gJ )YΞ,ξ,ψ (g1 ẇ)dg1 ,
G1
where f200 is any element in Cc∞ (G1 ) with pr1 (f200 ) = Fψ0 (f2 )|G1 .
Fix g2 ∈ G2 and let f200 ∈ Cc∞ (G1 ) be a smooth function whose support is contained in K1
and pr1 (f200 ) = Fψ0 (π2 (g2 )fΞ )|G1 . Then
Z Z
−1
0
SΞ,ξ,ψ (g2 ) = w
K1
=w
−1
Z
Z
K1
=w
−1
=w
−1
Z
YΞ,ξ,ψ (k1 ẇkJ )Fψ0 (π2 (g2 )fΞ )(k1 )dkJ dk1
KJ
YΞ,ξ,ψ (k1 ẇkJ )f200 (k1 )dkJ dk1
KJ
Z
K1
YΞ,ξ,ψ (k1 ẇ)f200 (k1 kJ )dkJ dk1
KJ
0
BI J (ξ,ψ) (fξ,ψ , lΞ,ξ,ψ
(π2 (g2 )fΞ )).
0
. We have
Therefore in order to prove the lemma, we only need to show w · lΞ,ξ,ψ = lΞ,ξ,ψ
dim HomNr−1 oJ (I(Ξ), I J (ξ −1 , ψ) ⊗ ψr−1 ) = 1.
This is proved in [40] in the case Sp, but the proof works equally well in the case Mp as it uses
only the decomposition Gi = Bi Ki . Therefore we only have to find a function ϕ ∈ I(Ξ) such
that lΞ,ξ,ψ (ϕ)(1) 6= 0 and show that
0
lΞ,ξ,ψ
(ϕ)/lΞ,ξ,ψ (ϕ) = w.
(1)
Let Ki
(0)
be the Iwahori subgroup of Ki . Let Ti
(0)
reduction map Ti → Ti (oF /$). Let B i = Ti N i be the opposite
(1)
(1)
(1)
(1)
(1)
−1
N i = N i ∩ Ki . Let Ni = wi,long
N i wi,long . Let Nr−1 = Nr−1 ∩
Let ϕ = pr2 (1K (1) η ) ∈ Cc∞ (G2 ). Then
2
Z
lΞ,ξ,ψ (1K (1) η )(1) =
2
(1)
K2
33
(1)
= Ti (oF ) and Ti
YΞ,ξ,ψ (k2 η)dk2 .
be the kernel of the
Borel subgroup of Gi and
(1)
N2 .
(1)
By the Iwahori decomposition of K2 , it is not hard to check that
(1)
(0)
(1)
(1)
(1)
K2 η = T2 N (0) w2,long h(l0∗ , 0)T0 NJ Nr−1 .
(4.2.2)
(1)
Therefore YΞ,ξ,ψ (k2 η) = YΞ,ξ,ψ (η) = 1 for any k2 ∈ K2 . Thus
(1)
lΞ,ξ,ψ (1K (1) η )(1) = vol K2 .
2
0
We now compute lΞ,ξ,ψ
(pr2 (1K (1) η ))(1). First
2
Z
Z
1/2
1K (1) η (b2 g1 ẇu)(Ξ−1 δ2 )(b2 )ψr−1 (u)db2 du,
Fψ0 (pr2 (1K (1) η ))(g1 ) =
2
Nr−1
B2
g1 ∈ G1 .
2
(1)
(1)
By the decomposition (4.2.2) again, for any u ∈ Nr−1 , if b2 g1 ẇu ∈ K2 η, then u ∈ Nr−1 and
(1)
b2 g1 ẇ ∈ K2 η. Therefore
(1)
Fψ0 (pr2 (1K (1) η ))(g1 ) = vol Nr−1 ·
Z
2
(1)
1/2
B2
Z
= vol Nr−1 ·
1K (1) η (b2 g1 ẇ)(Ξ−1 δ2 )(b2 )db2
2
1/2
B1
1K (1) η (b1 g1 ẇ)((Ξ1 )−1 δ2 )(b1 )db1
2
Thus
(1)
0
(pr2 (1K (1) η ))(1) = vol Nr−1 ·
lΞ,ξ,ψ
2
(1)
Z
G1
1K (1) η (g1 ẇ)YΞ1 ,ξ,ψ (g1 ẇ)dg1
2
(1)
= vol Nr−1 · vol K1 .
(1)
(1)
(1)
The lemma then follows since vol Nr−1 · vol K1 = w vol K2 .
Lemma 4.2.3. We have
SΞ,ξ,ψ (1) =
∆G2
ζ(Ξ, ξ, ψ),
∆T2 ∆T0
SΞ0 ,ξ,ψ (1) =
∆G0
ζ(Ξ0 , ξ, ψ).
∆2T0
Proof. We claim that the restriction of the measure dg to the open subset B2 ηBJ Nr−1 decomposes as
∆G2
db2 dnr−1 dbJ ,
∆T2 ∆T0
where dbJ = db0 dldt if bJ = b0 h(l, t). In fact, on the one hand,
Z
∆G2
(1)
1K (1) η (g)dg = [K2 : K2 ]−1 = q − dim G2 +dim N2 +dim T2
.
∆T2
2
G2
dg|B2 ηBJ Nr−1 =
On the other hand, it follows from (4.2.2) that
Z Z
Z
1K (1) η (b2 ηbJ nr−1 )db2 dnr−1 dbJ = q − dim T0 −dim NJ −dim Nr−1 ∆T0 .
B2
Nr−1
BJ
2
The claim then follows. Therefore
Z Z
∆G2
−1
lΞ,ξ,ψ (fΞ )(gJ ) =
fΞ (ηbJ nr−1 gJ )(ξδJ 2 )(bJ )ψr−1 (nr−1 )dbJ dnr−1 .
∆T2 ∆T0 BJ Nr−1
34
We have
SΞ,ξ,ψ (1) =
∆G2
∆T2 ∆T0
Z
Z
L∗
Z
K0
Z
BJ
fΞ (w2,long h(l0∗ , 0)bJ nr−1 h(l∗ , 0)k)
Nr−1
1
(ξ −1 δJ2 )(bJ )ψr−1 (nr−1 )fξ,ψ (h(l∗ , 0)k)dnr−1 dbJ dkdl∗ .
We combine the integration over L, K0 and BJ as an integral over J and then conclude that
Z Z
∆G2
fΞ (w2,long h(l0∗ , 0)nr−1 gJ )ψr−1 (nr−1 )fξ,ψ (gJ )dnr−1 dgJ
SΞ,ξ,ψ (1) =
∆T2 ∆T0 J Nr−1
=
∆G2
ζ(Ξ, ξ, ψ).
∆T2 ∆T0
The equality
SΞ0 ,ξ,ψ (1) =
∆G0
ζ(Ξ0 , ξ, ψ)
∆2T0
can be proved similarly. In fact,
dgJ |BJ ηB0 =
∆G0
dbJ db0 .
∆2T0
Therefore
Z Z
SΞ0 ,ξ,ψ (1) =
J
=
=
1KJ (gJ ) 1K0 (g0 )YΞ0 ,ξ,ψ (gJ g0−1 )dgJ dg0
Z Z Z
1KJ (bJ ηb0 g0 ) 1K0 (g0 )YΞ0 ,ξ,ψ (bJ ηb0 )dbJ db0 dg0
G0
∆G0
∆2T0
G0
BJ
B0
∆G0
ζ(Ξ0 , ξ, ψ).
2
∆T0
Proof of Proposition 2.2.3. If r = 0, then Proposition 2.2.3 can be proved in exactly the same
way as [45, Appendix D.3]. We omit the details. See also Lemma 7.2.2.
Assume that r ≥ 1. Suppose that we are in the case Sp. It follows from Lemma 4.2.1 and
Lemma 4.2.3 that
I(1, Ξ, ξ, ψ) =
∆T2
∆G2

−1 ∆T0
∆G0

Lψ ( 21 , π0 × τ )

L(1, σ × τ )L(1, τ, ∧2 )

Y
1
L(1, Ξn+i Ξ−1
n+j )
1≤i<j≤r

Y
Lψ ( 12 , π0∨ × τ ∨ )
1
 I(1, ξ, Ξ0 , ψ).

−1
L(1, σ ∨ × τ ∨ )L(1, τ ∨ , ∧2 )
L(1, Ξn+i Ξn+j )
1≤i<j≤r

Proposition 2.2.3 in the case r ≥ 1 is then reduced to the case r = 0. The case Mp can be
proved in the same way. We only need to change notation at all necessary places.
35
Part 2. Compatibility with Ichino–Ikeda’s conjecture
The notation in this part of the paper is independent from Part I. We keep the notation and
convention in the Introduction. Additional notation will be fixed in each section.
5. Some assumptions and remarks
5.1. Parameters. We will prove that Conjecture 2.3.1(3) is compatible with Ichino–Ikeda’s
conjecture [22, Conjecture 2.1]. The most subtle part is the appearance of the size of the
centralizer of the global L-parameters in the formula. To address this issue, of course, one has
to assume that the Langlands correspondence exists and satisfies some expected properties. We
begin by setting down the precise hypotheses that we require. We remark that for orthogonal
groups and symplectic groups, they follow from the work of Arthur [2]. For metaplectic groups,
they should eventually follow from the on-going work of Wen-Wei Li (e.g. [32]).
We first state the hypothesis on the local Langlands correspondences.
Hypothesis LLC. We assume the Hypothesis (LLC), (Local factors), (Plancherel measures)
from [8, Appendix C] at all non-archimedean places v of F . Thus [8, Theorem C.5] holds if v is
non-archimedean. It also holds if v is archimedean by [36].
We note that if v is an archimedean place, then the Hypothesis (LLC) is established by Langlands [26]. Hypothesis (Local factors) is proved in [31]. Hypothesis (Plancherel measures) is
proved by [1]. If v is non-archimedean, then they should follow from [2, Theorem 1.5.1, Theorem 9.4.1, Conjecture 9.4.2].
Thus, if v is a place of F and πv is an irreducible admissible representation of G(Fv ), where
G = SO(2n + 1) (resp. SO(2n), resp. Sp(2n)) gives rise to a 2n (resp. 2n, resp. 2n + 1)
dimensional selfdual representation Ψπv of the Weil–Deligne group WD(Fv ) of sign −1 (resp.
+1, resp. +1). We call it the local L-parameter of πv .
Let πv be an irreducible admissible genuine representation of Mp(2n)(Fv ) and Θψv (πv ) be
the restriction to SO(V )(Fv ) of its theta lift to O(V )(Fv ) where V is a 2n + 1 dimensional
orthogonal space over Fv of discriminant 1. By [8, Theorem 1.1], the map πv 7→ Θψv (πv ) gives
a bijection between the set of irreducible admissible genuine representations of Mp(2n)(Fv ) and
the union of the sets of irreducible admissible representations of SO(V )(Fv ) where V ranges
over all 2n + 1 dimensional orthogonal spaces over Fv of discriminant 1. This bijection satisfies
several expected properties (c.f. [8, Theorem 1.3] for a list). The local L-parameter of πv is
defined to be ΨΘψv (πv ) . Note that the local L-parameter of πv depends on ψv .
We now turn to the global Langlands correspondences. We shall be concerned only with tempered cuspidal automorphic representations. To avoid mentioning the hypothetical Langlands
group LF , we use the following substitute of the global L-parameters following [2, Section 1.4]
and [7, Section 25, p. 103–105].
36
Let π be an irreducible cuspidal tempered automorphic representation of G(AF ), where G =
SO(2n + 1) (resp. SO(2n), Sp(2n), Mp(2n)). By the global L-parameter of π, we mean the
following data:
• a partition N = N1 + · · · + Nr , where N = 2n (resp. 2n, 2n + 1, 2n);
• a collection of pairwisely inequivalent selfdual irreducible cuspidal automorphic representations Πi of GLNi (AF ) of sign −1 (resp. +1, +1, −1), i = 1, · · · , r,
which satisfy the condition that for all places v of F , Ψπv ' ⊕ri=1 ΨΠi,v as representations of
WD(Fv ), where ΨΠi,v is an Ni dimensional representation of WD(Fv ) associated to Πi,v by
the local Langlands correspondences for GLNi (which is known due to [17] and [19]). By [24,
Theorem 4.4], the global L-parameter of π is unique if it exists. We write formally Ψπ = ri=1 Πi .
We now state the hypothesis on the global Langlands correspondences.
Hypothesis GLC. The global L-parameter of π exists.
For orthogonal and symplectic groups, this follows from [2, Theorem 1.5.2, Theorem 9.5.3].
For metaplectic groups, this should follow from the work of Wen-Wei Li.
With this reformulation of the L-parameter of π, we (re-)define the centralizer
Nr
1
Sπ = SΨπ = {(ai ) ∈ (Z/2Z)r | aN
1 · · · ar = 1}.
From now on, when we speak of the global L-parameters and their centralizers, we always
mean the one defined here.
We end this subsection by some discussions on the automorphic representations on the even
orthogonal groups. Suppose that π is an irreducible cuspidal tempered automorphic representation of O(2n)(AF ). We are interested in the restriction π|SO(2n)(AF ) . We summarize what we
need as the following hypothesis. We remark that all these assertions should follow from [2, Theorem 1.5.2, Theorem 9.5.3] and the discussion in [2, Section 8.3 and 8.4].
Hypothesis O. Each automorphic representation π appears with multiplicity one in the discrete spectrum of O(2n)(AF ). The following three cases exhaust all possibilities of π|SO(2n)(AF ) .
(1) π|SO(2n)(AF ) is irreducible and appears with multiplicity one in the discrete spectrum of
SO(2n)(AF ).
(2) π|SO(2n)(AF ) is irreducible and appears with multiplicity two in the discrete spectrum of
SO(2n)(AF ). In this case, π 6' π ⊗ det and π|SO(2n)(AF ) ⊕ (π ⊗ det)|SO(2n)(AF ) is the
π|SO(2n)(AF ) -isotipic component of the discrete spectrum.
(3) π|SO(2n)(AF ) = π + ⊕ π − where π + and π − are inequivalent automorphic representations
of SO(2n)(AF ). Both π + and π − appear with multiplicity one in the discrete spectrum
of SO(2n)(AF ). Moreover, Ψπ+ = Ψπ− .
In each case, let π 0 be an irreducible component of π|SO(2n)(AF ) . Then we define the Lparameter Ψπ of π by Ψπ = Ψπ0 . Suppose that Ψπ = Π1 · · · Πr where Πi is an irreducible
37
cuspidal automorphic representation of GLNi (AF ). Then in the first (resp. second and third)
case (resp. cases), at least one of Ni ’s is odd (resp. all Ni ’s are even).
Let be the nontrivial outer automorphism of SO(2n) and for any automorphic representation
σ of SO(2n)(AF ), we let σ be its twist by . In the first two cases, (π|SO(2n)(AF ) ) = π|SO(2n)(AF ) .
In the third case, (π ± ) = π ∓ . Here we use “=” to indicate that not only the automorphic
representations are isomorphic, but the spaces on which they realize are the same.
The automorphic representation π appears with multiplicity one in the discrete spectrum of
O(2n)(AF ), so the space on which it realizes is canonical. Suppose that π|SO(2n)(AF ) is irreducible
and appears with multiplicity two in the discrete spectrum of SO(2n)(AF ). The restrictions of π
and π ⊗ det to SO(2n)(AF ) are canonical subspaces of the discrete spectrum of SO(2n)(AF ) and
give a canonical decomposition of the π|SO(2n)(AF ) -isotypic component of the discrete spectrum
of SO(2n)(AF ) (we are not able to distinguish the restrictions of π and π ⊗det). Moreover, these
subspaces are characterized by the fact that they are invariant under the outer twist . In other
words, if π 0 (as an abstract representation) is an automorphic representation of SO(2n)(AF ) and
appears with multiplicity two in the discrete spectrum of SO(2n)(AF ), then there are precisely
two automorphic realizations V1 and V2 of π 0 that are invariant under the outer twist by . Both
V1 and V2 can be extended to automorphic representations of of O(2n)(AF ), say π1 and π2 . Then
π1 ⊗ det = π2 . Moreover, V1 and V2 are orthogonal in the discrete spectrum of SO(2n)(AF ) and
V1 ⊕ V2 is the π 0 -isotypic component of the discrete spectrum of SO(2n)(AF ).
Finally, assume that SO(2n) is quasi-split and π 0 is an irreducible cuspidal tempered generic
automorphic representation of SO(2n)(AF ) which appears with multiplicity two in the discrete
spectrum. Suppose that Ψπ0 = Π1 · · · Πr . Then (at least conjecturally) the descent construction [14] provides us with an automorphic realization of π 0 which is invariant under the
outer twist . We refer the readers to [28, Section 5] for some further discussions on the descent
construction.
Convention. We assume the hypothesis LLC, GLC and O from now on, unless otherwise
specified.
5.2. Theta correspondences. We are going to use the Rallis inner product formula in the
later sections of this paper. We will not recall the precise form of this formula in various cases,
but refer the readers to [47, 48] for the formula in the first term range and to [11] for the formula
in the second term range.
We now consider the behavior of the L-parameters under theta correspondences.
Lemma 5.2.1. Let V be a 2n dimensional orthogonal space over F and π an irreducible cuspidal
tempered automorphic representation of O(V )(AF ). Let Θψ (π) be its theta lift to Sp(2n)(AF )
38
with additive character ψ. Suppose that Θψ (π) is nonzero and cuspidal. Let Ψπ = ri=1 Πi be
the L-parameter of π. Then Πi 6= 1 (the trivial character of A×
F ) for all i.
Proof. Suppose that Πi = 1 for some i. We may assume that i = 1. Then by Hypothesis O,
π|SO(V )(AF ) is irreducible. We prove that π has a nonzero theta lift to Sp(2n − 2)(AF ). The
lemma then follows from the tower property of the theta lift [38].
If π has a nonzero theta lift to Sp(2n − 2r)(AF ) for some r > 1, then by the tower property
of the theta lift, π has a nonzero theta lift to Sp(2n − 2)(AF ). Thus we may assume that π does
not have a nonzero theta lift to any Sp(2n − 2r)(AF ) for any r > 1.
We fix a sufficiently large finite set S of places of F which contains all the archimedean places,
so that if v 6∈ S, then π (hence Πi ) is unramified. By the hypotheses LLC and GLC,
LS (s, π) =
r
Y
LS (s, Πi ),
i=1
where the left hand side is the standard L-function of π defined by the doubling method and
the right hand side is the standard L-function of Πi . If i 6= 1, then LS (s, Πi ) is holomorphic and
does not vanish at s = 1 [25] and LS (s, 1) have a simple pole at s = 1. Therefore LS (s, π) has
a simple pole at s = 1.
Let v be a place of F . By assumption, πv |SO(V )(Fv ) is irreducible. By [8, Theorem C.5], there
is an irreducible admissible representation σ of Sp(2n − 2)(Fv ) such that πv = Θψv (σ). This
means that πv has a nonzero theta lift to Sp(2n − 2)(Fv ).
It then follows from [48, Theorem 10.1] that π has a nonzero theta lift to Sp(2n − 2)(AF ).
This proves the lemma.
Lemma 5.2.2. Let V be a 2n + 1 (resp. 2n) dimensional orthogonal space over F and π be
an irreducible cuspidal tempered automorphic representation of O(V )(AF ). Let Θψ (π) be its
theta lift to Mp(2n)(AF ) (resp. Sp(2n)(AF )) with additive character ψ. Assume that Θψ (π) is
cuspidal and nonzero. Then
ΨΘψ (π) = Ψπ ⊗ χV ,
resp. ΨΘψ (π) = (Ψπ 1) ⊗ χV ,
where 1 stands for the trivial character of A×
F.
Proof. Let v be a place of F . By [8, Theorem C.5] and [12], we see that
ΨΘψv (πv ) = Ψπv ⊗ χV,v ,
resp. ΨΘψv (πv ) = (Ψπv ⊕ 1v ) ⊗ χV,v ,
By the previous lemma, in the case dim V = 2n, Ψπ does not contain 1. The lemma then follows
from [24, Theorem 4.4].
Lemma 5.2.3. Let π be an irreducible cuspidal tempered automorphic representation of O(V )(AF )
where V is a 2n dimensional orthogonal space over F . There is a canonical injective map
39
Sπ → SΘψ (π) . It is not bijective if and only if Ψπ = Π1 · · · Πr where Πi is an irreducible
cuspidal automorphic representation of GLNi (AF ) with Ni being even. In this case, Sπ is an
index two subgroup of SΘψ (π) .
Proof. Suppose Ψπ = ri=1 Πi , where Πi is an irreducible cuspidal automorphic representation
P
of GLNi (AF ) and ri=1 = 2n. By Lemma 5.2.2,
Nr
1
Sπ = {(ai ) ∈ (Z/2Z)r | aN
1 · · · ar = 1},
Nr
1
SΘψ (π) = {(ai ) ∈ (Z/2Z)r+1 | aN
1 · · · ar ar+1 = 1}.
The map (a1 , · · · , ar ) 7→ (a1 , · · · , ar , 1) is clearly injective. It is not bijective if and only if there
Nr
1
are elements (a1 , · · · , ar ) ∈ (Z/2Z)r so that aN
1 · · · ar = −1. This is equivalent to that at least
one of Ni ’s is odd.
6. Ichino–Ikeda’s conjecture for the full orthogonal group
We review in this section the conjecture of Ichino–Ikeda [22] and extend it to the full orthogonal group. The argument is very close to [9, § 2, 3]. Therefore we only state the result when
its proof is identical to that in [9].
6.1. Inner products. Let F be a number field and (U, qU ) be an n-dimensional orthogonal
group over F . Let H = O(U ) and H 0 = SO(U ). Recall that there is an exact sequence
1 → H 0 → H → µ2 → 1.
We view µ2 as an algebraic group over F . We write t for the nonidentity element in µ2 (F ) and
tv its image in µ2 (Fv ) for each place v of F . Note that if n is odd, then we may take t = −1.
The sequence splits canonically and gives an isomorphism H ' H 0 × µ2 .
Let dv be the measure on µ2 (Fv ) so that vol µ2 (Fv ) = 1. Then d =
gawa measure of µ2 (AF ). Let dh and
dh0
respectively. Then we have
Z
Z
v
dv is the Tama-
be the Tamagawa measure of H(AF ) and H 0 (AF )
Z
f (h)dh =
H(F )\H(AF )
Q
µ2 (F )\µ2 (AF )
f (h0 )dh0 d,
H 0 (F )\H 0 (AF )
for all f ∈ L1 (H(F )\H(AF )).
dhv where dhv is a measure on H(Fv ). Let dh0v = 2dhv |H 0 (Fv )
Q
be a measure on H 0 (Fv ). Then dh0 = v dh0v .
We fix a decomposition dh =
Q
v
Let π be an irreducible cuspidal automorphic representation of H(AF ). We denote by V the
space of automorphic functions on which π is realized. Let π 0 = π|H 0 (AF ) and V 0 = {f |H 0 (AF ) |
f ∈ V }. Let S be the set of places v of F such that πv |H 0 (Fv ) is reducible. This is also the set
of places v of F so that πv ⊗ detv ' πv . Let Bπ be the Petersson inner product on V given by
Z
0
Bπ (f, f ) =
f (h)f 0 (h)dh, f, f 0 ∈ V.
H(F )\H(AF )
40
We fix a decomposition Bπ =
Q
v
Bπv where Bπv is an inner product on πv .
We distinguish two cases.
Case I : S = ∅.
In this case, π 0 is irreducible and the restriction to H 0 (AF ) as functions induces an isomorphism V ' V 0 as representations of H 0 (AF ). Let Bπ0 be the Petersson inner product on V 0
(defined using the Tamagawa measure on H 0 (AF )).
Lemma 6.1.1. For any f, f 0 ∈ V , we have
Bπ0 (f |H 0 (AF ) , f 0 |H 0 (AF ) ) = 2Bπ (f, f 0 ).
Proof. This can be proved in the same way as [9, Lemma 2.1].
Case II : S 6= ∅.
We fix an isomorphism

!
V ' lim
−→
S
O
Vv
⊗

O
φv  ,
v6∈S
v∈S
where Vv is the space on which πv is realized and φv is an H(oF,v )-invariant vector in Vv for
v 6∈ S.
If v ∈ S, then πv ⊗ detv 6' πv and πv0 ' πv+ ⊕ πv− where πv± are irreducible admissible
representations of H 0 (Fv ). We have Vv0 ' Vv+ ⊕ Vv− where Vv∗ is the space on which πv∗ are
realized and ∗ = ± or 0. Note that Vv− ' πv (t)Vv+ . For almost all places v ∈ S, we have
−
±
0
±
−
+
φv = φ+
v + φv where φv is an H (oF,v )-invariant element in Vv and φv = πv (tv )φv . If v 6∈ S,
then πv0 is an irreducible admissible representation on the space Vv .
In this case, by the Hypothesis O, there are two irreducible cuspidal automorphic representations π + and π − so that π 0 ' π + ⊕ π − , π − ' π + ◦ Ad t, V 0 = V + ⊕ V − where V ± are the
spaces on which π ± are realized. We may label the two irreducible components of πv0 for v ∈ S
so that
!
π± '
O
πv±

⊗

O
πv0  ,
v6∈S
v∈S








O  O  O 
O ±  


 
Vv 
φ±
φv 
V ± = lim
V
⊗


v
v⊗

⊗

−→
S
v∈S
v∈S
v6∈S
v∈S
v∈S
v6∈S
v6∈S
v6∈S
Let Bπ+ be the Petersson inner product on V + with a fixed decomposition
Y
Y
Bπ+ =
Bπv+
Bπv ,
v6∈S
v∈S
where
41
• Bπv+ is an H 0 (Fv ) invariant pairing on Vv+ if v ∈ S and Bπv is an H(Fv ) invariant pairing
on Vv if v 6∈ S.
+
• Bπv+ (φ+
v , φv ) = Bv (φ, φ) = 1 for almost all v.
If v ∈ S, we define an H 0 (Fv ) invariant pairing Vv− by Bv− (φv , φv ) = Bv+ (πv (tv )φv , πv (tv )φv ).
−
Then for almost all v, we have Bv− (φ−
v , φv ) = 1. We then define an H(Fv ) invariant pairing on
Vv by
1
 (B + (φ+ , φ+ ) + B − (φ− , φ− )), if v ∈ S
v
v
v
v
v
v
\
Bv (φv , φv ) = 2

Bv (φv , φv ),
if v 6∈ S.
Then for almost all v, Bv\ (φv , φv ) = 1.
Lemma 6.1.2.
Bπ =
Y
Bv\ .
v
Proof. This can be proved in the same way as [9, Lemma 2.3].
6.2. Ichino–Ikeda’s conjecture for special orthogonal groups. We review Ichino–Ikeda’s
conjecture [22, Conjecture 2.1] in this subsection. In fact, there is a slight inaccuracy in its
orginal formulation in [22] when the multiplicity of the automorphic representation on the even
orthogonal group in the discrete automorphic spectrum is two. We will make some modifications
to the conjecture in this case.
Let Un+1 and Un be orthogonal spaces of dimension n+1 and n with an embedding Un ⊂ Un+1 .
Let Hi0 = SO(Ui ) (i = n, n + 1). Let dh be the Tamagawa measure on Hn0 (AF ) and we fix a
Q
decomposition dh = v dhv where dhv is a Haar measure on Hn0 (Fv ) and vol Hn0 (oF,v ) = 1 for
almost all v.
Let πn+1 = ⊗v πn+1,v and πn = ⊗v πn,v be irreducible cuspidal tempered automorphic repre0 (A ) and H 0 (A ) respectively. Let V
sentations of Hn+1
n+1 = ⊗v Vn+1,v and Vn = ⊗v Vn,v be
F
F
n
the space on which πn+1 and πn are realized respectively. Let Bπn+1 and Bπn be the Petersson
inner products on Vn+1 (resp. Vn ) respectively. We fix a decomposition
Y
Y
Bπn+1 =
Bπn+1,v , Bπn =
Bπn,v
v
v
where Bπn+1,v and Bπn,v are inner products on Vn+1,v and Vn,v respectively.
0
0
0
Let fn+1 = ⊗fn+1,v , fn+1
= ⊗fn+1,v
∈ Vn+1 and fn = ⊗fn,v , fn0 = ⊗fn,v
∈ Vn . Define
Z
Z
0
0
0
J (fn+1 , fn+1 , fn , fn ) =
fn+1 (h)fn (h)dh ·
fn+1
(h)fn0 (h)dh.
0 (F )\H 0 (A )
Hn
F
n
0 (F )\H 0 (A )
Hn
F
n
For each place v, we define
0
0
Jv (fn+1,v , fn+1,v
, fn,v , fn,v
)
Z
=
0 (F )
Hn
v
0
0
Bn+1,v (πn+1,v (hv )fn+1,v , fn+1,v
)Bn,v (πn,v (hv )fn,v , fn,v
)dhv .
42
Let S be a sufficiently large finite set of places of F containing all archimedean places so
0
0 ) are H 0 (o
0
that if v 6∈ S, then fn+1,v , fn+1,v
(resp. fn,v , fn,v
n+1 F,v ) (resp. Hn (oF,v )) fixed and
0 ) = 1. In particular, π
0
) = Bπn,v (fn,v , fn,v
Bπn+1,v (fn+1,v , fn+1,v
n+1,v and πn,v are both unramified
if v 6∈ S. Let {α1,v , · · · , α[ n+1 ],v } and {β1,v , · · · , β[ n2 ],v } be the Satake parameters of πn+1,v and
2
πn,v respectively. Let
−1
An+1,v = diag[α1,v , · · · , α[ n+1 ],v , α[−1
, · · · , α1,v
]
n+1
],v
2
An,v = diag[β1,v , · · ·
2
, β[ n2 ],v , β[−1
n
],v , · · ·
2
−1
, β1,v
].
Let
LS (s, πn+1 × πn ) =
Y
det(1 − An+1,v ⊗ An,v · qv−s )−1
v6∈S
be the tensor product L-function and LS (s, πn+1 , Ad) and LS (s, πn , Ad) be the adjoint Lfunctions.
Conjecture 6.2.1 (Ichino–Ikeda [22, Conjecture 2.1]). If πn+1 and πn appear with multiplicity
one the discrete spectrum, then
0
J (fn+1 , fn+1
, fn , fn0 ) =
LS ( 12 , πn+1 × πn )
1
∆SH 0
n+1 LS (1, πn+1 , Ad)LS (1, πn , Ad)
|Sπn+1 ||Sπn |
Y
0
0
Jv (fn+1,v , fn+1,v
, fn,v , fn,v
).
v∈S
If n is odd (resp. even) and πn+1 (resp. πn ) appears with multiplicity two in the discrete spectrum
0 (A ) (resp. H 0 (A )) and V
0
of Hn+1
n+1 (resp. Vn ) is invariant under the outer twist of Hn+1
F
F
n
(resp. Hn0 ), then
0
J (fn+1 , fn+1
, fn , fn0 ) =
LS ( 12 , πn+1 × πn )
2
∆SH 0
n+1 LS (1, πn+1 , Ad)LS (1, πn , Ad)
|Sπn+1 ||Sπn |
Y
0
0
Iv (fn+1,v , fn+1,v
, fn,v , fn,v
).
v∈S
Remark 6.2.2. The same inaccuracy also occurs in [34]. One also needs to modify [34, Conjecture 2.5] in a similar way when the automorphic representation on the even orthogonal group
has multiplicity two. In this case, the automorphic realization is required to be -invariant and
(in the notation of [34]) 1/|SΨ(π2 ) ||SΨ(π0 ) | needs to be replaced by 2/|SΨ(π2 ) ||SΨ(π0 ) |.
6.3. Ichino–Ikeda’s conjecture for full orthogonal groups. Let Un+1 and Un be orthogonal
spaces of dimension n + 1 and n with an embedding Un ⊂ Un+1 . Let Hi = O(Ui ) and Hi0 =
SO(Ui ) (i = n, n + 1). Let dh be the Tamagawa measure on Hn (AF ) and we fix a decomposition
Q
dh = v dhv where dhv is a Haar measure on Hn (Fv ) and vol Hn (oF,v ) = 1 for almost all v.
Let πn+1 = ⊗v πn+1,v and πn = ⊗v πn,v be irreducible cuspidal tempered automorphic representations of Hn+1 (AF ) and Hn (AF ) respectively. Let Vn+1 = ⊗v Vn+1,v and Vn = ⊗v Vn,v be
43
the space on which πn+1 and πn are realized respectively. Let Bπn+1 and Bπn be the Petersson
inner products on Vn+1 (resp. Vn ) respectively. We fix a decomposition
Y
Y
Bπn,v
Bπn+1,v , Bπn =
Bπn+1 =
v
v
where Bπn+1,v and Bπn,v are inner products on Vn+1,v and Vn,v respectively.
Let fn+1 = ⊗fn+1,v ∈ Vn+1 and fn = ⊗fn,v ∈ Vn . Define
Z
Z
fn+1 (h)fn (h)dh ·
I(fn+1 , fn ) =
Hn (F )\Hn (AF )
For each place v, we define
Z
Iv (fn+1,v , fn,v ) =
fn+1 (h)fn (h)dh.
Hn (F )\Hn (AF )
Bn+1,v (πn+1,v (hv )fn+1,v , fn+1,v )Bn,v (πn,v (hv )fn,v , fn,v )dhv .
Hn (Fv )
Let S be a sufficiently large finite set of places of F containing all archimedean places so that if
v 6∈ S, then fn+1,v (resp. fn,v ) is Hn+1 (oF,v ) (resp. Hn (oF,v )) fixed and Bπn+1,v (fn+1,v , fn+1,v ) =
Bπn,v (fn,v , fn,v ) = 1. In particular, πn+1,v and πn,v are both unramified if v 6∈ S. Let {α1,v , · · · , α[ n+1 ],v }
2
and {β1,v , · · · , β[ n2 ],v } be the Satake parameters of πn+1,v and πn,v respectively. Let
−1
]
An+1,v = diag[α1,v , · · · , α[ n+1 ],v , α[−1
, · · · , α1,v
n+1
],v
2
An,v = diag[β1,v , · · ·
2
, β[ n2 ],v , β[−1
n
],v , · · ·
2
−1
].
, β1,v
Let
LS (s, πn+1 × πn ) =
Y
det(1 − An+1,v ⊗ An,v · qv−s )−1
v6∈S
be the tensor product L-function and LS (s, πn+1 , Ad) and LS (s, πn , Ad) be the adjoint Lfunctions.
The Ichino–Ikeda’s conjecture for the full orthogonal group is the following.
Conjecture 6.3.1. We have
I(fn+1 , fn ) =
Y
LS ( 21 , πn+1 × πn )
2γ
∆SHn+1 S
Iv (fn+1,v , fn,v ).
|Sπn+1 ||Sπn |
L (1, πn+1 , Ad)LS (1, πn , Ad)
v∈S
where γ is given as follows. Suppose n is even (resp. odd). Let Ψπn = Πi (resp. Ψπn+1 = Πi )
where Πi is an irreducible cuspidal automorphic representation of GLNi (AF ). Then γ = 0 (resp.
1) if at least one of Ni ’s is odd (resp. all Ni ’s are even).
Proposition 6.3.2. Conjecture 6.3.1 follows from Conjecture 6.2.1.
Proof. We assume that n is odd. The case n being even can be handled similarly, with modifi0 is irreducible for all places
cations of notation at various places. Then Hn ' Hn0 × µ2 . So πn,v
0
v of F . Let S be the set of places of F such that πn+1,v
is reducible.
44
If v 6∈ S, then fn+1,v = φn+1,v is fixed by Hn+1 (oF,v ) and fn,v is fixed by Hn0 (oF,v ). We may
+
+
further assume that fn+1,v = fn+1,v
∈ Vn+1,v
if v ∈ S ∩ S. Thus
Y
Y
Y
+
fn+1,v =
fn+1,v
fn+1,v
φn+1,v .
v∈S,v6∈S
v∈S∩S
v6∈S
Put
S 0 = S\(S ∩ S),
s = |S ∩ S|,
s0 = |S 0 |.
If S 6= ∅, then
Z
fn+1 (h)fn (h)dh
Hn (F )\Hn (AF )
=
=
=
1
2s+s0 +1
fn+1 (h)fn (h)dh
0
0
∈µ2 (FS ) Hn (F )\Hn (AF )
1
Z
X
2s+s0 +1
1
(fn+1 (h)fn (h) + fn+1 (ht)fn (ht))dh
0
0
∈µ2 (FS 0 ) Hn (F )\Hn (AF )
X
2s+s0
Z
X
Z
fn+1 (h)fn (h)dh.
0
0
∈µ2 (FS 0 ) Hn (F )\Hn (AF )
If S = ∅, then
Z
fn+1 (h)fn (h)dh =
Hn (F )\Hn (AF )
1
2s+1
X
Z
fn+1 (h)fn (h)dh.
0
0
∈µ2 (FS ) Hn (F )\Hn (AF )
We fix a decomposition
Bπ+
n+1
=
Y
Y
Bπ+
n+1,v
,
Bπn+1,v
0
=2
resp. Bπn+1
0
Bπn+1,v
0
v
v6∈S
v∈S
Y
if v 6∈ S).
if S 6= ∅ (resp. S = ∅), so that Bπn+1,v = Bπ\ n+1,v if v ∈ S (resp. Bπn+1,v = Bπn+1,v
0
We fix a decomposition
Bπn0 = 2
Y
Bπn,v
0 ,
v
so that Bπn,v = Bπn,v
0 .
Then Conjecture 6.2.1 implies that
0
LS ( 12 , πn+1 × πn )
2m+γ
I(fn+1 , fn ) = 2s+2s0
∆SHn+1 S
L (1, πn+1 , Ad)LS (1, πn , Ad)
2
|Sπn+1 ||Sπn |
X
Y
Jv (πn+1 ()fn+1,v , πn+1 (0 )fn+1,v , πn ()fn,v , πn (0 )fn,v ),
,0 ∈µ2 (FS 0 ) v∈S
where
0
– γ 0 = 1 (resp. 0) if πn+1
is reducible (resp. irreducible).
0
– m = 1 (resp. 0) if πn+1
is irreducible and appears with multiplicity two (resp. any irre0 (A ).
ducible constituent appears with multiplicity one) in the discrete spectrum of Hn+1
F
45
We note that γ = m + γ 0 . In fact, in the first (resp. second, resp. third) case in Hypothesis
O, we have γ = γ 0 = m = 0 (resp. γ = 1, m = 1, γ 0 = 0, resp. γ = 1, m = 0, γ 0 = 1).
Therefore to deduce Conjecture 6.3.1 from Conjecture 6.2.1, we only need to prove the following
two identities. If v ∈ S, then
1
Jv (fn+1,v , fn+1,v , fn,v , fn,v ) = Iv (fn+1,v , fn,v ).
4
If v 6∈ S, then
X
1
4
Jv (πn+1 ()fn+1,v , πn+1 (0 )fn+1,v , πn ()fn,v , πn (0 )fn,v ) = Iv (fn+1,v , fn,v ).
,0 ∈µ2 (Fv )
These two identities can be proved in the same way as [22, Lemma 3.4]. Therefore Conjecture 6.3.1 follows from [22, Conjecture 2.1].
7. Compactibility with Ichino–Ikeda’s conjecture: Sp(2n) × Mp(2n)
7.1. The Theorem. The goal of this section is to study Conjecture 2.3.1 for Sp(2n) × Mp(2n).
We are going to show that Conjecture 2.3.1 is compatible with Ichino–Ikeda’s conjecture for
SO(2n + 1) × SO(2n) in some cases. Result of this sort for unitary groups appeared in [45,
Proposition 1.4.1]. The local counterpart of this argument has been used to establish the local
Gan–Gross–Prasad conjecture for the Fourier–Jacobi models [4, 10].
Let λ ∈ F × . Let (V, qV ) be a 2n + 1-dimensional orthogonal space and Vλ is a 2n-dimensional
subspace such that Vλ⊥ is a one dimensional orthogonal space of discriminant λ. Let H = O(V )
and Hλ = O(Vλ ) and ιλ : Hλ → H be the natural embedding.
e = Mp(W ). Let Ω
e ψ (resp.
Let W be a 2n-dimensional symplectic space and G = Sp(W ), G
e F ) × H(AF ) (resp. G(AF ) × Hλ (AF )) which is realized
Ωψ ) be the Weil representation of G(A
e F ) realized on
on S(V (AF )n ) (resp. S(Vλ (AF )n )). Let ωψλ be the Weil representation of G(A
S(AnF ). Then we have the theta series
e ψ (e
Θ
g , h, Φ),
Θψ (g, hλ , Φλ ),
θψλ (e
g , φ)
e F ) × H(AF ), G(A
e F ) × Hλ (AF ) and G(AF ) respectively, where Φ ∈ S(V (AF )n ), Φλ ∈
on G(A
S(Vλ (AF )n ) and φ ∈ S(AnF ).
e F ). Let
Let π be an irreducible cuspidal tempered genuine automorphic representation of G(A
e ψ (σ) be the theta lift of π to H(AF ), i.e. the automorphic representation generated by the
Θ
functions of the form
Z
e ψ (g, ·, Φ)dg,
ϕ(g)Θ
e ψ (ϕ, Φ)(·) =
Θ
ϕ ∈ π, Φ ∈ S(V (AF )n ).
G(F )\G(AF )
Let σ be an irreducible cuspidal tempered automorphic representation of Hλ (AF ). Let Θψ (σ)
be the theta lift of σ to G(AF ), i.e. the automorphic representation generated by the functions
46
of the form
Z
f (hλ )Θψ (·, hλ , Φλ )dhλ ,
Θψ (f, Φλ )(·) =
f ∈ σ, Φλ ∈ S(Vλ (AF )n ).
Hλ (F )\Hλ (AF )
e ψ (π) and Θψ (σ) are both cuspidal (possibly zero). If ConjecTheorem 7.1.1. Suppose that Θ
−1
e
ture 6.3.1 holds for (Θψ (π), σ), then Conjecture 2.3.1(3) holds for (π, Θψ (σ)) with the additive
−1
character ψ−λ .
Remark 7.1.2. We have shown in Proposition 6.3.2 that Conjecture 6.3.1 can be deduced from
the original conjecture of Ichino–Ikeda (Conjecture 6.2.1). The theorem thus says that Conjecture 2.3.1(3) and Ichino–Ikeda’s conjecture are compatible in this situation. The same remark
also applies to Theorem 8.1.1 in the next section.
7.2. A seesaw diagram. The proof of Theorem 7.1.1 is very similar to [45, Proposition 1.4.1].
It makes use of the following seesaw diagram.
e
G×G
H
e
G
Hλ × O(Vλ⊥ )
Suppose that f = ⊗fv ∈ σ, ϕ = ⊗ϕv ∈ π, Φλ = ⊗Φλ,v ∈ S(Vλ (AF )n ) and φ = ⊗φv ∈ S(AnF )
are all factorizable.
Lemma 7.2.1. We have
Z
FJ ψ−λ (ϕ, Θψ (f, Φλ ), φ) =
e ψ (ϕ, Φλ ⊗ φ)(ιλ (h))dh.
f (h)Θ
−1
Hλ (F )\Hλ (AF )
Proof.
FJ ψ−λ (ϕ, Θψ (f, Φλ ), φ)
Z
Z
=
G(F )\G(AF )
ϕ(g)f (h)Θψ (g, h, Φλ )θψ−λ (g, φ)dhdg
Hλ (F )\Hλ (AF )
Z
Z
e ψ (g, ιλ (h), Φλ ⊗ φ)f (h)dgdh
ϕ(g)Θ
=
Hλ (F )\Hλ (AF )
Z
=
G(F )\G(AF )
e ψ (ϕ, Φλ ⊗ φ)(ιλ (h))dh.
f (h)Θ
−1
Hλ (F )\Hλ (AF )
Let v be a place of F . We use B to denote the inner products on various unitary representations.
47
Lemma 7.2.2. The integral
Z
Z
B(σv (h)fv , fv )B(Ωψv (g, h)Φλ,v , Φλ,v )B(πv (g)ϕv , ϕv )B(ωψ−λ,v (g)φv , φv )dgdh
Hλ (Fv )
G(Fv )
is absolutely convergent.
Proof. To simplify notation, we suppress the subscript v from the notation in the proof. Put

 1,
|x| ≤ 1;
Υ(x) =
 |x|−1 , |x| > 1.
It is enough to prove that the double integral
Z
Z
n r
2n+1 Y Y
−1
−1
M
M
Υ(ai b−1
(7.2.1)
δHλ2 (b)δG 2 (a)|a1 · · · an | 2
j )ς(a) ς(b) dadb
A+
H
λ
A+
G
i=1 j=1
is convergent, where M is some positive real number, r is the Witt index of Vλ and
+
−1
a = diag[a1 , · · · , an , a−1
n , · · · , a1 ] ∈ AG ,
+
−1
b = diag[b1 , · · · , br , 1, · · · , 1, b−1
r , · · · , b1 ] ∈ AHλ .
We assume that r < n. The case r = n is very similar and needs only a slight modification.
We left it to the interested readers.
We have |b1 | ≤ · · · ≤ |br | ≤ 1. Let j = (j1 , · · · , jr ) be r nonnegative integers such that
+
j1 + · · · + jr ≤ n and let Ij be the subset of A+
G × AHλ consisting of elements
a1 ≤ · · · ≤ aj1 ≤ b1 ≤ aj1 +1 ≤ · · · ≤ aj1 +j2 ≤ b2 ≤ · · · ≤ br ≤ aj1 +···+jr +1 ≤ · · · ≤ an ≤ 1.
S
+
Then A+
j Ij . Thus it is enough to prove the convergence of (7.2.1) when the domain
G × AH λ =
is replaced by Ij .
Over the region Ij , the integrand of (7.2.1) equals
1
|a1 | 2 · · · |aj1 |
2j1 +1
2
|b1 |−j1 +1 |aj1 +1 |
· · · |br |−j1 −···−jr +r |aj1 +···+jr +1 |
2j1 +1
2
· · · |aj1 +j2 |
2(j1 +···+jr )+1−2r
2
2j1 +2j2 −3
2
· · · |an |
|b2 |−j1 −j2 +2
2n−1−2r
2
.
Then lemma then follows from the following elementary fact.
Fact. Fix D a positive real number. The integral
!D
Z
s
X
|x1 |n1 −1 · · · |xs |ns −1 −
log|xi |
dx1 · · · dxs
|x1 |≤···≤|xs |≤1
i=1
is convergent if n1 + · · · nt > 0 for all 1 ≤ t ≤ s.
We now prove Theorem 7.1.1. Let S be a sufficiently large finite set of places of F , such that
if v 6∈ S, then the following conditions hold.
(1) v is non-archimedean, 2 and λ are in o×
F,v , the conductor of ψv if oF,v .
(2) The group A is unramified with a hyperspecial subgroup A(oF,v ), where A = H, Hλ , G.
48
(3) fv is Hλ (oF,v ) fixed and ϕv is G(oF,v ) fixed. Moreover B(fv , fv ) = B(ϕv , ϕv ) = 1.
(4) Φλ is the characteristic function of Vλ (oF,v )n and φv is the characteristic function of onF,v .
(5) The volume of the hyperspecial subgroup KAv is 1 under the chosen measure on A(Fv ),
where A = G, G0 , H, Hλ .
e ψ−1 (π) 6= 0. If this is not the case, it follows from the computation
We may assume that Θ
below that both sides of Conjecture 2.3.1(3) vanish. Applying Conjecture 6.3.1 and the Rallis
e to H), we get
inner product formula (for theta lifting from G
|FJ ψλ (ϕ, Θψ (f, Φλ ), φ)|2 =
e ψ (π) × σ)
LSψ−1 ( 12 , π × χV )
LS ( 12 , Θ
2γ−1 ∆SH
−1
Qn
S
e ψ (π), Ad)LS (1, σ, Ad)
|SΘ
e ψ (π) ||Sσ | LS (1, Θ
i=1 ζF (2i)
−1
−1
YZ
v∈S
Hλ (Fv )
Z
B(σv (h)fv , fv )B(Ωψv (g, h)Φλ,v , Φλ,v )B(πv (g)ϕv , ϕv )B(ωψ−λ,v (g)φv , φv )dgdh,
G(Fv )
where γ is described as in Conjecture 6.3.1. The double integral is absolutely convergent by
Lemma 7.2.2. Thus we can change the order of integration by integrating over g ∈ G(Fv ) first.
Then we apply Rallis inner product formula (for theta lifting from Hλ to G), and get
−1
e ψ (π) × σ)
LS ( 12 , Θ
2γ−1 ∆SH
LS (1, σ)
−1
2
Qn
|FJ ψλ (ϕ, ξ, φ)| =
·
S
e ψ (π), Ad)LS (1, σ, Ad)
|SΘ
e ψ (π) ||Sσ | LS (1, Θ
i=1 ζF (2i)
−1
−1
LSψ−1 ( 12 , π × χV ) Y Z
Qn
B(Θψv (σv )(g)ξv , ξv )B(πv (g)ϕv , ϕv )B(ωψ−λ (g)φv , φv )dg,
S
i=1 ζF (2i)
v∈S Hλ (Fv )
where ξ = ⊗ξv ∈ Θψ (σ). By Lemma 5.2.3, |SΘ
eψ
−1
(π)
||Sσ | = 2γ−1 |Sπ ||SΘψ (σ) |. Theorem 7.1.1
then follows from Lemma 5.2.2.
7.3. Some remarks. We end this section by some remarks on Theorem 7.1.1.
Remark 7.3.1. We have proved in the theorem that we can deduce Conjecture 2.3.1(3) from
Conjecture 6.3.1 under the assumptions of the theorem. Similarly, we may also deduce Conjecture 6.3.1 from Conjecture 2.3.1(3). We only need to run the above argument backwards.
Remark 7.3.2. Instead of the seesaw diagram that has been used in the proof of Theorem 7.1.1,
we may consider the following seesaw diagram.
Mp(2n) × Mp(2n)
O(2n + 2)
Sp(2n)
O(2n + 1) × O(1)
Then we can go back and forth between Conjecture 2.3.1(3) for Sp(2n) × Mp(2n) and Ichino–
Ikeda’s conjecture for SO(2n + 2) × SO(2n + 1).
49
In particular, if n = 1, then the Ichino–Ikeda’s conjecture, hence Conjecture 6.3.1 is known.
In this case, without assuming Hypothesis LLC, GLC and O, [37, Theorem 4.5] proved Conjecture 2.3.1(3) (under some hypothesis) with |Sπ2 ||Sπ0 | replaced by 14 . This result is compatible
with our conjecture if we assume LLC, GLC and O.
Remark 7.3.3. Instead of the seesaw diagrams above, we may consider
, Mp(2n) × Mp(2n)
Mp(2n) × Sp(2n)
O(2n + 2r + 1)
Mp(2n)
O(2n + 2r) × O(1)
Sp(2n)
O(2n + 2r)
.
O(2n + 2r − 1) × O(1)
In this way, the Conjecture 2.3.1(3) for tempered representations on Sp(2n) × Mp(2n) will
be related to the Ichino–Ikeda’s conjecture for nontempered representations. Ichino [20] and
Ichino–Ikeda [21] made use of the following seesaw diagrams respectively.
f
SL(2) × SL(2)
O(5)
f
SL(2)
O(4) × O(1)
,
f
f
SL(2)
× SL(2)
O(6)
SL(2)
O(5) × O(1)
At this moment, there is no precise form of the refined Gan–Gross–Prasad conjecture for nontempered representations. We hope that Conjecture 2.3.1(3) together with the seesaw diagrams
as above would shed some light on the formulation this conjecture.
8. Compactibility with Ichino–Ikeda’s conjecture: Sp(2n + 2) × Mp(2n)
8.1. The theorem. The goal of this section is to study Conjecture 2.3.1(3) for Sp(2n + 2) ×
Mp(2n).
Let W be an 2n + 2 dimensional symplectic space and G = Sp(W ). We choose a basis
{e1 , · · · , en+1 , e∗1 , · · · , e∗n+1 } of W so that symplectic form on W is given by the matrix
1n
−1n
!
.
Let X = hen+1 i, X ∗ = he∗n+1 i and W0 = he1 , · · · , en , e∗1 , · · · , e∗n i. With this choice of basis, we
identify W with F 2n+2 and W0 with F 2n . Let L = he1 , · · · , en i ' F n and L∗ = he∗1 , · · · , e∗n i '
F n . Then W0 = L + L∗ is a complete polarization of W0 . We represent elements in G as
matrices.
50
Let R = R(W0 ) = N G0 be the Jacobi group associated to W0 , where N is the unipotent
radical and G0 ' Sp(W0 ). The group R takes the form


x



where x, y ∈
F n,
κ ∈ F and
a b


a
b


 1

κ




tx
d 
 c

1
1
ty
1n
1
y
1n
!
c d
∈ G0 . We write the first matrix as n = n(x, y, κ). Let
f0 = Mp(W0 ) and R
e = RG
f0 .
G
Let (V, qV ) be a 2n + 2 dimensional orthogonal space and H = O(V ). Let λ ∈ F × and vλ0 ∈ V
such that qV (vλ0 , vλ0 ) = λ. Let Vλ be the orthogonal complement of hvλ0 i and Hλ = O(Vλ ).
e F ) which is realized on S(An ). Let Ωψ be the
Let ωψλ be the Weil representation of R(A
F
Weil representation of G(AF ) × H(AF ) which is realized on S(V (AF )n+1 ). Let Ω0ψ be the
e ψ be the Weil
Weil representation of G0 (AF ) × H(AF ) which is realized on S(V (AF )n ). Let Ω
f0 (AF ) × Hλ (AF ) which is realized on S(Vλ (AF )). Suppose that φ ∈ S(An )
representation of G
F
e ∈ S(Vλ (AF )n )). Then we have the theta series on R(A
e F)
(resp. Φ ∈ S(V n+1 (AF )), resp. Φ
f0 (AF ) × Hλ (AF ))
(resp. G(AF ) × H(AF ), resp. G
θψλ (r, φ),
resp. Θψ (g, h, Φ),
e ψ (e
e
resp. Θ
g , hλ , Φ).
Let π be an irreducible cuspidal tempered automorphic representation of H(AF ). We denote
by Θψ (π) the global theta lifting of π to G(AF ), i.e. the automorphic representation of G(AF )
generated by the functions of the form
Z
Θψ (f, Φ)(·) =
f (h)Θψ (·, h, Φ)dh,
f ∈ π, Φ ∈ S(V (AF )n+1 ).
H(F )\H(AF )
f0 (AF ) and
Let σ be an irreducible cuspidal tempered genuine automorphic representation of G
e ψ (σ) be the theta lifting of σ to Hλ (AF ), i.e. the automorphic representation of Hλ (AF )
Θ
generated by the functions of the form
Z
e ψ (g, ·, Φ)dg.
e
ϕ(g)Θ
e ψ (ϕ, Φ)(·)
e
Θ
=
G0 (F )\G0 (AF )
e ψ (σ) are both cuspidal. If Conjecture 6.3.1 holds for
Theorem 8.1.1. Assume that Θψ (π) and Θ
e ψ (σ)), then Conjecture 2.3.1(3) holds for (Θψ (π), σ) (with the additive character ψλ ). In
(π, Θ
particular, if n = 1, then Conjecture 2.3.1(3) holds for (Θψ (π), σ) (with the additive character
ψλ ).
51
The proof of this theorem will occupy the following four subsections. The last assertion follows
from the fact that Ichino–Ikeda’s conjecture is known for SO(4) × SO(3). Thus Conjecture 6.3.1
holds for O(4) × O(3).
e ψ (σ) is not zero. In fact, if Θ
e ψ (σ) is zero, then it follows
Remark 8.1.2. We don’t assume that Θ
from the computation below that both sides of the identity in Conjecture 2.3.1(3) are zero.
Remark 8.1.3. By assumption, there is a vλ0 ∈ V such that qV (vλ0 , vλ0 ) = λ. If follows from
the computation below that if such a vλ0 does not exist, then both sides of the identity in
Conjecture 2.3.1(3) are zero.
8.2. Measures. Without saying to the contrary, we always take the Tamagawa measure on
the group of adelic points of an algebraic group. Note that vol A(F )\A(AF ) = 1 where A =
f0 (AF ) = 1. Suppose that A = G, G0 , H, Hλ or G
f0 .
G, G0 , H, Hλ . Note also that vol G0 (F )\G
Q
We fix a decomposition dg = v dgv where dgv is a measure on A(Fv ) so that for almost all
places v, vol Kv = 1 where Kv = A(oF,v ) is a hyperspecial maximal compact subgroup of A(Fv ).
Lemma 8.2.1. Let f ∈ S(V (AF )). Then
!
Z
Z
Z
f (v)ψ(κqV (v, v))dv ψ(−λκ)dκ =
(8.2.1)
AF
V (AF )
f (h−1 vλ0 )dh.
Hλ (AF )\H(AF )
Proof. Suppose that V is not a four dimensional split quadratic space. Then the lemma follows
(s)
from the Siegel–Weil formula for SL2 ×H. Let E(g, Φf ) be the Eisenstein series on SL2 (AF )
(s)
SL2 (AF )
where Φf ∈ IndB
χV |·|s is the Siegel–Weil section where B is the standard upper triangular
(s)
Borel subgroup of SL2 . Then the left hand side of (8.2.1) is the ψλ -Fourier coefficient of E(g, Φf )
at s = s0 = n. The right hand side of (8.2.1) is the ψλ -Fourier coefficient of the theta integral
Z
θψ (g, h, f )dh,
H(F )\H(AF )
where θψ (g, h, f ) is the theta series on SL2 (AF ) × H(AF ). The lemma then follows from the
(convergent) Siegel–Weil formula
(s)
E(g, Φf )|s=s0
Z
=
θψ (g, h, f )dh.
H(F )\H(AF )
Suppose that V is split and dim V = 4. Without loss of generality, we may assume that λ = 1.
Then V is identified with the space of 2 × 2 matrices over F and the quadratic form is given
by the determinant. We may assume v10 = 12 ∈ V . Under this identification, H1 (AF )\H(AF ) is
identified with SL2 (AF ) and the quotient measure is identified with the Tamagawa measure on
SL2 (AF ). This is because the volume of H(F )H1 (AF )\H(AF ) equals one.
52
We write an element in V as
Z
AF
x1 x2
x3 x4
!
. The left hand side of the desired identity equals
Z
A4F
f (x1 , x2 , x3 , x4 )ψ(κ(x1 x4 − x2 x3 ) − κ)dx1 dx2 dx3 dx4 dκ.
By the Fourier inversion formula, it equals
Z
Z
A2F
f (x01 + ax3 , x02 + ax4 , x3 , x4 )dadx3 dx4 ,
AF
where (x01 , x02 ) ∈ A2F is a fixed vector of norm one and perpendicular to (x3 , x4 ) under the usual
Euclidean inner product on A2F . The choice of (x01 , x02 ) is not unique, but the above formula
does not depend on the choice. The measure dadx3 dx4 gives a measure on SL2 (AF ) which is
invariant under the right multiplication of SL2 (AF ). It is clear that it gives SL2 (F )\ SL2 (AF )
volume one, hence it is the Tamagawa measure on SL2 (AF ). The lemma then follows.
8.3. Global Fourier–Jacobi periods of Theta liftings. The goal of this subsection is to
compute
Z
Z
Z
f (h)Θψ (ng, h, Φ)θψλ (ng, φ)ϕ(g)dhdndg.
(8.3.1)
G0 (F )\G0 (AF )
N (F )\N (AF )
H(F )\H(AF )
Suppose n = n(x, y, κ), x = (x1 , · · · , xn ) ∈ (F \AF )n and y = (y1 , · · · , yn ) ∈ (F \AF )n . By
definition, we have
X
θψλ (ng, φ) =
ωψλ (g)φ(l1 + x1 , · · · , ln + xn )ψ(λy1 (x1 + 2l1 ) + · · · + λyn (xn + 2ln ) + λκ).
l1 ,··· ,ln ∈F
Suppose Φ = Φ0 ⊗ Φn+1 where Φ0 ∈ S(V (AF )n ) and Φn+1 ∈ S(V (AF )). Then we have
Θ(ng, h, Φ) =
X
Ω0ψ (g)Φ0 (h−1 (v1 + x1 vn+1 ), · · · , h−1 (vn + xn vn+1 ))Φn+1 (h−1 vn+1 )
v1 ,··· ,vn ,vn+1 ∈V
ψ(2y1 qV (v1 , vn+1 ) + · · · + 2yn qV (vn , vn+1 ) + (κ + y t x)qV (vn+1 , vn+1 )).
Therefore
Z
Θ(ng, h, Φ)ψλ (κ)dκ =
F \AF
X
Ω0ψ (g)Φ0 (h−1 (v1 + x1 vn+1 ), · · · , h−1 (vn + xn vn+1 ))
v1 ,··· ,vn ∈V
qV (vn+1 ,vn+1 )=λ
Φn+1 (h−1 vn+1 )ψ(2y1 qV (v1 , vn+1 ) + · · · 2yn qV (vn , vn+1 ) + y t xλ)).
53
From this we get
Z
Θ(ng, h, Φ)θψλ (nι(g), φ)dn
N (F )\N (AF )
Z
X
=
(F \AF
v1 ,··· ,vn ∈V
qV (vn+1 ,vn+1 )=λ
l1 ,··· ,ln ∈F
)2n
Ω0ψ (g)Φ0 (h−1 (v1 + x1 vn+1 ), · · · , h−1 (vn + xn vn+1 ))Φn+1 (h−1 vn+1 )
ωψλ (ι(g))φ(l1 + x1 , · · · , ln + xn )ψ(2y1 (qV (v1 , vn+1 ) − l1 λ) + · · · 2yn (qV (vn , vn+1 ) − ln λ)))dxdy.
Let Λλ = {v ∈ V | qV (v, v) = λ}. Then the group H(F ) acts transitively on Λλ (F ) and
identifies Hλ (F )\H(F ) with Λλ (F ) by h 7→ h−1 vλ0 . It follows that
Z
Z
X Z
f (h)
(8.3.1) =
v1 ,··· ,vn ∈V
l1 ,··· ,ln ∈F
(F \AF )2n
G0 (F )\G0 (AF )
Hλ (F )\H(AF )
Ω0ψ (g)Φ0 (h−1 (v1 + x1 vλ0 ), · · · , h−1 (vn + xn vλ0 ))Φn+1 (h−1 vλ0 )ωψλ (g)φ(l1 + x1 , · · · , ln + xn )
ψ(2y1 (qV (v1 , vλ0 ) − l1 λ) + · · · 2yn (qV (vn , vλ0 ) − ln λ)))ϕ(g)dhdgdxdy.
Then
(8.3.1) =
X
Z
Z
Z
Z
f (hλ h)
(F \AF )2n
v1 ,··· ,vn ∈V
l1 ,··· ,ln ∈F
G0 (F )\G0 (AF )
Hλ (AF )\H(AF )
Hλ (F )\Hλ (AF )
−1 0
−1 −1
−1 0
−1 0
Ω0ψ (g)Φ0 (h−1 h−1
λ v1 + x1 h vλ , · · · , h hλ vn + xn h vλ )Φn+1 (h vλ )ωψλ (g)φ(l1 + x1 , · · · , ln + xn )
ψ(2y1 (qV (v1 , vλ0 ) − l1 λ) + · · · 2yn (qV (vn , vλ0 ) − ln λ)))ϕ(g)dhλ dhdgdxdy.
Integrations over yi ’s yield
X
(8.3.1) =
Z
(F \AF )n
v1 ,··· ,vn ∈V
l1 ,··· ,ln ∈F
0 )=l λ,∀i
qV (vi ,vλ
i
Z
Z
G0 (F )\G0 (AF )
Z
Hλ (AF )\H(AF )
Hλ (F )\Hλ (AF )
−1 0
−1 −1
−1 0
f (hλ h)Ω0ψ (g)Φ0 (h−1 h−1
λ v1 + x1 h vλ , · · · , h hλ vn + xn h vλ )
Φn+1 (h−1 vλ0 )ωψλ (g)φ(l1 + x1 , · · · , ln + xn )ϕ(g)dhλ dhdgdx.
The variables vi have to be of the form li vλ0 + wi where wi ∈ Vλ . Therefore
Z
Z
Z
Z
X
(8.3.1) =
w1 ,··· ,wn ∈Vλ
l1 ,··· ,ln ∈F
(F \AF )n
G0 (F )\G0 (AF )
Hλ (AF )\H(AF )
Hλ (F )\Hλ (AF )
−1 0
−1 −1
−1 0
f (hλ h)Ω0ψ (g)Φ0 (h−1 h−1
λ w1 + (l1 + x1 )h vλ , · · · , h hλ wn + (ln + xn )h vλ )
Φn+1 (h−1 vλ0 )ωψλ (g)φ(l1 + x1 , · · · , ln + xn )ϕ(g)dhλ dhdgdx.
54
Thus
Z
X
(8.3.1) =
w1 ,··· ,wn ∈Vλ
Z
An
F
Z
Z
f (hλ h)
G0 (F )\G0 (AF )
Hλ (AF )\H(AF )
Hλ (F )\Hλ (AF )
−1 0
−1 −1
−1 0
Ω0ψ (g)Φ0 (h−1 h−1
λ w1 + x1 h vλ , · · · , h hλ wn + xn h vλ )
Φn+1 (h−1 vλ0 )ωψλ (g)φ(x1 , · · · , xn )ϕ(g)dhλ dhdgdx.
We define
Z
0
Φ ∗ φ(w1 , · · · , wn ) =
(8.3.2)
An
F
Φ0 (w1 + x1 vλ0 , · · · , wn + xn vλ0 )φ(x1 , · · · , xn )dx1 · · · dxn .
Then Φ0 ∗ φ ∈ S(Vλ (AF )n ).
It is straight forward to check that
e ψ (e
g )φ),
Ω
g , hλ )(Φ0 ∗ φ) = (Ω0ψ (g, hλ )Φ0 ) ∗ (ωψλ (e
f0 (AF ), hλ ∈ Hλ (AF ),
ge ∈ G
where g is the image of ge in G0 (AF ). With this definition, we have
!
Z
Z
e ψ (ϕ, (Ω0 (h)Φ0 ) ∗ φ)(hλ )dhλ
f (hλ h)Θ
ψ
(8.3.1) =
Hλ (AF )\H(AF )
Φn+1 (h−1 vλ0 )dh.
Hλ (F )\Hλ (AF )
We summarize the above computation in the following lemma.
Lemma 8.3.1.
Z
Z
G0 (F )\G0 (AF )
Z
Z
N (F )\N (AF )
H(F )\H(AF )
!
Z
e ψ (ϕ, (Ω0 (h)Φ0 )
f (hλ h)Θ
ψ
=
Hλ (AF )\H(AF )
f (h)Θψ (ng, h, Φ0 ⊗ Φn+1 )θψλ (ne
g , φ)ϕ(e
g )dhdndg
∗ φ)(hλ )dhλ
Φn+1 (h−1 vλ0 )dh.
Hλ (F )\Hλ (AF )
8.4. Local Fourier–Jacobi periods of theta liftings. We now switch to the local situation.
We fix a place v of F and suppress it from all notation. So F stands for a local field. We have the
local version of all the previous objects, e.g. Weil representations, the representations π, σ, and
e ψ (σ), the orbit Λλ of v 0 under the action of H(F ), which identified
the theta liftings Θψ (π), Θ
λ
with Hλ (F )\H(F ), etc. We denote by B the inner products on various unitary representations.
The goal is to compute
Z
Z
Z
(8.4.1)
G0 (F )
N (F )
B(π(h)f, f )B(Ωψ (ng, h)Φ, Φ)B(ωψλ (ng)φ, φ)B(σ(g)ϕ, ϕ)dhdndg,
H(F )
where Φ = Φ0 ⊗ Φn+1 where Φ0 ∈ S(V n ) and Φn+1 ∈ S(V ).
We begin with
Lemma 8.4.1. The integral (8.4.1) is absolutely convergent.
55
Proof. In view of Proposition 2.2.1 (the case r = 1), we only need to prove
Z
Ξ(h)|B(Ωψ (g, h)Φ, Φ)|dh Ξ(g)(1 + ς(g))A , g ∈ G(F ).
(8.4.2)
H(F )
Note that
Z
Ξ(h)B(Ωψ (g, h)Φ, Φ)dh Ξ(g)(1 + ς(g))A ,
H(F )
g ∈ G(F ),
since the left hand side is a matrix coefficient of a tempered representation.
Even though in general |Bψ (Ω(g, h)Φ, Φ)| is not a matrix coefficient of the Weil representation,
we claim that it is dominated by a matrix coefficient of the Weil representation. In fact, by the
+
Cartan decompositoin, we only need to prove this when g = a ∈ A+
G and h = b ∈ AH . Then
Z
|B(Ωψ (g, h)Φ, Φ)| ≤
|Φ(b−1 va)Φ(v)|dv.
V (F )n+1
We may find a Schwartz function Φ+ so that |Φ| ≤ Φ+ (pointwise). We have proved the claim
and hence the lemma.
We recall the following well-known lemma.
Lemma 8.4.2 ([34, Lemma 3.18]). There is a unique measure dh on Hλ (F )\H(F ), such that
for any f ∈ S(V ), we have
Z
Z
Z
f (h−1 vλ0 )dhdλ,
f (v)dv =
F×
V
Hλ (F )\H(F )
where dv is the self-dual measure on V and dλ is the self-dual measure on F .
We need the following integration formula.
Lemma 8.4.3. Let f ∈ S(V ). Then
R
V
f (v)ψ(κqV (v, v))dv is absolutely integrable as a function
of κ. Moreover,
Z Z
Z
(8.4.3)
f (v)ψ(κqV (v, v))dv ψ(−λκ)dκ =
F
Proof. The integral
V
R
V
f (h−1 vλ0 )dh.
Hλ (F )\H(F )
f (v)ψ(κqV (v, v))dv equals
!
!!
1
1 κ
n
Φf
,
−1
1
where Φnf is the Siegel–Weil section of IndSL2 (F ) χV |·|s at s = s0 = n. Then by the decomposition
!
!
!
!
1
1 κ
−κ−1 1
1
=
,
−1
1
−κ
κ−1 1
R
the order of magnitude of V f (v)ψ(κqV (v, v))dv is |κ|−n−1 when |κ| is large. The integrability
then follows.
56
By Lemma 8.4.2,
Z
Z
f (v)ψ(κqV (v, v))dv =
V
Since f is Schwartz,
R
F×
!
Z
Hλ0 (F )\H(F )
−1 0
Hλ0 (F )\H(F ) f (h vλ0 )dh
f (h−1 vλ00 )dh
ψ(−λ0 κ)dλ0 .
is integrable as a function of λ0 and is continuous
on F × . The lemma then follows from the Fourier inversion formula.
Thanks to Lemma 8.4.1, we may change the order of integrations in (8.4.1). We integrate
over N (F ) first. By definition,
Z
B(Ωψ (ng, h)Φ, Φ) =
V n+1
Ω0ψ (g)Φ0 (h−1 (v1 + x1 vn+1 ), · · · , h−1 (vn + xn vn+1 ))Φ0 (v1 , · · · , vn )
ψ(2y1 qV (v1 , vn+1 ) + · · · 2yn qV (vn , vn+1 ) + (κ + y t x)qV (vn+1 , vn+1 ))
Φn+1 (h−1 vn+1 )Φn+1 (vn+1 )dv1 · · · dvn+1 .
It follows from Lemma 8.4.3 that
(8.4.4)Z Z
ZF
=
V n+1
Ωψ (ng, h)Φ(v1 , · · · , vn , vn+1 )Φ(v1 , · · · , vn , vn+1 )ψ(−λκ)dv1 · · · dvn dvn+1 dκ
Z
Ω0ψ (g)Φ0 (h−1 (v1 + x1 h0−1 vλ0 ), · · · , h−1 (vn + xn h0−1 vλ0 ))Φ0 (v1 , · · · , vn )
Hλ (F )\H(F )
Vn
ψ(2y1 qV (v1 , h0−1 vλ0 ) + · · · 2yn qV (vn , h0−1 vλ0 ) + (x1 y1 + · · · xn yn )λ)
Φn+1 (h−1 h0−1 vλ0 )Φn+1 (h0−1 vλ0 )dv1 · · · dvn dh0 .
The integral on the right hand side is absolutely convergent. In fact, the integrand is bounded
by
C|Φ0 (v1 , · · · , vn )Φn+1 (h0−1 vλ0 )|,
where C is a constant which is independent of x and y.
By definition,
B(ωψλ (n(x, y, 0)g, h)φ, φ)
Z
=
φg (l1 + x1 , · · · , ln + xn )φ(l1 , · · · , ln )ψ(λy1 (x1 + 2l1 ) + · · · + λyn (xn + 2ln ))dl1 · · · dln .
Fn
We claim that
Z
Z
(8.4.5)
F 2n
|∗||B(ωψλ (n(x, y, 0)g, h)φ, φ)|dh0 dxdy
Hλ (F )\H(F )
57
is convergent, where ∗ stands for the right hand side of (8.4.4). Indeed, this integral is bounded
by the convergent integral
Z
Z
C×
Hλ (F )\H(F )
Vn
|Φ(v1 , · · · , vn , vn+1 )Φn+1 (h0−1 vλ0 )|dv1 · · · dvn dh0
Z
×
|B(ωλ (n(x, y, 0)g, h)φ, φ)|dxdy,
F 2n
where C is some constant.
Thanks to the convergence of (8.4.5), we can change the order of the integration of x, y ∈ F n
and h0 ∈ Hλ (F )\H(F ). We need to compute
Z
Z
F 2n
(8.4.6)
Z
Vn
Ω0ψ (g)Φ0 (h−1 (v1 + x1 h0−1 vλ0 ), · · · , h−1 (vn + xn h0−1 vλ0 ))Φ0 (v1 , · · · , vn )
Fn
ψ(2y1 qV (v1 , h0−1 vλ0 ) + · · · 2yn qV (vn , h0−1 vλ0 ) + (x1 y1 + · · · xn yn )λ)
ωψλ (ι(g))φ(l1 + x1 , · · · , ln + xn )φ(l1 , · · · , ln )ψ(−λy1 (x1 + 2l1 ) − · · · − λyn (xn + 2ln ))
dl1 · · · dln dv1 · · · dvn dydx.
We need the following integration formula.
Lemma 8.4.4. Let f be a Schwartz function on V n and φ a Schwartz function on F n . Let
v 0 ∈ V with qV (v 0 , v 0 ) = λ and {v 0 }⊥ be its orthogonal complement. Then
Z
Z
Fn
V
Z
n
ψ(2y1 qV (v1 , v 0 ) + · · · 2yn qV (vn , v 0 ) − 2y1 l1 λ − · · · − 2yn ln λ)
Fn
f (v1 , · · · , vn )φ(l1 , · · · , ln )dl1 · · · dln dv1 · · · dvn dy1 · · · dyn .
equals
−n
Z
Z
f (l1 v 0 + w1 , · · · ln v 0 + wn )φ(l1 , · · · , ln )dl1 · · · dln dw1 · · · dwn .
|2λ|
({v 0 }⊥ )n
Fn
Proof. Let fb and φb be the Fourier transform of f and φ respectively (with respect to ψ). Then
the first integral in the lemma equals
Z
Fn
b 1 λ, · · · , 2yn λ)dy1 · · · dyn .
fb(2y1 v 0 , · · · , 2yn v 0 )φ(2y
The lemma then follows from the fact that the Fourier transform preserves the inner product of
Schwartz functions.
58
Applying this lemma, we see that
(8.4.6)
−n
Z
Z
Z
=|2λ|
Vλn
Fn
Fn
Ω0ψ (g)Φ0 (h−1 h0−1 (w1 + l1 vλ0 + x1 vλ0 ), · · · , h−1 h0−1 (wn + ln vλ0 + xn vλ0 ))
Φ0 (w1 + l1 h0−1 vλ0 , · · · , wn + ln h0−1 vλ0 )ωψλ (ι(g))φ(l1 + x1 , · · · , ln + xn )φ(l1 , · · · , ln )
dw1 · · · dwn dl1 · · · dln dx.
This integral is absolutely convergent. We then make change of variables xi 7→ xi − li . Then
Z Z Z
−n
Ω0ψ (g)Φ0 (h−1 h0−1 (w1 + x1 vλ0 ), · · · , h−1 h0−1 (wn + xn vλ0 ))
(8.4.6) =|2λ|
Vλn
Fn
Fn
Φ0 (h0−1 (w1 + l1 vλ0 ), · · · , h0−1 (wn + ln vλ0 ))
ωψλ (ι(g))φ(x1 , · · · , xn )φ(l1 , · · · , ln )dl1 · · · dln dw1 · · · dwn dx.
We define a local analogue of (8.3.2), i.e.
Z
Φ0 (v1 + x1 vλ0 , · · · , vn + xn vλ0 )φ(x1 , · · · , xn )dx1 · · · dxn .
Φ0 ∗ φ(v1 , · · · , vn ) =
Fn
Then Φ0 ∗ φ ∈ S(Vλ ) and
e ψ (e
Ω
g , hλ )(Φ0 ∗ φ) = (Ω0ψ (g, hλ )Φ0 ) ∗ (ωψλ (e
g )φ),
f0 (F ), hλ ∈ Hλ (F ),
ge ∈ G
where g is the image of ge in G0 (F ).
Let dh0 be the quotient measure on Hλ (F )\H(F ) and c · dh0 = dh0 . Then we get
Z
Z Z
B(π(h)f, f )B(σ(g)ϕ, ϕ)B(Ω0ψ (g, h0 h)Φ ∗ ωψλ (g)φ, Ω0ψ (h0 )Φ ∗ φ)
(8.4.1) =c · |2λ|−n
Hλ \H
H
G0
Φn+1 (h−1 h0−1 vλ0 )Φn+1 (h0−1 vλ0 )dgdhdh0 .
We make a change of variable h 7→ h0−1 h and get
(8.4.1)
−n
ZZ
Z
=c · |2λ|
Hλ \H×H
B(π(h)f, π(h0 )f )B(σ(g)ϕ, ϕ)B(Ω0ψ (g, h)Φ ∗ ωψλ (g)φ, Ω0ψ (h0 )Φ ∗ φ)
G0
Φn+1 (h−1 vλ0 )Φn+1 (h0−1 vλ0 )dgdhdh0 .
The group Hλ embeds in H × H diagonally. This integral is absolutely convergent.
We further split the integration over h as hλ h where hλ ∈ Hλ and h ∈ Hλ \H. Then
Z
Z Z
−n
B(π(hλ h)f, π(h0 )f )B(σ(g)ϕ, ϕ)
(8.4.1) =c · |2λ|
(Hλ \H)2
Hλ
G0
e ψ (g, hλ )(Ω0 (h)Φ ∗ φ), (Ω0 (h0 )Φ ∗ φ))Φn+1 (h−1 v 0 )Φn+1 (h0−1 v 0 )dgdhλ dh0 dh.
B(Ω
ψ
ψ
λ
λ
We summarize the above computation into the following lemma.
59
Lemma 8.4.5. Suppose Φ = Φ0 ⊗ Φn+1 where Φ0 ∈ S(V n ) and Φn+1 ∈ S(V ). Then
Z
Z
Z
B(π(h)f, f )B(Ωψ (ng, h)Φ, Φ)B(ωψλ (ng)φ, φ)B(σ(g)ϕ, ϕ)dhdndg
G0 (F )
N (F )
H(F )
Z
−n
Z
Z
=c · |2λ|
(Hλ \H)2
Hλ
e ψ (g, hλ )(Ω0 (h)Φ ∗ φ), (Ω0 (h0 )Φ ∗ φ))dg
B(σ(g)ϕ, ϕ)B(Ω
ψ
ψ
G0
B(π(hλ h)f, π(h0 )f )Φn+1 (h−1 vλ0 )Φn+1 (h0−1 vλ0 )dhλ dhdh0 .
8.5. Proof of Theorem 8.1.1. By Lemma 8.3.1, we have
ZZ
2
Φn+1 (h−1 vλ0 )Φn+1 (h0−1 vλ0 )
|FJ ψλ (Θψ (f, Φ), ϕ, φ)| =
(Hλ (AF )\H(AF ))2
!
Z
e ψ (ϕ, (Ω0 (h)Φ0 )
f (hλ h)Θ
ψ
∗ φ)(hλ )dhλ
Hλ (F )\Hλ (AF )
!
Z
Hλ (F )\Hλ (AF )
e ψ (ϕ, (Ω0 (h0 )Φ0 )
f (h0λ h0 )Θ
ψ
∗
dhdh0 .
φ)(h0λ )dh0λ
We fix a sufficiently large finite set of places S of F so that if v 6∈ S, then the following
conditions hold.
(1) v is non-archimedean, 2 and λ are in o×
F,v , the conductor of ψ is oF,v .
(2) The group A is unramified with a hyperspecial maximal compact subgroup KAv =
A(oF,v ) where A = G, G0 , H, Hλ .
(3) fv and ϕv are KHv and KG0,v fixed respectively. Moreover, they are normalized so that
B(fv , fv ) = B(ϕv , ϕv ) = 1. In particular, πv and σv are both unramified.
(4) Φv is the characteristic function of V (oF,v )n+1 , φv is the characteristic function of onF,v .
(5) The volume of the hyperspecial maximal compact subgroup KAv is 1 under the chosen
measure on A(Fv ), where A = G, G0 , H, Hλ .
Lemma 8.5.1. If v 6∈ S, then cv = Lv (n + 1, χVv )−1 . Recall that dhv = cv · dhλ,v \dhv where
dhv is the measure defined in Lemma 8.4.2.
Proof. We denote temporarily by fv the characteristic function of V (oF,v ). Recall from the proof
of Lemma 8.4.3 that
Z
Hλ (Fv )\H(Fv )
fv (h−1 vλ0 )dh
Z
=
Fv
Φnfv
1
−1
!
!!
1 κ
1
ψv (−λκ)dκ,
where Φnfv is the Siegel–Weil section of IndSL2 (Fv ) χVv |·|s at s = s0 = n. It is well-known that
the right hand side equals Lv (n + 1, χVv )−1 .
0
We note that since λ ∈ o×
F,v , the orbit Λλ of vλ is defined over oF,v . The group H(oF,v ) acts
transitively on Vλ (oF,v ). Therefore Hλ (oF,v )\H(oF,v ) → Λλ (oF,v ) is a bijection. Thus under the
quotient measure dhλ,v \dhv , the volume of Λλ (Fv )∩V (oF,v ) is one. The lemma then follows. 60
Lemma 8.5.2.
Y
cv = LS (n + 1, χV ).
v∈S
Q
Proof. It follows from Lemma 8.2.1 that v cv = 1. Then
Y
Y
S
cv =
c−1
v = L (n + 1, χV ).
v∈S
v6∈S
f0 to Hλ ) and
Conjecture 6.3.1, the Rallis’ inner product formula (for theta lifting from G
Lemma 8.4.5 lead to
e ψ (σ))
LS ( 12 , π × Θ
2γ−1
e ψ (σ), Ad)
|Sπ ||SΘ
e ψ (σ) | LS (1, π, Ad)LS (1, Θ
Z
Z
Z
LSψλ ( 12 , σ × χVλ ) Y −1
S
cv
∆H(V ) · Qn
S
G0 (Fv ) N (Fv ) H(Fv )
j=1 ζF (2j) v∈S
|FJ ψλ (Θψ (f, Φ), ϕ, φ)|2 =
Bv (π(hv )fv , fv )Bv (Ωψv (hv , nv gv )Φv , Φv )Bv (ωψλ,v (nv gv )φv , φv )Bv (σv (gv )ϕv , ϕv )dhv dnv dgv ,
where γ is described as in Conjecture 6.3.1.
We then apply the Rallis inner product formula for the theta lifting from H to G. We conclude
that
e ψ (σ))
LS ( 12 , π × Θ
2γ−1
|FJ ψλ (ξ, ϕ, φ)| =
e ψ (σ), Ad)
|Sπ ||SΘ
e ψ (σ) | LS (1, π, Ad)LS (1, Θ
−1
LSψ ( 12 , σ × χVλ )
LS (1, π)
S
Qn
LS (n + 1, χV )−1
∆H · Qn
S (2j)
S (2i)LS (n + 1, χ )
ζ
ζ
V
j=1 F
i=1 F
Z
YZ
Bv (Θψv (πv )(nv gv )ξv , ξv )Bv (ωψλ,v (nv gv )φv , φv )Bv (σv (gv )ϕv , ϕv )dnv dgv ,
2
v∈S
G0 (Fv )
N (Fv )
γ−1 |S
where ξ = ⊗ξv ∈ Θψ (π). Note that |Sπ ||SΘ
e ψ (σ) | = 2
Θψ (π) ||Sσ | by Lemma 5.2.3. Conjec-
ture 2.3.1(3) then follows from Lemma 5.2.2.
8.6. A variant. So far we considered the case Sp(2n+2)×Mp(2n). The case Mp(2n+2)×Sp(2n)
is similar. We only mention the following theorem.
Let (V, qV ) be a 2n + 3 dimensional orthogonal space and H = O(V ). Suppose that λ ∈ F ×
and there is an element vλ0 ∈ V such that qV (vλ0 , vλ0 ) = λ. Let Vλ be the orthogonal complement
of vλ0 and Hλ = O(Vλ ). Let π be an irreducible cuspidal tempered automorphic representation
of H(AF ) and Θψ (π) its theta lift to Mp(2n + 2)(AF ) (with additive character ψ). Let σ be an
irreducible cuspidal tempered automorphic representation of Sp(2n)(AF ) and Θψ (σ) its theta
lift to Hλ (AF ).
61
Theorem 8.6.1. Suppose that Θψ (π) and Θψ (σ) are both cuspidal. If Conjecture 6.3.1 holds
for (π, Θψ (σ)), then Conjecture 2.3.1(3) holds for (Θψ (π), σ) (with the additive character ψλ ).
The proof of Theorem 8.6.1 is analogues to Theorem 8.1.1 and we leave the details to the
interested reader.
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School of Mathematics, Institute for Advanced Study, Einstein Drive, Princeton, NJ, 08540
E-mail address: [email protected]
64