Bull. Soc. math. France,
126, 1998, p. 181–244.
L-FUNCTIONS FOR SYMPLECTIC GROUPS
BY
David GINZBURG, Stephen RALLIS and
David SOUDRY (*)
ABSTRACT. — We construct global integrals of Shimura type, which represent the
standard (partial) L-function LS (π ⊗ σ, s), for π ⊗ σ, an irreducible, automorphic,
cuspidal and generic representation of Sp2n (A) × GLk (A). We present two different
constructions : one for the case n > k and one for the case n ≤ k. These constructions are, in a certain sense, dual to each other. We also study the (completely ana 2n (A). Here we
logous) case where π is a representation of the metaplectic group Sp
have to first fix a choice of a non-trivial additive character ψ, in order to define the
L-function LS
ψ (π ⊗ σ, s). The integrals depend on a cusp form of π, a theta series
2 (A) (
= min(n, k)) and an Eisenstein series on Sp2k (A) (or Sp
2k (A)) induced
on Sp
from σ.
RÉSUMÉ. — FONCTIONS L POUR GROUPES SYMPLECTIQUES. — On construit des
intégrales globales de type de Shimura, qui représentent la fonction L standard (partielle) LS (π ⊗ σ, s) pour une représentation π ⊗ σ, irréductible, automorphe, cuspidale
et générique de Sp2n (A) × GLk (A). On présente deux constructions différentes : une
pour le cas n > k, et une pour le cas n ≤ k. Ces constructions sont, dans un certain
sens, duales l’une de l’autre. On étudie de même le cas (tout à fait analogue) où π
2n (A). Ici, on doit d’abord fixer le
est une représentation du groupe métaplectique Sp
choix d’un caractère additif et non trivial ψ, pour définir la fonction L, LS
ψ (π ⊗ σ, s).
2 (A)
Les intégrales dépendent d’une forme cuspidale de π, d’une série thêta sur Sp
2k (A)) induite par σ.
(
= min(n, k)) et d’une série d’Eisenstein sur Sp2k (A) (ou Sp
(*) Texte reçu le 10 octobre 1997, accepté le 28 janvier 1998.
All three authors are supported by a grant from the Israel-USA Binational Science
Foundation.
D. GINZBURG and D. SOUDRY, School of Mathematical Sciences, Sackler Faculty of
Exact Sciences, Tel Aviv University, Tel Aviv 69978 (Israel).
Email : soudry@math.tau.ac.il, ginzburg@math.tau.ac.il.
S. RALLIS, Department of Mathematics, The Ohio State University, Columbus,
OH 43210 (U.S.A). Email : haar@math.ohio-state.edu.
AMS classification : 22 E 55, 11 F 85.
Keywords : L function, theta series, Eisenstein series, Whittaker model.
BULLETIN DE LA SOCIÉTÉ MATHÉMATIQUE DE FRANCE
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D. GINZBURG, S. RALLIS, D. SOUDRY
Introduction
We present a global integral of Shimura type, which interpolates the
standard L-function for generic cusp forms on Sp2n × GLk (resp. on
2n × GLk , where denotes the metaplectic cover.) The structure of
Sp
the global integral has one of two forms, according to whether n ≥ k
or n < k. However, these two forms of integrals are dual to each other,
in the sense that the roles of cusp forms and Eisenstein series are interchanged. This construction for Sp2n × GLn was already done in [GPS].
Let us give some more details on these global integrals. Let π (resp. π
)
be an irreducible automorphic, cuspidal representation of Sp2n (A) (resp.
2n (A) (A – the adèles of a global field F ). We assume that π (resp.
Sp
π
) is generic, i.e. π (resp. π
) has a nontrivial Whittaker-Fourier coefficient. Let σ be an irreducible, automorphic, cuspidal representation of
GLk (A). Let E(g, fσ,s ) be the Siegel-Eisenstein series on Sp2k (A) corresponding to a holomorphic K-finite section fσ,s in the representation of
Sp2k (A) induced from the Siegel parabolic subgroup and σ. In a similar
2k (A) inappropriate fashion, we may consider the representation of Sp
duced from σ (twisted by the Weil symbol), and construct, for a section
f˜σ,s ) on Sp
2k (A). Let ψ be
f˜σ,s , the corresponding Eisenstein series E(g,
()
a nontrivial character of F \ A. Denote by ωψ the Weil representation of
2 (A), where H is the corresponding Heisenberg group (of diH (A) · Sp
()
mension 2 + 1). Let θψ be the corresponding realization by theta series.
Assume that n ≥ k and let ϕ be a cusp form in the space of π. Then the
global integral for π × σ has the form
(k)
f˜σ,s ) dg.
I1 (s) =
ϕ(vg)θψ (v g)ψ (v) dv · E(g,
Sp2k (F )\Sp2k (A) V (n,k) \V (n,k)
F
A
V (n,k) is the unipotent radical in Sp2n , of the parabolic subgroup preserving a an (n − k)-dimensional isotropic flag. V (n,k) has projection v → v (n,k)
(n,k)
to the Heisenberg group Hk . ψ is a certain character of VF
\ VA
.
2k (A),
Sp2k is embedded in Sp2n in the “middle” 2k×2k block. For π
on Sp
we consider, for ϕ
in the space of π
,
(k) I2 (s) =
ϕ(vg)θ
ψ (v g)ψ (v) dv · E(g, fσ,s ) dg.
Sp2k (F )\Sp2k (A) V (n,k) \V (n,k)
F
A
In case n < k, we switch the roles of k and n and of ϕ (resp. ϕ)
and E
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L-FUNCTIONS FOR SYMPLECTIC GROUPS
(resp. E) and consider
J1 (s) =
ϕ(g)
Sp2n (F )\Sp2n (A)
(k,n)
VF
(n)
E(vg,
f˜σ,s )θψ (v g)ψ (v) dv dg,
(k,n)
\VA
E(vg, fσ,s )θψ (v g)ψ (v) dv dg.
(n)
ϕ(g)
J2 (s) =
Sp2n (F )\Sp2n (A)
183
(k,n)
VF
(k,n)
\VA
These integrals are of course meromorphic functions of s. We prove, for
decomposable data, that the integrals are Eulerian. The corresponding
local integrals satisfy the expected properties of the associated local theory
(meromorphic continuation, nonvanishing and local functional equation).
When all data is standard unramified at a place p the local integrals at p
give the following quotient
Case Sp2n × GLk :
L(πp ⊗ σp , s(k + 1) − 12 k)
,
L(σp , V 2 , 2s(k + 1) − k)
× GLk :
Case Sp
2n
πp ⊗ σp , s(k + 1) − 12 k)
Lψp (
·
L(σp , s(k + 1) − 12 (k − 1))L(σp , Λ2 , 2s(k + 1) − k)
Here V 2 and Λ2 are respectively the symmetric square and exterior square
representations of GLk (C). There is no canonical definition of the local
2n (Fp ) ⊗ GLk (Fp )). We have to first make a
L-factor for π
p ⊗ σp (on Sp
choice of a nontrivial character of Fp . Here we choose ψp . The choice of ψp
determines the unramified character η = η1 ⊗ · · · ⊗ ηn , of the diagonal
2n (F ), which corresponds to π
subgroup of Sp
p . See Section 3.1 for our
precise definition. We then have
Lψp (
πp ⊗ σp , s) =
n
L(σp ⊗ ηi , s)L(σp ⊗ ηi−1 , s).
i=1
We obtain these unramified computations, in cases n ≥ k, using
invariant theory in a direct fashion. We do not know how to do this
in case n < k. In this case, we prove, first, a certain
identity relating
Sp
(up to slight modifications) the local integrals for Ind P 2n σ × σ and
2n,n
Sp
Sp
Sp
Ind 2n σ × σ (resp. Ind 2n σ × σ and Ind P 2n σ × σ ) and
2n,n
2n,n
2n,n
P
P
then we use the unramified calculation of the first case (see [S1], where
similar identities are obtained for SO2n+1 × GLk .)
BULLETIN DE LA SOCIÉTÉ MATHÉMATIQUE DE FRANCE
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D. GINZBURG, S. RALLIS, D. SOUDRY
The importance of achieving L-functions via explicit integrals of the
above type is, apart from establishing their meromorphicity, the possibility
of locating their poles and relating their existence to functorial liftings
and nonvanishing of (generalized) periods. We already studied one such
example in case Sp2n × GL1 in [GRS1], where we showed that the only
possible pole of the partial L-function LS (π, s) is at s = 1, and this pole
occurs if and only if π has a nontrivial theta lift to a cuspidal generic
representation of SOn,n (A), and in this case a certain (generalized) period
is nontrivial on π. In forthcoming works [GRS2], [GRS3] we study the
existence of a pole at s = 1, when k > 1, and relate it to (explicit)
2n (A) and of GL2n (A)
functorial lifts between generic representations of Sp
and between generic representations of Sp2n (A) and of GL2n+1 (A). We
plan to study this explicit functorial lift for the nongeneric case as
well, using a similar theory of L-functions of arbitrary (not necessarily
2n (A)), due to the
generic) cuspidal representations of Sp2n (A) (resp. Sp
first two named authors. There “generic” is replaced by the existence
of a certain Fourier-Jacobi model. The advantage of having all these
different constructions is the possibility of relating generic and nongeneric
representations, having the same functorial lift to the appropriate GLn .
1. Notations
.1
.
.
1. — Let Jr denote the r × r matrix
. We define
Sp2r
= g ∈ GL2r : t g
1
−Jr
Jr g =
−Jr
Jr
.
2. — For our construction we will need the following subgroups of Sp2r .
For 0 ≤ k ≤ r let P2r,k denote the parabolic subgroup of Sp2r which
stabilizes a k-dimensional isotropic space. Thus
P2r,k = GLk × Sp2(r−k) U2r,k ,
where U2r,k is the unipotent radical of P2r,k . When k = 0 we understand
that P2r,0 = Sp2r and GL0 = U2r,0 = {1}. In terms of matrices, we
consider the embedding of P2r,k in Sp2r as follows :
h
g̃
(h, g) −→
, h ∈ GLk , g ∈ Sp2(r−k) ,
h∗
A B
where h∗ is such that the above matrix is in Sp2r , and given g =
C D
A 12 B
. The group U2r,k is embedded as all
in Sp2(r−k) , then g̃ =
2C D
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matrices in Sp2r of the form
Ik
I2(r−k)
∗
Ik
,
where I is the × identity matrix. Let Q2r,k ⊂ P2r,k denote the parabolic
subgroup of Sp2r whose Levi part is GLk1 × Sp2(r−k) . We shall denote its
unipotent radical by V2r,k .
3. — Let Hn denote the Heisenberg group with 2n + 1 variables. We
shall identify an element h ∈ Hn with a triple (x, y, z), where x, y ∈ M1×n
and z ∈ M1×1 . We define the following subgroups of Hn :
Xn = (x, 0, 0) ∈ Hn , Yn = (0, y, 0) ∈ Hn , Zn = (0, 0, z) ∈ Hn .
The product in Hn is given by
(x1 , y1 , z1 )(x2 , y2 , z2 ) = x1 + x2 , y1 + y2 , z1 + z2 + 12 (x1 Jn y2t − y1 Jn xt2 ) .
It is well-known that Hn is isomorphic to U2n+2,1 . We shall denote this
isomorphism by τ . In coordinates we have


τ (h) = τ (x, y, z) =
1 x 12 y z
y ∗ .
 In
I n x∗
1
Here x∗ and y ∗ are such that the above matrix is in Sp2n+2 . Thus,
y ∗ = 12 Jn y t
and x∗ = −Jn xt .
4. — Let F be a global field and A its ring of adèles. Given a linear
algebraic group G, we shall denote by G(A) the A points of G, etc.
Fix n and k in N. Let π (resp. σ) denote an irreducible automorphic
cuspidal representation of Sp2n (A) (resp. GLk (A)). We shall denote by Vπ
(resp. Vσ ) its realization in the space of cusp forms. We shall always
assume that π is generic. We recall this notation. Let ψ denote a nontrivial
additive character of F \ A. Given 1 ≤ ≤ n we define a character ψ
on V2n, as follows. Let v = (vij ) ∈ V2n, be a matrix realization of V2n,
as defined in Section 2. If 1 ≤ ≤ n we denote
ψ (v) = ψ
vi,i+1
i=1
BULLETIN DE LA SOCIÉTÉ MATHÉMATIQUE DE FRANCE
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and we shall also need
ψn (v) = ψ
n−1
i=1
vi,i+1 + 12 vn,n+1 .
Let Nk denote the maximal unipotent subgroup of GLk consisting of upper
triangular matrices. For n̄ = (n̄ij ) ∈ Nk we define
ψNk (n̄) = ψ
k−1
n̄i,i+1 .
i=1
We say that π is generic with respect to ψn if the space of functions
ϕπ (vg)ψn (v) dv
Wϕπ (g) =
V2n,n (F )\V2n,n (A)
is not identically zero. Here ϕπ ∈ Vπ and g ∈ Sp2n (A). We shall denote this
space of functions by W(π, ψn ). A similar definition holds for W(π, ψn ).
Similarly, for σ, we denote by W(σ, ψNk ) the space of functions of the
form
Wϕσ (h) =
ϕσ (n̄h)ψNk (n̄) dn̄
Nk (F )\Nk (A)
for ϕσ ∈ Vσ and h ∈ GLk (A). By Shalika’s well-known theorem, given σ
as above W(σ, ψNk ) ≡ 0.
2k (A) denote the metaplectic double cover of Sp2k (A).
5. — Let Sp
its full inverse image
Given a subgroup G of Sp2k we shall denote by G
2k . If a subgroup G of Sp
2k splits under the cover, we shall view G
in Sp
as a subgroup of Sp2k . Choosing the covering in a suitable way, it is wellknown that the groups Sp2k (F ) and V2k,k (A) split.
For a ∈ A∗ we let γa denote the Weil symbol. It depends on the choice
of ψ. When we want to mark this dependence, we write γa,ψ . Let δ P2k,k
denote the modular function of P2k,k . As usual, we view it as a function
of P2k,k by composing it with the projection P2k,k → P2k,k . For s ∈ C
and σ as in Section 4 we denote
2k (A) ˜ s) = Ind Sp
I(σ,
σ ⊗ δ sP2k,k ⊗ γ −1 .
2k,k (A)
P
(A) → Vσ
By definition, this is the space of all smooth functions f˜σ,s : Sp
2k
satisfying
(1.1)
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−1
˜
f˜σ,s (pg) = εγdet
m δ P2k,k (m)σ(m)fσ,s (g)
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2k (A), ε ∈ {±1} and p =
for all g ∈ Sp
m
∗
187
2k,k (A),
u, ε ∈ Sp
m
where m ∈ GLk (A) and u ∈ U2k,k (A). Here and henceforth we shall
as pairs g, ε, where g ∈ Sp and ε ∈ {±1} with
view elements of Sp
2k
2k
multiplication as defined, for example, in [BFH]. Let us remark that (1.1)
is well-defined since the subgroup GLk of Sp2k splits under the cover.
We shall also denote
Sp2k (A) s
I(σ, s) = Ind P2k,k
(A) σ ⊗ δ P2k,k .
Here σ and s are as before. Thus I(σ, s) is the space of all smooth functions
fσ,s : Sp2k (A) → Vσ , satisfying
fσ,s (pg) = δ sP2k,k (m)σ(m)fσ,s (g)
for all g ∈ Sp2k (A) and p = mu ∈ P2k,k (A), where m ∈ GLk (A) and
u ∈ U2k,k (A).
In both cases, we view the function fσ,s (g) (resp. f˜σ,s (g)) as complex
2k (A)) which are left U2k,k (A)
valued functions on Sp2k (A) (resp. Sp
2k (A)) the function m →
invariant, and for fixed g ∈ Sp2k (A) (resp. Sp
˜
fσ,s (mg) (resp. m → fσ,s (mg)), with m ∈ GLk (A), belongs to the
space of cusp forms on GLk (A) realizing the representation σ ⊗ δ sP2k,k
(resp. σ ⊗ δ sP2k,k ⊗ γ −1 ).
˜ s) define
Next, we define the Eisenstein series we need. For f˜σ,s ∈ I(σ,
f˜σ,s ) =
E(g,
f˜σ,s (δg),
δ∈P2k,k (F )\Sp2k (F )
2k (A). The above summation converges absolutely for Re(s)
where g ∈ Sp
large and admits a meromorphic continuation to the whole complex plane
with at most finitely many poles in Re(s) ≥ 12 . A similar definition holds
for E(g, fσ,s )-the Siegel Eisenstein series on Sp2k (A).
˜ s). We denote
Let f˜σ,s ∈ I(σ,
f˜Wσ,s (g) =
Nk (F )\Nk (A)
f˜σ,s (n̄g)ψNk (n̄) dn̄
2k (A). Thus, for g ∈ GLk (A), the function f˜Wσ ,s (g) is in
for g ∈ Sp
W(σ, ψNk ). Similarly we define the function fWσ,s (g).
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6. — Let ωψ denote the Weil representation. It is a representation of
2k (A) and it is realized on the space S(Ak ) — the
the group Hk (A)Sp
Schwartz space of Ak . The following formulas are well-known (see [P])
(1.2)
ωψ (0, y, z)(x, 0, 0), ε φ(ξ) = εψ(z + ξJk y t )φ(x + ξ),
(1.3)
ωψ
(1.4)
ωψ
m
m
1
, ε φ(ξ) = εγdet m | det m| 2 φ(ξm),
∗
Ik T
, ε φ(ξ) = εψ 12 ξT Jk ξ t φ(ξ).
Ik
Here φ ∈ S(Ak ), (0, y, z)(x, 0, 0) ∈ Hk (A), ε ∈ {±1}, m ∈ GLk (A) and
Ik T
∈ U2k,k (A). In the above formulas, we view ξ ∈ Ak as a row
Ik
vector.
We now define the theta function,
θ̃φ (hg) =
ωψ (hg)φ(ξ)
ξ∈F k
(A). This function is an
for φ ∈ S(Ak ), h ∈ Hk (A) and g ∈ Sp
2k
automorphic function of Hk (A)Sp2k,k (A) i.e. it is slowly increasing and
invariant under the rational points Hk (F ) Sp2k (F ).
2. The Global Integrals (n > k)
We keep the notations of Section 1. Assume that n > k. In this
section we shall describe the global integral which will represent the tensor
product L-function. Let

w0 = 
0
In−k
Ik
0

0 In−k
Ik 0

and define j(g) = w0 gw0−1 for all g ∈ Sp2n (A). The global integral we
consider is
I1 (ϕ, φ, f˜σ,s ) =
Sp2k (F )\Sp2k (A) Hk (F )\Hk (A) V2n,n−k−1 (F )\V2n,n−k−1 (A)
f˜σ,s )ψn−k−1 (v) dv dh dg.
ϕ j(vτ (h)g) θ̃φ (hg)E(g,
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˜ s), and ψn−k−1 is as defined
Here ϕ ∈ Vπ , φ ∈ S(Ak ) and f˜σ,s ∈ I(σ,
in Chapter 1, Section 4. Also g embeds in Sp2n as described in Section 1,
part 2.
A similar construction is valid for irreducible cuspidal representations π
k
(A).
Indeed,
given
ϕ
∈
V
,
φ
∈
S(A
)
and
f
∈
I
we
define
of Sp
σ,s
σ,s
2n
π
I2 (ϕ,
φ, fσ,s ) =
Sp2k (F )\Sp2k (A) Hk (F )\Hk (A) V2n,n−k−1 (F )\V2n,n−k−1 (A)
ϕ
j(vτ (h)g) θ̃φ (hg)E(g, fσ,s )ψn−k−1 (v) dv dh dg.
Let us remark that in both cases the integrals are well defined
in the
f˜σ,s ) and ϕ
sense that the functions θ̃φ (hg)E(g,
j(vτ (h)g) θ̃φ (hg) are
2k (A).
non-genuine functions of Sp
Let R ⊂ V2n,n be the unipotent subgroup consisting of all matrices of
the form




 In−k r
where
the
r
∈
M
Ik
(n−k)×n
:
R= 
∗
Ik r


bottom row is zero
In−k
We are now ready to prove :
THEOREM 2.1. — The integral I1 (ϕ, φ, f˜σ,s ) converges absolutely for
all s, except for those s for which the Eisenstein series has a pole.
For Re(s) large :
I1 (ϕ, φ, f˜σ,s ) =
Wϕ j(rτ (x, 0, 0)g)
V2k,k (A)\Sp2k (A) R(A) Xk (A)
ωψ (g)φ(x)f˜Wσ ,s (g) dx dr dg.
Here Wϕ ∈ W(π, ψn ) and f˜Wσ,s is as defined in Chapter 1, Section 5.
Proof. — The proof of the absolute convergence follows as in [G]
using the growth conditions of ϕ. We show the unfolding. Unfolding the
Eisenstein series and the theta series, we obtain that I1 (ϕ, φ, f˜σ,s ) equals
ωψ (hg)φ(ξ)f˜σ,s (g)ψn−k−1 (v) dv dh dg,
ϕ j(vτ (h)g)
P2k,k (F )\Sp2k (A)
ξ∈F k
where h and v are integrated as before. From (1.2) it follows that
ωψ (hg)φ(ξ) = ωψ (ξ, 0, 0)hg φ(0).
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Write h ∈ Hk (A) as h = (0, y, z)(x, 0, 0) (see Chapter 1, Section 3).
Changing variables using the left invariant properties of ϕ and collapsing
the sum over F k with the integration over Xk (F ) \ Xk (A), I1 (ϕ, φ, f˜σ,s )
equals
ϕ j(vτ ((0, y, z))τ ((x, 0, 0))g) ωψ (0, y, z)(x, 0, 0)g φ(0)
f˜σ,s (g)ψn−k−1 (v) dv dy dz dx dg.
Here y is integrated over Yk (F )\Yk (A), z is integrated over Zk (F )\Zk (A),
x over Xk (A) and v and g are integrated as before. From (1.2) we obtain
ωψ (0, y, z)(x, 0, 0)g φ(0) = ψ(z)ωψ (g)φ(x)
Thus I1 (ϕ, φ, f˜σ,s ) equals
ϕ j(vτ ((0, y, z))τ ((x, 0, 0))g) ωψ (g)φ(x)
f˜σ,s (g)ψn−k−1 (v)ψ(z) dv dy dz dx dg,
where all variables are integrated as before. We have the following Lemma :
LEMMA 2.2. — For data as above
ϕ j(vτ ((0, y, z))ug) ψn−k−1 (v)ψ(z) dv dy dz du
=
δ
Wϕ
R(A) δ∈Nk (F )\GLk (F )
I2(n−k)
δ∗
j(rg) dr.
Here u is integrated over U2k,k (F ) \ U2k,k (A), v over V2n,n−k−1 (F ) \
V2n,n−k−1 (A), y over Yk (F ) \ Yk (A) and z over Zk (F ) \ Zk (A).
The proof of Lemma 2.2 is as the proof of the Lemma in [G, p. 172].
We will give the details later.
Returning to the proof of the theorem, write P2k,k = GLk U2k,k and
=
P2k,k (F )\Sp2k (A)
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L-FUNCTIONS FOR SYMPLECTIC GROUPS
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Using this and Lemma 2.2., I1 (ϕ, φ, f˜σ,s ) equals
GLk (F )U2k,k (A)\Sp2k (A) R(A) Xk (A) δ∈Nk (F )\GLk (F )
Wϕ j(δ̂rτ (x, 0, 0)g) ωψ (g)φ(x)f˜σ,s (g) dx dr dg,
where j(δ̂) =
δ
I2(n−k)
δ∗
. Using the definition of j we see that
δ̂ ∈ Sp2k (F ). Hence, conjugating δ̂ across r and τ (x, 0, 0), and a change of
variables in r and x, we may collapse the summation with the integration
over GLk (F )U2k,k (A) \ Sp2k (A) to obtain
Wϕ j(rτ (x, 0, 0)g) ωψ (g)φ(x)f˜σ,s (g) dx dr dg,
where g is integrated over Nk (F )U2k,k (A)\Sp2k (A) and all other variables
as before. Write
=
.
Nk (F )U2k,k (A)\Sp2k (A)
Nk (A)U2k,k (A)\Sp2k (A) Nk (F )\Nk (A)
We have Nk U2k,k = V2k,k . After a change of variables in r and x,
I1 (ϕ, φ, f˜σ,s ) equals
Wϕ j(rτ (x, 0, 0)g) ωψ (g)φ(x)
f˜σ,s (n̄g)ψNk (n̄) dn̄ dx dr dg,
Nk (F )\Nk (A)
where g is integrated over V2k,k (A) \ Sp2k (A) and r and x as before. From
this the Theorem follows.
Proof of Lemma 2.2. — As mentioned before the proof of this lemma
follows the same pattern as the proof of the lemma in [G, p. 173–175]. We
give some details. Let Nn−k−1 ⊂ Nn be embedded in Nn in the lower right
corner. We define the following two unipotent subgroups of Sp2n (A). First



0
0
 Ik t̂

t̂ ∈ Mk×(n−k)
0
In−k 0
∗  :
T = 
such
that
the
first
In−k t̂


column is zero
Ik
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and second


I

 ˆk I


n−k
L= 
:
In−k

∗

Ik
ˆ


ˆ

∈ M(n−k)×k
such that the last  .

row is zero
Notice that L = j(R). We now prove the lemma. By definition
Wϕ (g) =
ϕ(vg)ψn (v) dv.
V2n,n (F )\V2n,n (A)
As in [G], and using the references given there we have
Wϕ
δ
I2(n−k)
δ∈Nk (F )\GLk (F )
δ
!
g =
∗
ϕ(utn̄g)ψn (n̄u) du dt dn̄.
Nn−k−1 (F )\Nn−k−1 (A) T (F )\T (A) U2n,n (F )\U2n,n (A)
Here we view n̄ as an Sp2n matrix via the embedding of GLn in Sp2n .
Also ψn (n̄u) is defined by restriction. Let T1 ⊂ T be defined by
T1 = t ∈ T : all columns of t̂ are zero except the second one .
Thus T1 Mk×1 . Also, let L1 ⊂ L be defined by
L1 = ∈ L : all rows of ˆ are zero except the first one .
Thus L1 M1×k . Following (3.5)–(3.8) in [G] we obtain
L1 (A)
Wϕ
δ∈Nk (F )\GLk (F )
δ
I2(n−k)
δ
!
1 g
∗
=
Nn−k−1 (F )\Nn−k−1 (A) L1 (F )\L1 (A) T (F )T1 (A)\T (A) U2n,n (F )\U2n,n (A)
ϕ(ut1 n̄g)ψn (n̄u) du dt d1 dn̄.
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Here n̄ and u are integrated as before. Continuing this process, (as in [G]),
we obtain
!
δ
I2(n−k)
Wϕ
g
∗
δ
L(A) δ∈Nk (F )\GLk (F )
=
ϕ(un̄g)ψn (n̄u) du d dn̄.
Nn−k−1 (F )\Nn−k−1 (A) L(F )\L(A) U2n,n (F )\U2n,n (A)
The lemma follows from the fact that
LNn−k−1 U2n,n = j V2n,n−k−1 U2k,k τ (0, Yk , Zk ) .
Here Nn−k−1 is embedded in Sp2n via the embedding of Nn−k−1 =→ Nn =→
Sp2n , where the first embedding is as described in the beginning of the
proof.
A similar statement holds for I2 (ϕ,
φ, fσ,s ).
φ, fσ,s ) converges absolutely for all
THEOREM 2.3. — The integral I2 (ϕ,
s except for those s for which the Eisenstein series has a pole. For Re(s)
large :
φ, fσ,s ) =
Wϕ j(rτ ((x, 0, 0))g
I2 (ϕ,
V2k,k (A)\Sp2k (A) R(A) Xk (A)
ωψ (g)φ(x)fWσ,s (g) dx dr dg.
Here Wψ ∈ W(
π , ψn ) and fWσ ,s is as defined in Chapter 1, Section 5.
Theorems 2.1 and 2.3 show that both global integrals are Eulerian. In
the next chapter we shall study the local integral obtained from these
global integrals.
3. The Local Theory
In this chapter we shall study the local integral in question. We shall
compute the unramified integrals and prove some nonvanishing results.
Let F be a local field. Let π, π
, σ be irreducible representations of
(F ) and GLk (F ) resp. As before we shall assume that n > k.
Sp2n (F ), Sp
2n
When convenient, we shall write Sp2n for Sp2n (F ), etc. Let ψ be a nontrivial additive character of F . We shall denote by W(π, ψn ), W(
π , ψn )
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and W(σ, ψNk ) the Whittaker space associated with π, π
and σ resp. The
local Weil representation ωψ of the group Hk Sp2k acts on S(F k ) — the
Schwartz space of F k . We have the local version of Chapter 1, Section 6
of (1.2)–(1.4), for the action of ωψ .
Let ( , ) denote the local Hilbert symbol and γa (a ∈ F ∗ ) the local Weil
constant. Let
Sp
˜
I(W(σ,
ψNk ), s) = Ind 2k W(σ, ψNk ) ⊗ δ sP2k,k ⊗ γ −1 .
2k,k
P
˜
ψNk ), s) is a smooth
Thus a function f˜Wσ,s , or f˜s in short, in I(W(σ,
function on the group Sp2k which takes values in W(σ, ψNk ). More
g
2k there is a function Wσ,s
∈ W(σ, ψNk ) such that
precisely, given g ∈ Sp
f˜s
m ∗
−1
g
g = δ sP2k,k (m)γdet
m Wσ,s (m),
m∗
where m ∈ GLk and
m ∗
m∗
∈ P2k,k . Similarly we define
Sp2k I W(σ, ψNk ), s = Ind P2k,k
W(σ, ψNk ) ⊗ δ sP2k,k .
The local integrals we study are
˜
I1 (W, φ, fs ) =
W j(rτ ((x, 0, 0))g)
V2k,k \Sp2k
R Xk
V2k,k \Sp2k
R Xk
ωψ (g)φ(x)f˜s (g) dx dr dg
and
, φ, fs ) =
I2 (W
j(rτ ((x, 0, 0))g
W
ωψ (g)φ(x)fs (g) dx dr dg.
∈ W(
˜
π , ψn ), φ ∈ S(F k ), f˜s ∈ I(W(σ,
ψNk ), s)
Here W ∈ W(π, ψn ), W
and fs ∈ I(W(σ, ψNk ), s).
If F is a finite place we let p denote a generator of the maximal ideal
in the ring of integers of F . Also |p| = q −1 . For any local field, K(G) will
denote the standard maximal compact subgroup of G. It is well-known
and we shall identify K(Sp ) with its image
that K(Sp2k ) splits in Sp
2k
2k
in Sp2k .
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3.1. The Unramified Computations. —Let F be a nonarchimedean
field. Assume all data is unramified. More precisely, we assume that there
˜
ψNk ), s) which
are W ∈ W(π, ψn ), Wσ ∈ W(σ, ψNk ) and f˜s ∈ I(W(σ,
are fixed under the corresponding maximal compact subgroup. In other
words, W (k̄) = W (e) = 1 for all k̄ ∈ K(Sp2n ), and Wσ (k̄) = Wσ (e) = 1
for all k̄ ∈ K(GLk ) and f˜s (k̄) = f˜s (e) = 1 for all k̄ ∈ K(Sp2k ). In these
cases ψ is an unramified character of F . Let φ ∈ S(F k ) satisfy φ(x) = 1
if x ∈ Ok and zero otherwise. Here O is a ring of integers in F . Similarly,
∈ W(
we assume the existence of W
π , ψn ) and fs ∈ I(W(σ, ψNk ), s) with
similar properties.
Next we describe the L-functions we study. From general theory π
1
Sp
is a quotient of Ind Bn2n (αδ B2 n ), where Bn is the Borel of Sp2n and α
−1
an unramified character of Bn . Thus if t = diag(t1 , . . . , tn , t−1
n , . . . , t1 )
denotes the maximal torus of Sp2n then α(t) = α1 (t1 ) · · · αn (tn ), where
αi are unramified characters of F ∗ . Let
−1
Ap = diag α1 (p), . . . , αn (p), 1, α−1
n (p), . . . , α1 (p)
be the semisimple conjugacy class of SO2n+1 (C) attached to π. Similarly,
we may associate with σ a semisimple conjugacy class in GLk (C) denoted by
Bp = diag β1 (p), . . . , βk (p) .
The local tensor product L-function is defined
−1
L(π ⊗ σ, s) = det I(2n+1)k − Ap ⊗ Bp q −s
.
We also define the local symmetric square L-function of σ by
L(σ, V 2 , s) =
k
−1
.
1 − βi (p)βj (p)q −s
i,j=1
We prove
THEOREM 3.1. — For all unramified data and for Re(s) large
I1 (W, φ, f˜s ) =
L(π ⊗ σ, s(k + 1) − 12 k)
·
L(σ, V 2 , 2s(k + 1) − k)
Proof. — Let T denote the maximal torus of GLk . We parametrize T
as follows :
t = diag(a1 , a2 , . . . , ak ), ai ∈ F ∗ .
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Let t̂ denote the image of t in Sp2k under the embedding described in
Chapter 1, Section 2. Thus
−1
t̂ = diag(a1 , . . . , ak , a−1
k , . . . , a1 ).
It is easy to check that j(t̂) embeds in Sp2n as
−1
j(t̂) = diag(a1 , . . . , ak , 1, . . . , 1, a−1
k , . . . , a1 ).
Given a split algebraic matrix group G, we shall denote by B(G) its
Borel subgroup consisting of upper triangular matrices. We have
δ B(Sp2k ) (t̂) = |a1 |2k |a2 |2(k−1) · · · |ak |2 ,
δ B(Sp2n ) j(t̂) = |a1 |2n |a2 |2(n−1) · · · |ak |2(n−k+1) ,
δ B(GLk ) (t) = |a1 |k−1 |a2 |k−3 · · · |ak |−(k−1) ,
δ P2k,k (t̂) = |a1 · · · ak |k+1 .
To compute our integral, we apply the Iwasawa decomposition to obtain
that I1 (W, φ, f˜s ) equals
W j rτ ((x, 0, 0))t̂ ωψ (t̂)φ(x)f˜s (t̂)δ −1
B(Sp ) (t̂) dx dr dt.
2k
T
R Xk
Using the definition of f˜s this equals
s
−1
−1
W j rτ ((x, 0, 0))t̂ ωψ (t̂)φ(x)Wσ (t)γdet
t δ P2k,k (t̂)δ B(Sp
2k )
(t̂) dr dx dt.
Next we use the local version of Chapter 1, Section 6, and a change of
variables in r and x to obtain
1
W j t̂rτ ((x, 0, 0) Wσ (t)φ(x)| det t| 2 −(n−k)
δ sP2k,k (t̂)δ−1
B(Sp ) (t̂) dr dx dt.
2k
Since φ is supported on Ok we may ignore the x integration. Also,
as
in [G, p. 176] we may show that the support of the function r → W j(t̂r)
is in R(O). Hence we may ignore the r integration as well to obtain
1
W j(t̂) Wσ (t)| det t| 2 −(n−k) δ sP2k,k (t̂)δ −1
B(Sp ) (t̂) dt.
2k
T
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Denote
− 12
K j(t̂) = δ B(Sp
2n )
j(t̂) W j(t̂) ,
−1
2
Kσ (t) = δ B(GL
(t)Wσ (t).
k)
Plugging this to the above integral and using the fact that
1
2
δ B(Sp
2n )
j(t̂) δ −1
B(Sp
2k )
we obtain
T
1
k
1
1
2
(t̂)δ B(GL
(t)| det t| 2 −(n−k) = | det t|− 2 k
k)
1
K j(t̂) Kσ (t)| det t|s(k+1)− 2 k dt.
From the support properties of the Whittaker function on Sp2n this equals
K j(pn1 , . . . , pnk )∧ Kσ (pn1 , . . . , pnk ) xn1 +···+nk ,
n1 ≥n2 ≥···≥nk ≥0
1
where x = q −s(k+1)+ 2 k and
(pn1 , . . . , pnk ) = diag(pn1 , . . . , pnk ),
j(pn1 , . . . , pnk )∧ = diag(pn1 , . . . , pnk , 1, . . . , 1, p−nk , . . . , p−n1 ).
For any k positive numbers m1 ≥ m2 ≥ · · · ≥ mk ≥ 0 let
tr(m1 , m2 , . . . , mk , 0, . . . 0 | m1 , m2 , . . . , mk )
denote the trace of the irreducible finite dimensional representation of
SO2n+1 (C) × GLk (C) applied to a semisimple representative corresponding to π and σ, whose highest weight is (m1 , . . . , mk , 0, . . . , 0) in the
SO2n+1 component and (m1 , . . . , mk ) in the GLk component. Using the
Casselman-Shalika formula [CS], I1 (W, φ, f˜s ) equals
tr(n1 , . . . , nk , 0, . . . 0 | n1 , . . . , nk )xn1 +n2 +···+nk .
n1 ≥n2 ≥···≥nk ≥0
Next, we use the Poincaré identity
∞
L π ⊗ σ, s(k + 1) − 12 k =
tr S (Ap ⊗ Bp ) x .
=0
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Here S denote the symmetric -th power operation. It follows from [T1]
that the right-hand-side of the above identity equals
∞
tr S m (S 2 (Bp )) x2m
m=0
tr n1 , . . . , nk , 0, . . . , 0 | n1 , . . . , nk xn1 +···+nk
n1 ≥···≥nk ≥0
From this the theorem follows.
, φ, fs ) at unramified places. The definition of
Next we compute I2 (W
the local L-factor which corresponds to π
⊗ σ is not canonical. We have to
first make a choice of a nontrivial additive character ψ of F . We use ψ to
write the unramified character of the torus of Sp2n (F ) which corresponds
n and the
to π
. Thus, π
is a quotient of the representation induced from B
character
−1
diag(t1 , . . . , tn , t−1
n , . . . , t1 ), ε −→ εη1 (t1 ) · · · ηn (tn )γt1 ···tn ,ψ ,
where η1 , . . . , ηn are unramified characters of F ∗ . Let
Cp = diag η1 (p), . . . , ηn (p), ηn−1 (p), . . . , η1−1 (p) .
We define
π ⊗ σ, s) = det(I2nk − q −s Cp ⊗ Bp )−1 .
Lψ (
Note that
⊗ (σ ⊗ χa ), s ,
π ⊗ σ, s) = Lψ π
Lψa (
where χa (t) = (t, a), the quadratic character, which corresponds to a.
Note also that
π ⊗ σ, s) = L θψ (
π ) ⊗ σ, s ,
Lψ (
where θψ (
π ) is the image of π
under the local theta correspondence from
Sp2n (F ) to SO2n+1 (F ), with respect to ψ . This means that π = θψ (
π)
is such that
(n(2n+1))
⊗π
, π) = 0,
Hom ωψ
(n(2n+1))
where ωψ
is the Weil representation of
2n(2n+1) (F )
Sp
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as a dual pair.
We also define the exterior square L function of σ
L(σ, Λ2 , s) =
−1
1 − βi (p)βj (p)q −s
1≤i<j≤k
and the standard L function of σ
−1
L(σ, s) = det Ik − Bp q −s
We prove :
THEOREM 3.2. — For all unramified data and for Re(s) large
, φ, fs ) =
I2 (W
π ⊗ σ, s(k + 1) −
Lψ (
L(σ, s(k + 1) −
1
(k
2
1
k)
2
− 1))L(σ, Λ2 , 2s(k + 1) − k)
Proof. — Using similar notations as in the proof of Theorem 3.1 we
, φ, fs ) equals
obtain that I2 (W
j(t̂) Wσ (t)| det t| 12 −(n−k) δ sP (t̂)δ −1
W
B(Sp ) (t̂)γdet t dt,
2k,k
2k
T
where γdet t is obtained from the formulas in Chapter 1, Section 6. Denote
1
j(t̂) = δ − 2
K
B(Sp
2n )
−1
j(t̂) γdet
t W j(t) .
, φ, fs , s) equals
As in Theorem 3.1 we obtain that I2 (W
n1 ≥n2 ≥···≥nk ≥0
j(pn1 , . . . , pnk )∧ Kσ (pn1 , . . . , pnk )
K
χ(p)n1 +···+nk xn1 +···+nk
Here we used the relation γdet t γdet t = (det t, det t) and that (ε, ε) = 1 if
|ε| = 1. We denote
χ(x) = (x, x) = (−1, x).
To prove the Theorem we will follow [BFG, Section 5], using the formula
as established in [BFH]. In order to use [BFH] we parametrize π
for K
as in (1.9) of [BFH], such that αn+1−i = µi (p). Thus, the unramified
character, corresponding to π
in (1.9) of [BFH] is
−1
diag(t1 , . . . , tn , t−1
n , . . . , t1 ), ε −→ µ1 (t1 ) · · · µn (tn )γt1 ···tn ,ψ .
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−1
−1
Since γt = (t, t)γt,ψ
= χ(t)γt,ψ
, it follows that
−1
Lψ (
π ⊗ σ, s) = det I2nk − q −s χ(p)Cp ⊗ Bp
,
−1
where Cp = diag(µ1 (p), . . . , µn (p), µ−1
n (p), . . . , µ1 (p)).
Let A (resp. B) denote the alternator in the group algebra
"
#
C µ1 (p)±1 , . . . , µ±1
n (p)
(resp. C[β1 (p), . . . , βk (p)]) corresponding to the Weyl group of Sp2n (C)
(resp. GLk (C)). As in [BFG, p. 53] we have
1 +n 2 +n−1
µ2
· · · µnn
χSp(2n) (1 , . . . , k ) = ∆−1
Sp(2n) A µ1
and
m1 +k−1 m2 +k−2
χGL(k) (m1 , . . . , mk ) = ∆−1
β2
· · · βkmk
GL(k) B β1
Here
1 ≥ 2 ≥ · · · ≥ n ≥ 0,
m1 ≥ m2 ≥ · · · ≥ mk ≥ 0,
∆Sp(2n) = χSp(2n) (0, . . . , 0) and ∆GL(k) = χGL(k) (0, . . . , 0).
To express our integral in terms of the alternator, we need to use the
formula given in [BFH, Thm 1.2]. We have
j(pn1 , . . . , pnk )∧
(3.1) K
=
∆−1
Sp(m) A
µn1 µn−1
2
· · · µn
k
i=1
µni i
n
1
(1 − (p, p)q − 2 µ−1
j ) .
j=1
We recall that the difference between the above identity and the one stated
in [BFH, Thm 1.2], is due to a different way of the parametrization of the
semisimple conjugacy class associated with π
.
Next, using the Poincaré identity,
∞
1
tr S (Bp ⊗ Cp ) χ(p) x ,
L π
⊗ σ ⊗ χ, s(k + 1) − 2 k =
=0
using Theorem 2.5 in [T2], the right-hand-side equals
∞
tr S m Λ2 (Bp ) x2m C(µ, β; x),
m=0
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where
C(µ, β; x) =
χGL(k) (n1 , . . . , nk )χSp(2n) (n1 , . . . , nk , 0, . . . , 0)
n1 ≥···≥nk ≥0
χ(p)n1 +···+nk xn1 +···+nk .
We have
(3.2)
C(µ, β; x) =
k
1
(1 − βi q − 2 x)−1 ∆−1
Sp(2n)
i=1
χGL(k) (n1 , . . . , nk )
n1 ≥n2 ≥···≥nk ≥0
n
k
1
· · · µn
µni i
(1 − (p, p)q − 2 µ−1
)
A µn1 µn−1
2
j
i=1
j=1
χ(p)n1 +···+nk xn1 +···+nk .
Indeed, this is the analog of our case to identity (5.4) in [BFG]. The proof
of our formula is exactly the same. We omit the details. Using (3.1) we
see that (3.2) becomes
C(µ, β; x) = L σ, s(k + 1) − 12 (k − 1)
j(pn1 , . . . , pnk )∧ Kσ (pn1 , . . . , pnk )
K
n1 ≥n2 ≥···≥nk ≥0
χ(p)n1 +···+nk xn1 +···+nk .
Here we used the [CS] formula
Kσ (pn1 , . . . , pnk ) = χGL(k) (n1 , . . . , nk ).
On the other hand
∞
tr S m Λ2 (Bp ) x2m = L σ, Λ2 , 2s(k + 1) − k
m=0
Hence,
, φ, fs ) =
I2 (W
j(pn1 , . . . , pnk )∧
K
n1 ≥···≥nk ≥0
Kσ (pn1 , . . . , pnk ) χ(p)n1 +···+nk xn1 +···+nk
=
=
C(µ, β; x)
L(σ, s(k + 1) − 12 (k − 1))
Lψ (
π ⊗ σ, s(k + 1) −
L(σ, s(k + 1) −
1
(k
2
1
k)
2
− 1))L(σ, Λ2 , 2s(k + 1) − k)
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3.2. A Nonvanishing Result. — Let F be a local field. The main
result in this section is to prove that given s0 ∈ C there is a choice of data
such that our local integrals are nonzero. As in [JS], [S1] and [S2] we have
the following asymptotic expansion for the Whittaker spaces. Let
t = diag t1 · · · tn , t2 · · · tn , . . . , tn , t−1
n . . .)
be a parametrization of the maximal torus Tn of Sp2n . We have
(F ∗ )n such
LEMMA 3.3. — There is a finite set ∆ of finite functions
of
n
that for all W in W(π, ψn ) and for α ∈ ∆ there is φα ∈ S F × K(Sp2k )
such that
W (tk) =
φα (t1 , . . . , tn , m)α(t1 , . . . , tk ).
α∈∆
Here t ∈ T parametrized as above and m ∈ K(Sp2n ).
The asymptotic expansion for functions in W(σ, ψNk ) is given in [JS].
We use this to prove :
LEMMA 3.4. — For all W in W(π, ψn ), φ in S(F k ) and f˜s in
I(W(σ, ψNk ), s), the integral I1 (W, φ, f˜s ) converges absolutely for Re(s)
large.
Proof. — Let T denote the maximal torus of GLk parameterized by
t = (t1 · · · tk , t2 · · · tk , . . . , tk ) .
We let t̂ denote the embedding of t in Sp2k . Following the same steps as
in the unramified computation (Theorem 3.1), I1 (W, φ, f˜s ) equals
W j t̂rτ ((x, 0, 0))m ωψ (m)φ(x)
T
R Xk K(Sp2k )
f˜s (t̂m)| det t|α1 s+α2 dr dx dm dt.
be the subgroup of GLn defined by
Here α1 ∈ N and α2 ∈ Z. Let R
Ik
=
: r̃ ∈ M(n−k)×k .
R
r̃ In−k
in Sp2n via the embedding of GLn in Sp2n . One
We embed R
Ignoring the compact it is enough to
that j(Rτ (Xk )) = R.
absolute convergence of
W j(t̂)r̃ φ(r̃n−k )Wσ (t)| det t|α3 s+α4 dr̃ dt.
T R
Here α3 ∈ N, α4 ∈ Z and r̃n−k denotes the last row of
⊂ GLn . Now we can proceed as in Proposition 4.2
that T R
Proposition 7.1 in [JS].
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prove the
r̃. Notice
of [S1] or
L-FUNCTIONS FOR SYMPLECTIC GROUPS
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LEMMA 3.5. — With notations as in Lemma 3.4 the integrals I1 (W, φ, f˜s )
admits a meromorphic continuation to the whole complex plane. Moreover
this continuation is continuous in W , φ and f˜s .
Proof. — Using the notations of Lemma 3.4, we consider first the
integral
W j(t̂)r̃ φ(r̃n−k )Wσ (t)| det t|α1 s+α2 dr̃ dt.
(3.3)
T R
defined by
n−k−1 be the subgroup of R
Let R
: r̃n−k = 0 .
n−k−1 = r̃ ∈ R
R
Thus the above integral equals
W1 j(t)r̃ Wσ (t)| det t|α1 s+α2 dr̃ dt,
n−k−1
T R
where
W1 (m) =
Fk
W (mr̃n−k )φ(r̃n−k ) dr̃n−k .
with F k . Thus to prove
Here m ∈ Sp2n and we identified the last row of R
the meromorphic continuation of (3.3) it is enough to prove it for
W j(t)r̃ Wσ (t)| det t|α1 s+α2 dr̃ dt.
(3.4)
n−k−1
T R
Denote
k
i ein : i ∈ F ⊂ GLn .
Ln = In +
i=1
The function W is a linear combination of functions of the form
W2 =
φ()π()W1 d,
Ln
where φ ∈ S(Ln ) and W1 ∈ W(π, ψ). This is clear if F is nonarchimedean,
and follows from [DM] if F is archimedean. Here, as before we view Ln
as a subgroup of Sp2n via the embedding of GLn in Sp2n . Thus to prove
the meromorphic continuation of (3.4) we may consider
W j(t)r̃ φ()Wσ (t)| det t|α1 s+α2 d dr̃ dt.
n−k−1 Ln
T R
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Conjugating to the left this equals
$ n−k−1 )Wσ (t)| det t|α1 s+α2 dr̃ dt,
W j(t)r̃ φ(r̃
T Rn−k−1
where r̃n−k−1 denotes the n − k − 1 row of r̃ and φ$ the Fourier transform
of φ. Let
n−k−2 = r̃ ∈ R
n−k−1 : r̃n−k−1 = 0 .
R
Hence to prove the meromorphic continuation of the above integral it is
enough to consider
W j(t)r̃ Wσ (t)| det t|α1 s+α2 dr̃ dt.
n−k−2
T R
Continuing this process we get rid of the unipotent integration and
reduce to
W j(t) Wσ (t)| det t|α1 s+α2 dt.
T
The meromorphic continuation of this integral is obtained by Lemma 3.3.
This establishes the meromorphic continuation of (3.3). Moreover, it
follows from Theorem 2, Section 4, and Lemma 2, Section 5, in [S2]
that (3.3) admits a meromorphic continuation to all s ∈ C which is
continuous in W, Wσ and φ. Hence, as in Lemma 1 of [S2] we obtain
the meromorphic continuation of
W j t̂rτ ((x, 0, 0) m ωψ (m)φ(x)f˜s (t̂m) dr dx dm dt.
T
R Xk K(Sp2k )
From this the Lemma follows.
Finally we prove :
PROPOSITION 3.6. — Given s0 ∈ C there is W ∈ W(π, ψn ), φ ∈ S(F k )
˜
and a K(Sp2k )-finite function f˜s ∈ I(W(σ,
ψNk ), s) such that I1 (W, φ, f˜s )
is nonzero at s0 .
˜ s). Let Φ
Proof. — We construct the following family of sections in I(σ,
be an arbitrary smooth function on (P2k,k ∩ K(Sp2k )) \ K(Sp2k ), and
Sp
f˜ ∈ Ind 2k W(σ, ψNk ) ⊗ γ −1 . We denote
2k,k
P
f˜sΦ (pm) = δ sP2k,k (p)f˜(pm)Φ(m),
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Φ
˜ s). Denote
∈ I(σ,
where p ∈ P2k,k and m ∈ K(Sp2k ). Thus f˜σ,s
W j rτ ((x, 0, 0))gm
I11 (W, φ, s; m) =
Nk \GLk
ωψ (gm)φ(x)δ −1
P2k,k (g) dx dr dg.
R Xk
Then, as can be checked, we have for Re(s) large
Φ
˜
(3.5)
I1 (W, φ, fs , s) =
I11 (W, φ, s; m)Φ(m) dm.
K1 \K(Sp2k )
As in Lemma 3.5 we can show that I11 (W, φ, s; m) which converges for
Re(s) large, admits a meromorphic continuation to the whole complex
plane. Moreover, I11 (W, φ, s; m) is a continuous function in m. Suppose
that I1 (W, φ, fs ) is zero at s = s0 for all choice of data. This implies
that the right-hand-side of (3.5) is zero at s0 . Since Φ is arbitrary
we may deduce that the meromorphic continuation of I11 (W, φ, s; m) is
zero at s = s0 for all choice of data. Plugging m = 1 and applying the
definition of f˜ we may deduce that the meromorphic continuation of
W j rτ ((x, 0, 0) g) ωψ (g)φ(x)
Nk \GLk R Xk
−1
Wσ (g)| det g|(s−1)(k+1) γdet
g dx dr dg
is zero at s = s0 , for all choice of data. Conjugating g to the left, and
changing variables we may assume that the meromorphic continuation of
W j grτ ((x, 0, 0) φ(x)
(3.6)
Nk \GLk R Xk
Wσ (g)| det g|αs+β dx dr dg
is zero at s = s0 for all choice of data. Here α ∈ N and β ∈ Z. Recall that
φ ∈ S(F k ). Consider, for x ∈ F k
I12 (W, Wσ , s; x) =
W j grτ ((x, 0, 0) Wσ (g)| det g|αs+β dr dg.
Nk \GLk R
As before, we can prove that I12 (W, Wσ , s; x) converges absolutely for Re(s)
large and admits a meromorphic continuation to the whole complex plane.
Thus (3.6) equals
Xk
I12 (W, Wσ , s; x)φ(x) dx.
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Hence we may deduce that the meromorphic continuation of the above
integral is zero at s = s0 for all choice of φ ∈ S(F k ). Thus I12 (W, Wσ , s; x)
is zero at s = s0 for all choice of data. Let x = 0, and hence we may
assume that the meromorphic continuation of
W j(gr) Wσ (g)| det g|αs+β dr dg
Nk \GLk
R
is zero at s = s0 for all W and Wσ . At this point we will use a similar
process as in the proof of Lemma 3.5.
Using the notations there we may rewrite the above integral as
W j(g)r̃ Wσ (g)| det g|αs+β dr̃ dg.
n−k−1
Nk \GLk R
Let φ ∈ S(Ln ). Replace W by
φ()π()W d.
Ln
Thus the above integral equals
W (j(g)r̃)φ()Wσ (g)| det g|αs+β d dr̃ dg
Nk \GLk R
L
n−k−1 n
$ n−k−1 )Wσ (g)| det g|αs+β dr̃ dg,
=
W j(g)r̃ φ(r̃
n−k−1
Nk \GLk R
where φ$ is the Fourier transform of φ and r̃n−k−1 denotes the n − k − 1
n−k−1 . Arguing as before, we may deduce that the meromorphic
row of R
continuation of
W j(g)r̃ Wσ (g)| det g|αs+β dr̃ dg
n−k−2
Nk \GLk R
is zero at s = s0 for all choice of data. Continuing this process, we may
assume that the meromorphic continuation of
W j(g) Wσ (g)| det g|αs+β dg
N k\GLk
is zero at s = s0 for all choice of data.
Arguing as in [JS] this implies that W (e)Wσ (e) is zero for all W and
Wσ which is a contradiction.
, φ, fs ). As in the case of I1 (W, φ, f˜s )
We have a similar result for I2 (W
we may prove that I2 (W , φ, fs ) converges absolutely for Re(s) large and
admits a meromorphic continuation to the whole complex plane, which is
, φ and fs . From this as in Proposition 3.6 we have :
also continuous in W
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∈ W(
π , ψn ), φ ∈ S(F k )
PROPOSITION 3.7. — Given s0 ∈ C there is W
, φ, fs )
and a K(Sp2k )-finite function fs ∈ I(W(σ, ψNk ), s) such that I2 (W
is nonzero at s = s0 .
4. A Double Coset Decomposition
In this Chapter we shall prove a Lemma and fix some notations needed
later. We fix n and k with k > n. Denote Q02k,k−n = Sp2n V2k,k−n . Clearly
Q02k,k−n is a subgroup of Q2k,k−n . We embed GLk−n in Sp2k as
g −→
g
I2n
g
,
∗
g ∈ GLk−n
and all subgroups of GLk−n will be embedded in Sp2k via this embedding.
Here g ∗ is such that the above matrix is in Sp2k .
For given 0 ≤ i ≤ k − n denote by Mi the maximal parabolic of GLk−n
defined by
Mi = GLi × GLk−n−i Li .
We let M0 = Mk−n = GLk−n . Here Li is the unipotent radical of Mi and
we choose it to consist of lower unipotent, that is,
Li =
Ii
A Ik−n−i
: A ∈ M(k−n−i)×i .
For 0 ≤ i ≤ k − n let Wi denote the Weyl group of GLi . We shall
identify Wi with all i × i permutation matrices. Finally for 0 ≤ i ≤ k − n
we set
γi =
Ii
I2(k−i)
−Ii
.
Thus γi is an element of the Weyl group of Sp2k .
LEMMA 4.1. — A set of representatives for P2k,k \ Sp2k /Q02k,k−n is
contained in the set of all matrices of the form γi w, where 0 ≤ i ≤ k − n
and w ∈ (Wi × Wk−n−i ) \ Wk−n .
Proof. — Clearly P2k,k−n ⊃ Q02k,k−n . It is not hard to check that γi ,
0 ≤ i ≤ k − n is a set of representatives for P2k,k \ Sp2k /P2k,k−n . Indeed,
the space P2k,k \ Sp2k can be identified with the set of all k dimensional
isotropic subspaces. Hence, we can parametrize the above double cosets
with all possible intersections of k − n dimensional isotropic subspaces
with all k dimensional isotropic subspaces and for these we can choose
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as representatives γi . Hence every representative of P2k,k \ Sp2k /Q02k,k−n
can be chosen in the set of all elements γi w, where
w ∈ (γi−1 P2k,k γi ∩ P2k,k−n ) \ P2k,k−n /Q02k,k−n .
Since γi−1 P2k,k γi ∩ P2k,k−n ⊃ Mi then we may choose a set of representatives for the set
(γi−1 P2k,k γi ∩ P2k,k−n ) \ P2k,k−n /Q02k,k−n
to be contained in the set Mi \ GLk−n /Nk−n . (The group Nk−n is
defined in Chapter 1, Section 4. Clearly, a set of representatives for Mi \
GLk−n /Nk−n can be chosen to be contained in (Wi ×Wk−n−i )\Wk−n .)
Next we prove :
LEMMA 4.2. — For 0 ≤ i ≤ n−k, each representative of (Wi ×Wk−n−i )\
Wk−n can be chosen to be a symmetric matrix.
Proof. — Let w1 , . . . , wk−n−1 denote the simple reflections of Wk−n so
that w1 , . . . , wi are in Wi and wi+2 , . . . , wk−n−1 are in Wk−n−i . Given
w ∈ Wk−n we shall denote by w(r, j) its (r, j) entry. Let
w=
A B
C D
∈ Wk−n ,
where
A ∈ Mi×i , B ∈ Mi×(k−n−i) , C ∈ M(k−n−i)×i , D ∈ M(k−n−i)×(k−n−i) .
By multiplying from the left by elements of Wi × Wk−n−i we need to
bring w to a symmetric matrix. Recall that left multiplication is just
changing the rows of w. We claim that by multiplication on the left by Wi
we can bring w to the following form. First if w(r, j) = 1, where r ≤ i
and j ≤ i then r = j. In other words, all nonzero entries of A are on the
main diagonal. Secondly, suppose that w(r1 , j1 ) = w(r2 , j2 ) = 1, where
r1 < r2 ≤ i and i + 1 ≤ j1 , j2 ≤ k − n then j1 < j2 . In other words B can
be put in row echelon form. Indeed, if w(1, j) = 1, where j ≤ i, we can
use w1 , . . . , wi−1 to assume that j = 1 and then proceed by induction.
If w(1, j) = 0 for all j ≤ i, then clearly A contains at most i − 1 ones and
hence B is nonzero. Let
p1 = min w(r, j) = 1 where 1 ≤ r ≤ i, j ≥ i + 1 .
j
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Using w1 , . . . , wi−1 we may assume that w(1, p1 ) = 1. Now we proceed by
induction i.e. using w2 , . . . , wi−1 we may arrange all rows in w between
two and i in the desired way. Hence we may assume that A and B in w
has the above pattern.
Next, using Wk−n−i we apply similar arguments to C and D. More
precisely, suppose that w(r1 , p1 ) = 1, where 1 ≤ r1 ≤ i, i + 1 ≤ p1
and if w(r2 , p2 ) = 1 with 1 ≤ r2 ≤ i and i + 1 ≤ p2 then p1 < p2 .
This means that w(r, j) = 0 for all 1 ≤ r ≤ i and i + 1 ≤ j < p1 .
Hence, using wi+2 , . . . , wk−n−i we may assume that w(r, r) = 1 for all
i + 1 ≤ r < p1 . Since w(r1 , p1 ) = 1 then w(r1 , r1 ) = 0 and hence, from
the fact that all nonzero entries of A are on the diagonal we may assume
that if w(r, r1 ) = 1 then r > i. But since w(j, j) = 1 for all i + 1 ≤ j < p1
then if w(r, r1 ) = 1 then r ≥ p1 . Hence using wp1 , . . . , wk−n−i we may
assume that w(p1 , r1 ) = 1. Continuing this process we may assume that
if w(r, p) = 1 then w(p, r) = 1 and all nonzero entries of A and D are on
the main diagonal. In particular w is symmetric.
Let α1 , . . . , αk−n−1 denote the simple roots of GLk−n corresponding
to Nk−n . Let xα (t) denote the one parameter unipotent subgroup of Nk−n
corresponding to the root α. Thus,
xαj (t) = Ik−n + tej,j+1 .
Here er,j denotes the (k − n) × (k − n) matrix whose only nonzero entry
is one in the (r, j) position.
We shall agree that w = e is the representative of the coset Wi ×Wk−n−i
in Wk−n .
LEMMA 4.3. — Let w ∈ (Wi × Wk−n−i ) \ Wk−n such that w = e. Then
there exists a simple root αj such that wxαj (t)w−1 ∈ Li .
Proof. — Let e = w ∈ (Wi × Wk−n−i ) \ Wk−n . We assume that w is as
A B
described in the proof of Lemma 4.2. In other words, if w =
C D
then w is symmetric and all nonzero entries of A and D are on the
main diagonal. Since w = e, C = 0. Let r1 be the smallest integer such
that w(r1 , j1 ) = 1, where r1 > i and 1 ≤ j1 ≤ i. If r1 > i + 1 then
w(r1 − 1, r1 − 1) = 1. Hence
wxαr1 −1 (t)w−1 = I + twer1 −1,r1 = I + ter1 −1,j1 ∈ Li .
Hence we may assume that r1 = i+1. Thus w(i+1, j1 ) = 1. Let r2 > r1 be
the smallest integer such that w(r2 , j2 ) = 1, where 1 ≤ j2 ≤ i. Of course
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such an r2 may not exist. However if it does, then arguing as with r1 ,
we may assume that r2 = i + 2. We claim that j2 = j1 + 1. Indeed,
from the shape of w we may deduce that j2 > j1 . If j2 = j1 + 1, then
w(j1 + 1, j1 + 1) = 1 and hence
wxαj1 (t)w−1 = I + twej1 ,j1 +1 w−1 = I + tei+1,j1 +1 ∈ Li .
Thus we may conclude that w has the shape

I
p

w=
0
Is
Iq
Is
0

,
Ik
where p + q + s = i and k = k − p − q − 2s. p, q or k can be zero but not s,
since w = e. If q = 0, then w(p + s + 1, p + s + 1) = w(i + s, p + s) = 1
and then
wxαp+s (t)w−1 = I + twep+s,p+s+1 w−1 = I + tei+s,p+s+1 ∈ Li .
If q = 0 then wxαi (t)w−1 ∈ Li . We are done.
5. The Global Integrals (k > n)
In this chapter, we introduce the global integrals which will represent
the tensor product L-function in the case when k > n. These integrals
are “dual” to the ones introduced in Chapter 2 in the sense that the cusp
form and the Eisenstein series are interchanged. We define
J1 (ϕ, φ, f˜σ,s ) =
Sp2n (F )\Sp2n (A) Hn (F )\Hn (A) V2k,k−n−1 (F )\V2k,k−n−1 (A)
(h)g)ψk−n−1 (v) dv dh dg.
ϕ(g)θ̃φ (hg)E(vτ
˜ s). We have the covering version,
Here ϕ ∈ Vπ , φ ∈ S(An ) and f˜σ,s ∈ I(σ,
φ, fσ,s ) =
J2 (ϕ,
Sp2n (F )\Sp2n (A) Hn (F )\Hn (A) V2k,k−n−1 (F )\V2k,k−n−1 (A)
ϕ(g)
θ̃φ (hg)E(vτ (h)g)ψn−k−1 (v) dv dh dg.
Here ϕ
∈ V
and fσ,s ∈ I(σ, s). In the next Theorem we shall show that
π
these integrals are Eulerian.
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First, let

γ = I
0
0
k−n
0
In
0
0
0
211

0
0
0 −Ik−n 
.
0
0
0
In
We shall also need to consider the following unipotent subgroup of V2k,k−n .
Let




 Ik−n 0 t 0∗
such
that
the
t
∈
M
t 
In 0
(k−n)×n
:
Tk−n = 
In
0


bottom row of t is zero
Ik−n
and let ξ0 = (0, . . . , 1) ∈ F n . Due to the embedding of Hn in Sp2n as
described in Chapter 1 Section (3), we need to change the character ψNk .
More precisely, let ψNk be the character of Nk defined by
ψNk (n̄) = ψ
n−1
k−1
n̄i,i+1 + 2n̄n,n+1 +
i=1
n̄i,i+1 .
i=n+1
Here n̄ = (n̄ij ) ∈ Nk . Since σ is a cuspidal representation of GLk ,
W(σ, ψNk ) = W(σ, ψNk ).
˜ s),
As in Chapter 1, Section 5 we denote for f˜σ,s ∈ I(σ,
f˜σ,s (n̄g)ψNk (n̄) dn̄.
f˜Wσ,s (g) =
Nk (F )\Nk (A)
A similar definition applies to fσ,s ∈ I(σ, s).
We have :
THEOREM 5.1. — The integral J1 (ϕ, φ, f˜σ,s ) converges absolutely for
all s except for those s for which the Eisenstein series has a pole. For
Re(s) large :
J1 (ϕ, φ, f˜σ,s ) =
V2n,n (A)\Sp2n (A) Yn (A)\Hn (A) Tk−n (A)Nk−n (A)\V2k,k−n−1 (A)
Wϕ (g)ωψ (hg)φ(ξ0 )f˜Wσ ,s (γvτ (h)g) dv dh dg.
Here Wϕ ∈ W(π, ψn ) and as always we view Nk−n as a subgroup of Sp2k
via the embedding of GLk−n in Sp2k .
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Proof. — We prove that J1 (ϕ, φ, f˜σ,s ) is Eulerian. Unfolding the Eisentstein series and interchanging summation with integration we obtain that
J1 (ϕ, φ, f˜σ,s ) equals
(5.1)
δ
δ (F )\Sp
δ
δ
Sp2n
2n (A) Hn (F )\Hn (A) V2k,k−n−1 (F )\V2k,k−n−1 (A)
ϕ(g)θ̃φ (hg)f˜σ,s (δvτ (h)g)ψk−n−1 (v) dv dh dg,
where δ ∈ P2k,k (F ) \ Sp2k (F )/Q02k,k−n and where, for a given group
G ⊂ Sp2k ,
G δ = δ −1 P2k,k δ ∩ G and Hnδ = τ −1 δ −1 P2k,k δ ∩ τ (Hn ) .
Recall that Q02k,k−n = Sp2n · τ (Hn )V2k,k−n−1 . From Lemma 4.1 we may
choose δ = γi w, where 0 ≤ i ≤ k − n and w ∈ (Wi × Wk−n−i ) \ Wk−n .
Let δ = γi w with 0 ≤ i ≤ k − n − 1 and w = e as above. It follows
from Lemma 4.3 that there exists a simple root αj of GLk−n such that
wxαj (t)w−1 ∈ Li . (See Chapter 4 for notations.) It is easy to check
that γi Li γi−1 ⊂ U2k,k . However f˜σ,s is left invariant under elements in
U2k,k (A). Thus γi wxαj (t)(γi w)−1 %∈ U2k,k (A). Since ψk−n−1 restricted
to xαj (t) is nonzero we end up with F \A ψ(t) dt as an inner integral. Hence
the contribution of such δ to J1 (ϕ, φ, f˜σ,s ) is zero. Assume δ = γi with
0 ≤ i ≤ k−n−1. It is not hard to check that γi τ (Zn )γi−1 = τ (Zn ) ⊂ U2k,k .
However θ̃φ ((0,
% 0, z)hg) = ψ(z)θ̃φ (hg) for all z ∈ Zn (A). Hence, once
again we get F \A ψ(z) dz as an inner integral. Hence in (5.1) we are left
with δ = γk−n . Since we may change γk−n on the left with any Weyl
element of GLk , we may replace γk−n by γ. Simple matrix multiplication
shows that
Spγ2n = P2n,n ;
γ
V2k,k−n−1
= Nk−n Tk−n .
Hnγ = Yn ;
Indeed, recall that if g =
A B
C D
∈ Sp2n , its embedding in Sp2k is
g −→
Ik
g
Ik
,
and after conjugating with γ, the image of the element above is
&
'
A
C
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such an element lies in P2k,k if and only if C = 0, and hence Spγ2n = P2n,n .
The other identities follow similarly. Hence J1 (ϕ, φ, f˜σ,s ) equals
ϕ(g)θ̃φ (hg)f˜σ,s γvτ (h)g ψk−n−1 (v) dv dh dg.
Here g is integrated over P2n,n (F ) \ Sp2n (A), h is integrated over Yn (F ) \
Hn (A) and v over Tk−n (F )Nk−n (F ) \ V2k,k−n−1 (A).
0
Let Mn−1
⊂ GLn denote the mirabolic subgroup of GLn . Thus
0
Mn−1
=
∗ ∗
0 1
∈ GLn .
Using Lemma 4.3.2 in [GPS] we have
θ̃φ (hg) = ωψ (hg)φ(0) +
ωψ (δhg)φ(ξ0 ),
0
δ∈Mn−1
(F )\GLn (F )
where we recall that ξ0 = (0, . . . , 0, 1) ∈ F n . We plug this expansion in
the above integral. We consider the contribution of each summand. The
first term contributes
(5.2)
ϕ(g)ωψ (hg)φ(0)f˜σ,s γvτ (h)g ψk−n−1 (v) dv dh dg.
We claim that this integral is zero. Indeed, let P2n,n = GLn U2n,n . Write
=
P2n,n (F )\Sp2n (A)
.
GLn (F )U2n,n (A)\Sp2n (A) U2n,n (F )\U2n,n (A)
Thus we obtain as an inner integral to (5.2)
ϕ(ug)ωψ (hug)φ(0)f˜σ,s γvτ (h)ug ψk−n−1 (v) du dh dv,
where u is integrated over U2n,n (F ) \ U2n,n (A) and v and h as before.
Conjugating u to the left, using the left invariant properties of ωψ and f˜σ,s
and changing variables, we obtain
ϕ(ug) du
U2n,n (F )\U2n,n (A)
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as an inner integral, which is zero by cuspidality. Hence J1 (ϕ, φ, f˜σ,s )
equals
ωψ (δhg)φ(ξ0 )f˜σ,s γvτ (h)g ψk−n−1 (v) dv dh dg.
ϕ(g)
0
δ∈Mn−1
(F )\GLn (F )
Collapsing the summation with the integration this equals
ϕ(g)ωψ (hg)φ(ξ0 )f˜σ,s γvτ (h)g ψk−n−1 (v) dv dh dg,
0
(F )U2n,n (F ) \ Sp2n (A). Write
where g is integrated over Mn−1
=
0
Mn−1
(F )U2n,n (F )\Sp2n (A)
0
Mn−1
(F )U2n,n (A)\Sp2n (A)
.
U2n,n (F )\U2n,n (A)
Thus J1 (ϕ, φ, f˜σ,s ) equals
ϕ(ug)ωψ (hug)φ(ξ0 )f˜σ,s γvτ (h)ug ψk−n−1 (v) dv dh du dg.
0
(F )U2n,n (A) \ Sp2n (A), u over U2n,n (F ) \
Here g is integrated over Mn−1
U2n,n (A) and h and v as before. We have, using Chapter 1, Section 6,
formula (1.4),
ωψ (uhg)φ(ξ0 ) = ψn (u)ωψ (hg)φ(ξ0 ),
u ∈ U2n,n (A).
Here ψn is as defined in Chapter 1, Section 4. Conjugating u to the left,
changing variables in v and h the above integral equals
ϕ(ug)ψn (u) du ωψ (hg)φ(ξ0 )f˜σ,s γvτ (h)g ψk−n−1 (v) dv dh dg.
Let GLn−1 ⊂ GLn be embedded as
g −→
As in [PS] we have
U2n,n (F )\U2n,n (A)
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ϕ(ug)ψn (u) du =
δ∈Nn−1 (F )\GLn−1 (F )
◦
2
Wϕ (δg).
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L-FUNCTIONS FOR SYMPLECTIC GROUPS
0
we may collapse the summation with
From the definition of Mn−1
integration to obtain that J1 (ϕ, φ, f˜σ,s ) equals
Wϕ (g)ωψ (hg)φ(ξ0 )f˜σ,s γvτ (h)g ψk−n−1 (v) dv dh dg,
where g is integrated over Nn (F )U2n,n (A)\Sp2n (A) and v and h as before.
Write
=
.
Yn (F )\Hn (A)
Yn (A)\Hn (A)
Yn (F )\Yn (A)
Let y = (y1 , . . . , yn ) ∈ Yn . It follows from Chapter 1, Section 6,
formula (1.2) that
ωψ (0, y, 0)hg φ(ξ0 ) = ψ(y1 )ωψ (hg)φ(ξ0 ).
Write (recall that V2n,n = Nn U2n,n )
=
Nn (F )U2n,n (A)\Sp2n (A)
V2n,n (A)\Sp2n (A)
.
Nn (F )\Nn (A)
−1
We have Wϕ (n̄g) = ψN
(n̄)Wϕ (g) for all n̄ ∈ Nn . Also, write
n
Tk−n (F )Nk−n (F )\V2k,k−n−1 (A)
=
.
Tk−n (A)Nk−n (A)\V2k,k−n−1 (A) Tk−n (F )Nk−n (F )\Tk−n (A)Nk−n (A)
Combining all this, we obtain as an inner integration
−1
f˜σ,s γv τ (0, y, 0) n̄vτ (h)g ψk−n−1 (v )ψN
(n̄)ψ(y1 ) dv dy dn̄.
n
Here v is integrated over Tk−n (F )Nk−n (F ) \ Tk−n (A)Nk−n (A), y over
Yn (F ) \ Yn (A) and n̄ over Nn (F ) \ Nn (A). It follows from matrix multiplication that
γTk−n Nk−n τ (Yn )Nn γ −1 = Nk
and that if γv τ (0, y, 0)n̄γ −1 = nk ∈ Nk then
ψk−n−1 (v )ψNn (n̄)ψ(y1 ) = ψNk (nk ).
Hence the above integral equals to f˜Wσ ,s (γvτ (h)g). From this the Theorem
follows.
We also have the same Theorem for the integral J2 (ϕ,
φ, fσ,s ).
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THEOREM 5.2. — The J2 (ϕ,
φ, fσ,s ) converges absolutely for all s except
for those s for which the Eisenstein series has a pole. For Re(s) large,
φ, fσ,s ) =
J2 (ϕ,
V2n,n (A)\Sp2n (A) Yn (A)\Hn (A) Tk−n (A)Nk−n (A)\V2k,k−n−1 (A)
Wϕ(g)ωψ (hg)φ(ξ0 )fWσ ,s γvτ (h)g dv dh dg,
where Wϕ ∈ W(
π , ψn ).
6. The Local Theory (k > n)
In this section we shall study the local integrals in the case where
k > n. The approach here will be based on the method developed in [S1]
and [JPSS]. We shall keep the notations introduced in Section 3. The local
integrals to be studied are the ones which come from the factorization of
the global integrals introduced in Section 5. More precisely, define
J1 (W, φ, f˜s ) =
V2n,n \Sp2n Yn \Hn Tk−n Nk−n \V2k,k−n−1
W (g)ωψ (hg)φ(ξ0 )f˜Wσ,s γvτ (h)g dv dh dg.
φ ∈ S(F n ) and f˜W ∈ I(W(σ,
˜
ψ −1 ), s). We have a
Here W ∈ W(π, ψ),
σ,s
similar integral for the covering case. That is, let
, φ, fs ) =
J2 (W
V2n,n \Sp2n Yn \Hn Tk−n Nk−n \V2k,k−n−1
(g)ωψ (hg)φ(ξ0 )fWσ ,s (γvτ (hg) dv dh dg,
W
and fW ∈ I(W(σ, ψ −1 ), s). As before we shall
∈ W(
where now W
π , ψ)
σ,s
˜
˜
write fσ,s for fWσ,s , etc. Section 6.1, which follows, is presented in the
formal level (i.e. ignoring convergence issues.) All justifications and basic
properties of the local integrals are deferred to Section 6.3.
6.1. — We start our local analysis by proving a formal identity. This
identity is analogous to the one proved in [S1, Section 11.4] and is based
on similar ideas. We start with a few assumptions and notations.
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Let P 2n,n denote the opposite parabolic to P2n,n . We shall assume that
Sp2n
π = Ind P
2n,n
1
+ζ σ ⊗ δ P2
,
2n,n
where σ is an admissible generic representation of GLn . We shall denote
1
+ζ Sp
its central character by ωσ . Given ϕσ ,ζ ∈ Ind P 2n W(σ , ψ) ⊗ δ P2
2n,n
2n,n
we set
(6.1)
W (g) =
ϕσ ,ζ (ug)ψn (u) du.
U2n,n
Here g ∈ Sp2n . The integral converges for Re(ζ) large and as in [S1] it has
an analytic continuation to the whole ζ plane and hence W ∈ W(π, ψ).
there is a function ϕσ ,0 such that (6.1) holds
Also, given W ∈ W(π, ψ)
in the sense of analytic continuation. Recall that
2k Sp
˜
I(W(σ,
ψ −1 ), s) = Ind P2k,k
W(σ, ψ −1 ) ⊗ δ sP2k,k ⊗ γ −1 .
Another induced space we need is
2k Sp
˜
I(W(σ
⊗ χ, ψ −1 ), s) = Ind P2k,k
W(σ ⊗ χ, ψ −1 ) ⊗ δ sP2k,k ⊗ γ −1 ,
m
= (det m, det m) and ( , ) denotes the local Hilbert
m∗
−1
symbol. Due to the relation γdet
m · (det m, det m) = γdet m ,
where χ
2k Sp
I˜ W(σ ⊗ χ, ψ −1 ), s = Ind P2k,k
W(σ, ψ −1 ) ⊗ δ sP2k,k ⊗ γ .
˜
ψ −1 ), s) we associate to it a function
Given a function f˜Wσ ,s ∈ I(W(σ,
−1
˜
˜
fWσ ,s,χ ∈ I(W(σ ⊗ χ, ψ ), s) satisfying
(
)
m ∗
˜
f˜Wσ ,s,χ
∗ g = χ(m)fWσ,s (g).
m
We shall write f˜σ,s,χ for f˜Wσ,s,χ . We shall denote

0 −2In
−In
 Ik−n 0
and let w̄0 = 
wn =
In

0
− 12 In

Ik−n .
0
Also, given g ∈ Sp2n we set j (g) = w̄0 g w̄0−1 . Finally we shall denote
by γ(σ × σ , s, ψ −1 ) the gamma factor attached to σ and σ as defined
in [JPSS]. We shall prove :
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PROPOSITION 6.1. — With the above notations let W (g) be given
by (6.1). We have
ωσ (−1)k−2 γ(σ × σ , (k + 1)s − (n + 1)ζ − 12 k, ψ −1 )J1 (W, φ, f˜σ,s )
= ϕσ ,ζ (wn g)ωψ (g)φ(2x)
f˜σ,s,χ wk uj rτ ((x, 0, 0))g w̄0 ψk (u) du dx dr dg.
Here g is integrated over V2n,n \ Sp2n , x over F n , u over U2k,k and r
over R (see Chapter 2 for the definition of R). Each integral converges in
some (s, ζ) domain and admit a meromorphic continuation to all values
of (s, ζ). The above equality is understood in the analytic continuation
sense.
Proof. — At this point we shall ignore convergence issues and prove
formally the above identity. The convergence properties will be dealt with
in Section 6.3. As can be seen, we may identify the product of the groups
Tk−n Nk−n \ V2k,k−n−1 and τ (Yn \ Hn ) with the group for all matrices in
Sp2k of the form


(A, 0, B) := 
Ik−n A
In
0
0
In
B
0
A∗
,
Ik−n
where A ∈ M(k−n)×n , and B ∈ M(k−n)×k−n) satisfying B t Jk−n = Jk−n B.
Also, A∗ = −Jn At Jk−n . Under the above identification (x, 0, 0) in Hn is
identified with the last row of A and (0, 0, z) is identified with bk−n,k−n .
Hence we can write
J1 (W, φ, f˜σ,s ) = W (g)ωψ (x, 0, z)g φ(ξ0 )f˜σ,s γ(A, 0, B)g dA dB dg.
Here g is integrated over V2n,n \Sp2n and A and B as above. Plugging (6.1)
in the above integral and collapsing integrations, J1 (W, φ, f˜σ,s ) equals
ϕσ ,ζ (g)ωψ (x, 0, z)g φ(ξ0 )f˜σ,s γ(A, 0, B)g dA dB dg,
where now g is integrated over Nn \ Sp2n . Proposition 6.8 says that the
last integral converges absolutely in a domain of the form

(n + 1) Re ζ+ 12 +C < (k + 1) Re(s),



(k + 1) Re(s) < (1 + ε0 )(n + 1) Re ζ+ 12 + Q,



R < Re(ζ),
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where Q, C, R, ε0 are constants which depend on (σ, σ , k, n) and ε0 > 0.
We continue the calculation in this domain. See, also, the remark prior
to Proposition 6.8.
Factor
=
.
Nn \Sp2n
GLn \ Sp2n
Nn \GLn
Hence J1 (W, φ, f˜σ,s ) equals
m
m
ϕσ ,ζ
∗ g ωψ (x, 0, z)
∗ g φ(ξ0 )
m
m
&
&
' '
Ik−n
m
˜
g dA dB dm dg,
fσ,s γ(A, 0, B)
∗
m
Ik−n
where m is integrated over Nn \ GLn and g over GLn \ Sp2n . We have
δ P 2n,n
m
m∗
= | det m|−(n+1) .
From the definition of ϕσ ,ζ we can write
ϕσ ,ζ
m
m
∗
1
−(n+1)( +ζ)
2
g = | det m|
Wσ ,g,ζ (m),
where Wσ ,g,ζ ∈ W(σ , ψ). We also have
m
ωψ (x, 0, z)
m
1
∗ g φ(ξ0 ) = | det m| 2 γdet m ωψ (xm, 0, z)g φ(ξ0 m).
Finally conjugating m to the left in f˜σ,s and changing variables in A,
J1 (W, φ, f˜σ,s ) equals
Wσ ,g,ζ (m)ωψ (x, 0, z))g φ(ξ0 m)
m
˜
I2(k−n)
fσ,s
γ(A, 0, B)g
∗
m
1
γdet m | det m|−(n+1)ζ+ 2 n−k dA dB dm dg,
where all the integration remains as before. ¿From the definition of f˜σ,s
there is a function F ∈ W(σ, ψ) such that
f˜σ,s
m ∗
(k+1)s −1
γdet m Fσ,h (m),
∗ , ε = | det m|
0 m
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2k . To simplify notations, we shall write
where m ∈ GLk and h ∈ Sp
f˜σ,s (h; m) for Fσ,h (m). Thus, J1 (W, φ, f˜σ,s ) equals
(6.2)
Wσ ,g,ζ (m)ωψ (x, 0, z)g φ(ξ0 m)
m
f˜σ,s γ(A, 0, B)g;
Ik−n
1
| det m| 2 n−k+(k+1)s−(n+1)ζ dm dA dB dg,
where the integrations are as before. To proceed let Wσ (h) ∈ W(σ, ψ),
and given φ1 ∈ S(F n ) define
I et
0
n
1
0
σ(φ1 )Wσ (h) =
Wσ h
φ1 (e) de.
Ik−n−1
Fn
We also denote φ$1 (ξ) =
We have
Fn
$
φ1 (e)ψ(2ξ et ) de and hence φ$1 (ξ) = φ1 (−ξ).
m
Wσ (m) σ(φ$1 )Wσ
(6.3)
Nn \GLn
=
Wσ (m)
Fn
Nn \GLn
=
Wσ (m)Wσ
Ik−n
Nn \GLn
Wσ (m)Wσ
Ik−1n
| det m|α dm
In e
m
1
Ik−n
m
Nn \GLn
=
φ$1 (e)Wσ
m
Ik−n
α
| det m|
Fn
!
0
0
Ik−n−1
| det m|α de dm
φ$1 (e)ψ(2ξ0 m et ) de dm
φ1 (−ξ0 m)| det m|α dm.
In the above equalities Wσ ∈ W(σ, ψ), Wσ ∈ W(σ, ψ) and α ∈ C with
Re(α) large. Recall that
(
Wσ
In
y
Ik−n
)
h = ψ(2yn,k−n )Wσ (h)
which explains the reason for the presence of the number 2. In (6.2)
changing variables m → −m, we obtain as an inner integration
m
| det m|α dm,
Wσ ,g,ζ (m)φ1 (−ξ0 m)f˜σ,s h;
ωσ (−1)
I
k−n
Nn \GLn
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where φ1 = ωψ ((x, 0, z)g)φ and h = γ0 (A, 0, B)g. Here


0
γ0 =  I 0
k−n
0
−In 0
0
0
0 −Ik−n 
0
0
0
0 −In
0
and α = 12 n − k + (k + 1)s − (n + 1)ζ. Using the equalities in (6.3) the
above integral equals
m
| det m|α dm.
Wσ ,g,ζ (m) σ(φ$1 )f˜σ,s h;
ωσ (−1)
Ik−n
Nn \GLn
Next we shall use the local functional equation for GLk × GLn . We shall
apply it as in [S1, p. 70]. We have
ωσ (−1)k−1 γ(σ × σ , β, ψ −1 )
m
Wσ ,g,ζ (m) σ(φ$1 )f˜σ,s h;
Nn \GLn
Ik−n
1
=
| det m|β− 2 (k−n) dm
0 1 0 Wσ ,g,ζ (m) σ(φ$1 )f˜σ,s h; 0 0 Ik−n−1
m0
Nn \GLn Mn×(k−n−1)
y
1
| det m|β− 2 (k+n)+n dy dm.
Here β = (k + 1)s − (n + 1)ζ −
1
k.
2
Multiplying (6.2) by
ωσ (−1)k−2 γ σ × σ , (k + 1)s − (n + 1)ζ − 12 k, ψ −1 ,
using the definition of (σ(φ$1 )f˜σ,s ), we obtain first formally that
ωσ (−1)k−2 γ σ × σ , (k + 1)s − (n + 1)ζ − 12 k, ψ −1 J1 (W, φ, f˜σ,s )
equals
Wσ ,g,ζ (m) ωψ (x, 0, z)g φ(e)
0 1 0 I et
n
˜
1
fσ,s γ0 (A, 0, B)g; 0 0 Ik−n−1
m0
y
(k+1)s−(n+1)ζ−k+ 12 n
| det m|
Ik−n−1
dy de dA dB dm dg.
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Here e is integrated over F n , y over Mn×(k−n−1) and the other variables
as before. Recall that
m
−(n+1)( 12 +ζ)
Wσ ,g,ζ (m).
ϕσ,ζ
∗ g = | det m|
m
The interpretation of this passage is as in [S1, 11.8]. We first prove, in
Proposition 6.9, that the last integral converges absolutely in a certain
domain (see (6.39)). Next we note that in this domain, the last integral
must be proportional to J1 , and actually is a rational function in q −s . The
reason is that they both satisfy certain equivariance conditions, which hold
uniquely up to scalars for almost all values of q −s . In Proposition 6.10,
we use a special substitution to calculate the proportionality factor which
indeed is
ωσ (−1)k−2 γ σ × σ , (k + 1)s − (n + 1)ζ − 12 k, ψ −1 .
Write
0 1
0
0 0 Ik−n−1
m0
y
0 1
=
0
0 0 Ik−n−1
m0
0
I 0 m−1 y n
1
0
.
Ik−n−1
Changing variables y → my, the above integral equals
m
ϕσ ,ζ
∗ g ωψ ((x, 0, z)g)φ(e)
m
0 1 0 I et
y
n
˜
1
0
fσ,s γ0 (A, 0, B)g; 0 0 Ik−n−1
m0
Ik−n−1
0
(k+1)s− 12
| det m|
dy de dA dB dm dg.
It will be convenient to write C = ( et , y). Thus C is an n × (k − n) matrix
whose first column is et . Given an n × (k − n) matrix D we shall denote


∗
(0, D, 0) := 
Ik−n 0 −D
In 0
In
0
D
0
,
Ik−n
where D∗ = −Jk−n Dt Jn . Thus the above integral equals
m
ϕσ ,ζ
∗ g ωψ ((x, 0, z)g)φ(e)
m
0 Ik−n
m
f˜σ,s γ0 (0, C, 0)(A, 0, B)g;
In 0
Ik−n
1
| det m|(k+1)s− 2 dA dB dc dm dg.
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Next we bring the m across to obtain
ϕσ ,ζ
m
m
∗
g ωψ (xm, 0, z)g φ(e ·t m−1 )
m
f˜σ,s γ0 (0, C, 0)(A, 0, B)
m
∗
g;
Ik−n
In
− 12
| det m|
γdet m dA dB dC dm dg.
Here we also performed a change of variables in A and C which explains
the presence of xm and e · tm−1 in ωψ (xm, 0, z)g)φ(e · tm−1 ). Also, the
factor γdet m appears from the definition of f˜σ,s . Let
&
γ1 =
Ik−n
−In 0
0 −In
−Ik−n
'
.
We also have
ωψ ((xm, 0, z)g)φ(e · tm−1 ) = ωψ
(
= ωψ
−Jn
1
Jn
m−1
= | det m| 2 γdet m ωψ
(
Jn mt Jn
(
−Jn
Jn
Jn
)
(xm, 0, z)g φ(2e · tm−1 )
−Jn
m
(x, 0, z)
(x, 0, z)
m
m
m
)
t −1
)
∗ g φ(2e · m
)
g φ(2e).
∗
Plugging this in the above integral, we obtain
(
)
m
Jn
m
(x, 0, z)
ϕσ ,ζ
∗ g ωψ
∗ g φ(2e)
−Jn
m
m
m
f˜σ,s γ1 (0, C, 0)(A, 0, B)
∗ g (det m, det m) dA dB dC dm dg,
m
where we used the identity γdet m γdet m = (det m, det m). Recall that g is
integrated over GLn \ Sp2n and m over Nn \ GLn . Hence we may collapse
the two integrations to obtain
(
)
Jn
ϕσ ,ζ (g)ωψ
(x, 0, z)g φ(2e)
−Jn
f˜σ,s,χ (γ1 (0, C, 0)(A, 0, B)g) dA dB dC dg,
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where now g is integrated over Nn \ Sp2n . We have
(
ωψ
−Jn
Jn
)
(
(x, 0, z)g φ(2e) = ωψ (0, −xJn , z)
= ψ(z − 2ext )ωψ
−Jn
(
−Jn
Jn
)
g φ(2e)
Jn
)
g φ(2e).
Plug this to the above integral and also conjugate (0, C, 0) across (A, 0, B)
we obtain
(
)
Jn
ϕσ ,ζ (g)ωψ
g φ(2e)
−Jn
f˜σ,s,χ (γ1 (A, 0, B)(0, C, 0)g)ψ(bk−n,k−n ) dA dB dC dg.
Here we changed variables in B and we remind the reader that by our
Jn
Jn
=
wn−1 . Changing
notations bk−n,k−n was z. Write
−Jn
Jn
variables g → wn g we obtain
ϕσ ,ζ (wn g)ωψ (g)φ(2eJn )

 
I
k−n
f˜σ,s,χ γ1 (A, 0, B)(0, C, 0)
0 − 12 In
2In
0
g 
Ik−n
ψ(bk−n,k−n ) dA dB dC dg.
Denote

γ2 = 
Ik−n
−2In
1
I
2 n
−Ik−n

.
Then, conjugating the Weyl element to the left the above integral equals
ϕσ ,ζ (wn g)ωψ (g)φ(2eJn )
f˜σ,s,χ γ2 (0, − 12 A∗ , B)(−2C ∗ , 0, 0)g ψ(bk−n,k−n ) dA dB dC dg.
Recall that g is integrated over Nn \ Sp2n . Write
=
Nn \Sp2n
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We have
225
(
)
In T
g φ(2eJn ) = ψ(2eJn T et )ωψ (g)φ(2eJn ).
In
Also, one can check that conjugating the matrix


ωψ


Ik−n


In 12 T
In
Ik−n
∗
across (−2C , 0, 0), bk−n,k−n is changed to bk−n,k−n + 2eJn T et . Hence,
after a change of variables, we obtain
ϕσ ,ζ (wn g)ωψ (g)φ(2eJn )


I

k−n
f˜σ,s,χ γ2 (0, − 12 A∗ , B)
In 12 T
In
(−2C ∗ , 0, 0)g 
Ik−n
ψ(bk−n,k−n ) dA dB dC dT dg,
where now g is integrated over V2n,n \ Sp2n and T over all n × n matrices
satisfying T t Jn = Jn T . Now γ2 = wk · w̄0 . Conjugate w̄0 to the right. It
is not hard to check that the groups of all matrices of the form
I

u = w̄0 (0, − 12 A∗ , B)
k−n
In 12 T
In
w̄0−1
Ik−n
equals U2k,k and that ψ(bk−n,k−n ) = ψk (u). Recall that C = ( et , y). Using
the definition of the group R and changing variables e → eJn , we obtain
ϕσ ,ζ (wn g)ωψ (g)φ(2e)f˜σ,s,χ wk uj rτ ((e, 0, 0))g w̄0 ψk (u) du de dr dg.
Here g is integrated over V2n,n \Sp2n , r over R, u over U2k,k and e over F n .
From this the proposition follows.
, φ, fs ). More precisely, we shall
A similar identity holds for J2 (W
assume that
1
+ζ
Sp
⊗ γ −1 ,
π
= Ind P 2n σ ⊗ δ P2
2n,n
2n,n
σ ,ζ in
where σ is an admissible generic representation of GLn . Given ϕ
1
Sp2n −1
2 +ζ
W(σ , ψ) ⊗ δ P
we get
⊗γ
Ind P
2n,n
2n,n
W (g) =
ϕ
σ ,ζ (ug)ψn (u) du.
U2n,n
We have :
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(g) as above, we have :
PROPOSITION 6.2. — With W
, φ, fσ,s )
ωσ (−1)k−2 γ σ × σ , (k + 1)s − (n + 1)ζ − 12 k, ψ −1 J2 (W
= ϕ
σ ,ζ (wn g)ωψ (g)φ(2x)fσ,s,χ wk uj rτ ((x, 0, 0))g w̄0
ψk (u) du dx dr dg.
Here all variables are integrated as in Proposition 6.1. Each integral
converges in some (s, ζ) domain and admits a meromorphic continuation
to all values of (s, ζ). The above equality is understood in the analytic
continuation sense.
6.2 The Unramified Computations (k > n). — We keep the notations of Section 3.1. Following [S1], we shall use the identities established
in Section 6.1 to compute the local unramified integrals.
THEOREM 6.3. — Let p be an odd prime. For all unramified data and
for Re(s) large
J1 (W, φ, f˜σ,s ) =
L(π ⊗ σ, s(k + 1) − 12 k)
·
L(σ, V 2 , 2s(k + 1) − k)
1
+ζ
Proof. — We may assume that π = Ind P 2n (σ ⊗ δ P2
) and σ is a
2n
2n,n
generic unramified representation of GLn . To use Proposition 6.1 we first
need to normalize identity (6.1). It follows from [GS], Section 2.3 that
−1
ϕσ ,ζ (u)ψn (u) du = L σ$ , ( 12 + ζ)(n + 1) − 12 (n − 1)
U2n,n
−1
L σ$ , Λ2 , (1 + 2ζ)(n + 1) − n
Sp
Here σ$ denotes the contragredient representation of σ . Hence we can
write
W (g) = L σ$ , ( 12 + ζ)(n + 1) − 12 (n − 1) L σ$ , Λ2 , (1 + 2ζ)(n + 1) − n
ϕσ ,ζ (ug)ψn (u) du
U2n,n
and with this normalization W (e) = 1. Note that we obtain the L
functions of σ$ since we induce from a lower parabolic. In a similar way
the integral
,
f˜σ,s,χ (wk uh)ψk (u) du, h ∈ Sp
(6.4)
2k
U2k,k
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˜
defines a Whittaker functional for the induced space I(W(σ
⊗ χ, ψ −1 ), s).
Again, it follows as in [GS], Section 2.3 that the normalizing factor for (6.4)
is L(σ, V 2 , 2s(k + 1) − k)−1 . Thus the function
2
Wσ,s,χ (h) = L σ, V , 2s(k + 1) − k
f˜σ,s,χ (wk uh)ψk (u) du
U2k,k
˜
is the normalized unramified vector in I(W(σ
⊗ χ, ψ −1 ), s). Applying
Proposition 6.1 we have,
γ σ × σ , (k + 1)s − (n + 1)ζ − 12 k, ψ −1 J1 (W, φ, f˜σ,s )
= L σ$ , ( 12 + ζ)(n + 1) − 12 (n − 1) L σ$ , Λ2 , (1 + 2ζ)(n + 1) − n
ϕσ ,ζ (wn g)ωψ (g)φ(2x)f˜σ,s,χ wk uj rτ ((x, 0, 0))g w̄0
ψk (u) du dr dx dg,
where the domain of integration is as in Proposition 6.1. Since f˜σ,s,χ
is unramified we may ignore w̄0 and using Iwasa decomposition for
V2n,n \Sp2n , we may replace j by j and replace φ(2x) by φ(x). Multiplying
the above identity by L(σ, V 2 , 2s(k + 1) − k) we obtain
(6.5) γ(σ × σ , (k + 1)s − (n + 1)ζ − 1 h, ψ −1 )J1 (W, φ, f˜σ,s )
2
σ , Λ2 , (1 + 2ζ)(n + 1) − n)
L($
σ , ( 12 + ζ)(n + 1) − 12 (n − 1))L($
=
L(σ, V 2 , 2s(k + 1) − k)
σ,s,χ j rτ ((x, 0, 0) g dx dr dg,
ϕσ ,ζ (wn g)ωψ (g)φ(x)W
where r, x and g are integrated as before. Notice that
1
+ζ Sp2n $
σ ⊗ δ P22n,n .
Fσ ,ζ (g) = ϕσ ,ζ (wn g) ∈ Ind P2n,n
σ,s,χ , φ, Fσ ,ζ , 1 + ζ). Applying
Hence the above integral equals I2 (W
2
Theorem 3.2 for the representations σ$ and
2k Sp
−1
π
= Ind P2k,k
σ ⊗ δ sP2k,k ⊗ χ ⊗ γdet
the above integral equals
2k
Sp
−1
Lψ−1 Ind P2k,k
(σ ⊗ δ sP2k,k ⊗ χ ⊗ γdet
) ⊗ σ$ , (n + 1)( 12 + ζ) −
1
n
2
s− 12
⊗ σ$ , (n + 1)( 12 + ζ) − 12 n
= L σ ⊗ δ P2k,k
−s+ 1
×L σ
$ ⊗ δ P2k,k2 ⊗ σ$ , (n + 1)( 12 + ζ) − 12 n
−1
× L σ$ , (n + 1)( 12 + ζ) − 12 (n − 1)
−1
× L σ$ , Λ2 , 2(n + 1)( 12 + ζ) − n
.
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Plugging this to (6.5) we obtain
γ σ × σ , (k + 1)s − (n + 1)ζ −
=
1
L(σ, V
2 , 2s(k
1
k, ψ −1
2
J1 (W, φ, f˜σ,s )
L σ ⊗ σ$ , (k + 1)s + (n + 1)ζ − 12 k
+ 1) − k)
×L σ
$ ⊗ σ$ , −(k + 1)s + (n + 1)ζ + 12 k + 1 .
It follows from [JPSS] that for α ∈ C
γ(σ × σ , α, ψ −1 ) =
L($
σ × σ$ , 1 − α)
·
L(σ × σ , α)
Hence
J1 (W, φ, f˜s )
L σ ⊗ σ$ , (k + 1)s + (n + 1)ζ − 12 k
L(σ, V
+ 1) − k)
× L σ ⊗ σ , (k + 1)s − (n + 1)ζ − 12 k
1
L σ ⊗ σ$ ⊗ δ −ζ
=
, (k + 1)s − 12 k
P 2n,n
L(σ, V 2 , 2s(k + 1) − k)
× L σ ⊗ σ ⊗ δ ζP
, (k + 1)s − 12 k
=
1
2 , 2s(k
2n,n
L(π ⊗ σ, (k + 1)s − 12 k)
·
=
L(σ, V 2 , 2s(k + 1) − k)
In a similar way, using Proposition 6.2 and Theorem 3.1, we prove :
THEOREM 6.4. — Let p be an odd prime. For all unramified data and
for Re(s) large
, φ, fσ,s ) =
J2 (W
π ⊗ σ, s(k + 1) − 12 k)
Lψ (
·
L(σ, s(k + 1) − 12 (k − 1))L(σ, Λ2 , 2s(k + 1) − k)
6.3. Justifications (for Section 6.1). — We first establish the
convergence, in a right half plane, of the integrals J1 , J2 . Since their
structure is similar, it is enough to consider one family of integrals, say J1 .
PROPOSITION 6.5. — The integrals J1 (W, φ, f˜σ,s ) converge absolutely in
a right half plane Re(s) ≥ s0 .
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Proof. — Let us write J1 in explicit coordinates (performing already
the conjugation γvτ (h)γ −1 )
˜
J1 (W, φ, fσ,s ) =
(6.5)
W (g)ωψ (g)φ(ξ0 + uk−n )
V2n,n \Sp2n

X k,n
 
In
0 Ik−n
˜


γg ψ(vk−n,1 ) dx̄ dg,
fσ,s
u
v Ik−n
0 u
0 In
where


(6.6.)
X k,n =


x̄ =

In
 0 Ik−n

u
v Ik−n
0 u
0 In
∈ Sp2k



.
Assume, first, that the local field F is non-archimedean. Using the Iwasawa
decomposition in Sp2n (F ), it is enough to establish the convergence of
˜
W (a)δ −1
(6.7)
Bn (a)ωψ (a)φ(ξ0 + uk−n )fσ,s (x̄γa)ψ(vk−n,1 ) dx̄ dg.
An X k,n
Here An is the diagonal subgroup of Sp2n and x̄ is the element appearing
b
in (6.6). Bn is the standard Borel subgroup of Sp2n . Write a =
∗ ,
b
where b ∈ GLn (F ) is diagonal. Then (6.7) equals
b
b
−1 b
˜
(6.8)
δ
φ(ξ
x̄γ;
W
+
u
)
f
0
k−n
σ,s
b∗
Ik−n
b∗ Bn
(F ∗ )n X k,n
1
| det b|(k+1)s+n−k+ 2 ψ(vk−n,1 ) dx̄ db.
In general, we have (as in [S1, p. 22, Lemma 4.4.])
*
*
t
*
*˜
(k+1) Re(s)
(6.9)
cj,s ηj (t)
*fσ,s (u
∗ r)* ≤ | det b|
t
for u ∈ V2k,k , r ∈ K(Sp2k ). Here t is diagonal in GLn (F ) and lies in the
support of a gauge on GLn (F ), which is independent of u and r. The ηj are
positive quasi-characters which depend on σ. Thus, (6.8) is majorized by
b
(k+1) Re(s)+n−k+ 12
δ −1
·
ξ
ηj (b) db
(6.10) cφ
cj,s
∗ | det b|
Bn
b
∗
n
(F )
H(x̄)(k+1) Re(s) Ej (x̄) dx̄.
X k,n
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Here ξ is a gauge on Sp2n (F ), which majorizes W . cφ is a bound for φ,
and H and Ej are defined on Sp2k (F ) as follows. In the notation of (6.9)
(6.11)
t
H u
(6.12)
t
Ej u
t
∗ r = | det t|,
t
∗
r = ηj (t).
The dx̄-integral in (6.10) converges for Re(s) 0 (as in [S1, Lemma 4.5]).
Each db-integral in (6.10) is a linear combination of integrals of the form
φ(b1 , . . . , bn )χ1 (b1 ) · · · χn (bn )
(6.13)
(F ∗ )n
|b1 b22 · · · bnn |(k+1) Re(s) d∗ (b1 , . . . , bn ),
where φ ∈ S(F n ) (is positive) and χ1 , . . . , χn are positive quasi-characters
of F ∗ (depending on π, σ). The integrals (6.13) converge in a right half
plane (which depends on χ1 , . . . , χn , k).
Now assume that F is archimedean. The absolute convergence in a right
half plane is obtained similarly, only that in the Iwasawa decomposition,
which leads to (6.7), we get rid of the compact integration, not by Kfiniteness, which we do not assume here, but rather by the majorizations
(6.14)
*
*
*W (a · h)* ≤ ξ(a),
(6.15)
*
t
*˜
*fσ,s (u
t
a ∈ An , h ∈ K(Sp2n ),
*
*
(k+1) Re(s)
N
|ωσ (tn )| · t−1
∗ r)* ≤ cs | det t|
n t .
In (6.14), ξ is a gauge on Sp2n (F ). In (6.15), cs is a constant which depends
on s. Here t has the form diag(t1 t2 · · · tn , t2 · · · tn , . . . , tn−1 tn ) and ωσ is
the central character of σ. The integer N depends on σ. Finally,
n−1
n−1
+
+
+diag(a1 , a2 , . . . , an−1 , 1)+ = 1 +
|ai |2 +
|ai |−2 .
i=1
i=1
With these majorizations the proof now continues as before without
change.
Next, we show that the integrals J1 , J2 can be made to be identically 1
(for all s), for a choice of data (W, φ, f˜σ,s ) in case F is non-archimedean.
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PROPOSITION 6.6. — Let F be non-archimedean. There is a choice of
W ∈ W (π, ψ), φ ∈ S(F n ) and a section f˜σ,s , such that
J1 (W, φ, f˜σ,s ) = 1,
∀s ∈ C.
(A similar proposition holds for J2 , with exactly the same proof.)
Proof. — Write the integral (6.5) as follows
˜
(6.16) J1 (W, φ, fσ,s ) =
W (az̄)δ −1
Bn (a)ωψ (az̄)φ(ξ0 + uk−n )
An ×V 2n,n X k,n



In
0 Ik−n
˜


γaz̄ ψ(vk−n,1 ) dx̄ da dz̄.
fσ,s
u
v Ik−n
0 u
0 In
Here
,
1 . 0
z̄1 0
.
∈ Sp2n ; z̄1 =
V 2n,n = z̄ =
∈ GLn .
.
y z̄1∗
∗
1
b
Write again a =
∗ , b-diagonal. Then (6.16) equals
b
bz̄1
In
−1 b
(k+1) Re(s)+n−k+ 12
W
δ
(6.17)
∗ ∗
∗ | det b|
B
n
b z̄n
y In
b
In
ωψ
φ(bn ξ0 z̄1 + uk−n )ψ(vk−n,1 )
y In
&& I
'
'
n
bz̄
1
0
I
k−n
f˜σ,s
γ,
d(· · ·).
u
v Ik−n
Ik−n
y
u
0
In
Choose the right γ-translate of f˜σ,s to have support in P2k,k ·Ω, where Ω is
a small neighbourhood of I2k , such that f˜σ,s (ωγ; m) = W (m), for ω ∈ Ω.
W is a given function in the Whittaker model of σ. With this choice
(u, v, y) in (6.17) must lie in a small neighbourhood V of zero, and we can
I
I
φ = φ and W g n
= W (g),
choose Ω so small that ωψ n
y In
y In
for y, such that (u, v, y) lies in V. Up to a constant, (6.17) becomes
bz̄1
−1 b
(6.18)
W
∗ ∗ δ Bn
∗ φ(ξ0 bz̄1 )
b z̄1
b
bz̄1
dz̄1 db.
| det b|s W Ik−1
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Here s = (k + 1) Re(s) + n − k + 12 . Also v above is so small that
φ(ξ+uk−n ) = φ(ξ), if (u, v, y) ∈ V. Now, it is easy to choose data in (6.18),
to make it constant. We can simply choose W , such that the function
bz̄1
is the characteristic function of any given small
(b, z̄1 ) → W Ik−n
neighbourhood of (In , In ).
PROPOSITION 6.7. — Assume that F is archimedean. For each com(j)
plex number s0 , there are choices of data (Wj , φj , f˜σ,s ), such that
.
(j)
J1 (Wj , φj , f˜σ,s ) is meromorphic and nonzero at s0 . (Similar proposij
tion for J2 ).
Proof. — Write J1 in the form (6.16). Let the right γ-translate of f˜σ,s
have support in P2k,k · U 2k,k and assume
m
f˜σ,s v
m
∗
ū; e = γ −1 (det m)| det m|(k+1)s ϕ(ū)W (em),
where v ∈ U2k,k , ū ∈ U 2k,k (the opposite to U2k,k ) ; e, m ∈ GLn (F ), and
ϕ ∈ Cc∞ (U 2k,k ). With this choice
(6.19)
J1 (W, φ, f˜σ,s ) =
W
bz̄1
b∗ z̄1∗
In
y In
I
b
φ(ξ0 bz̄1 + uk−n )ϕ(u, v, y)δ −1
ωψ n
Bn
y In
b∗
1
bz̄1
ψ(Vk−n,1 )| det b|(k+1)s+n−k+ 2 d(· · ·).
W
Ik−n
Notation being as in Proposition 6.6. ϕ(u, v, y) is short for


In
I
ϕ u k−n
v Ik−n
y
u
.
In
Choose ϕ of the form ϕ(u, v, y) = ϕ1 (u)ϕ2 (v)ϕ3 (y).% The dv-integration
in (6.19) is carried separately and gives a constant ϕ2 (v)ψ(vk−n,1 ) dv.
Consider the dy-integration
ϕ3 (y)π
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L-FUNCTIONS FOR SYMPLECTIC GROUPS
This is a convolution of ϕ3 against W ⊗ φ ∈ W (π, ψ) ⊗ ωψ . By [DM], this
represents, up to linear combinations, a general element of W (π, ψ) ⊗ ωψ .
Thus, a suitable linear combination of integrals of the form (6.19) gives
bz̄1
(6.20)
W
φ(ξ0 bz̄1 + uk−n )ϕ1 (u)
b∗ z̄1∗
bz̄1
b
δ −1
W
∗ | det b| du db dz̄1 .
B
n
Ik−n
b
Here s = (k + 1)s + n − k + 12 . Again, by [DM],
ϕ1 (u)φ(ξ + uk−n ) du
represents, up to linear combinations, a general element of S(F n ). Thus,
a suitable linear combination of integrals of the form (6.20) becomes
b
bz̄1
bz̄1
φ(ξ0 bz̄1 )δ −1
| det b|s db dz̄1 .
W
∗ ∗ W
∗
B
n
b
b z̄1
Ik−1
b
−(n+1)
Note that since δ −1
= δ −1
, the last integral
∗
Bn
BGLn (b)| det b|
b
becomes
m
m
φ(ξ0 m)| det m|s −(n+1) dm
(6.21)
W
∗ W
I
m
k−n
Nn \GLn
and for
W =
α(y)σ
I y
n
1
Ik−n−1
W dy,
where α is a Cc∞ -function,
m
m
(6.22)
W =α
$(ξ0 m)W .
Ik−n
Ik−n
Choosing φ, such that φ = α
$, we see that (6.21) becomes
m
m
| det m|s −(n+1) dm.
W
∗ W
I
m
k−n
Nn \GLn
Now, as in [S2, Section 4], this last integral is meromorphic in s. Fix now
s = s0 . If the integral (6.20) is identically zero for all data, then, since φ
is arbitrary, it follows that the following integral is identically zero


bz̄
1
W bz̄
(6.23)
W
1
Ik−n+1
b∗ z̄ ∗


b
 1
δ −1
Bn
1
| det b|s db dz̄,
b∗
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&1
..
where b = diag(b1 , . . . , bn−1 ) and z̄ =
I
W =
α(y)σ
W Thus

W
n−1
y
1
Ik−n+1
=α
$(ξ0 m)W 
bz̄
W bz̄
1
1
b∗ z̄ ∗
1
dy, where α is a Cc∞ -function. As in (6.22),
m
∈ GLn−1 . Replace W by
.
∗
Ik−n
'
Ik−n+1
m
Ik−n+1
.
α
$(ξ0 bz̄)


b
 1
δ −1
Bn
1
| det b|s db dz̄ ≡ 0.
b∗
Since α is arbitrary, we conclude that
b
bz̄
bz̄
I4
I
δ −1
W
| det b|s db dz̄ ≡ 0,
W
4
B
n
Ik−n+2
b∗ z̄ ∗
b∗
etc. Finally we get that W (I2n )W (In ) is identically zero, which is
absurd.
REMARK. — In case F is nonarchimedean, with residue field having q
elements, the integrals J1 , J2 are rational functions in q −s . They satisfy
certain equivariance properties for trilinear forms which guarantee their
uniqueness up to scalar multiples for almost all values of q −s . This will
appear in greater generality in a work of Baruch and Rallis (compare
[S1, Sec. 8]). A general principle of Bernstein implies the rationality of
J1 , J2 (See [GPS, 1.2.3]). Here is a description of these equivariance
properties.
Let π be an irreducible, ψn -generic representation of Sp2n (F ) (resp.
2n (F )) and σ an irreducible, generic representation of GLk (F ). The
Sp
integral J1 (resp. J2 ) belongs to the space E of trilinear forms J on
N (π, ψn ) × S(F n ) × VI(σ,s) (resp. W (π, ψn ) × S(F n ) × VI(σ,s)
), which
˜
satisfy
−1
J W, φ, I(σ, s)(v)fσ,s = ψk−n−1
(v)J(W, φ, fσ,s ),
J W, ωψ (h)φ, I(σ, s)(τ (h))fσ,s = J(W, φ, fσ,s ),
J π(g)W, ωψ (g)φ, I(σ, s)(g)fσ,s = J(W, φ, fσ,s )
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π , ψn )), φ ∈ S(F n ), fσ,s ∈ VI(σ,s) ,
for W ∈ W (π, ψn ), (resp. W ∈ W (
˜
˜
2n (F ) (resp.
(resp. fσ,s ∈ I(σ, s)), v ∈ Vk−n−1 (F ), h ∈ Hn (F ) and g ∈ Sp
g ∈ Sp2n (F )). We may also replace π by a representation fully induced
from a parabolic subgroup and an irreducible (generic) representation of
the Levi part. The proof of the one-dimensionality of the space of E, for
almost all values of q −s , follows closely the proof of Theorem 5.1.
The following three propositions justify the formal steps taken in
Proposition 6.1. From now on, F is assumed to be non-archimedean.
PROPOSITION 6.8. — The integral
(6.24)
ϕσ ,ζ (g)ωψ ((x, 0, z)g)φ(ξ0 )f˜σ,s γ(A, 0, B)g dA dB dg
Nn \Sp2n
converges absolutely in a domain of the form
(6.25)

(n + 1) Re(ζ + 12 ) + C < (k + 1) Re(s),



(k + 1) Re(s) < (1 + ε0 )(n + 1) Re(ζ + 12 ) + Q,



R < Re(ζ),
where Q, C, R, ε0 are constants which depend on σ, σ , n, k and ε0 is
positive.
Proof. — Using the Iwasawa decomposition in Sp2n (F ) (and Kfiniteness). It is enough to consider
(6.26)
*
*
*
*ϕσ ,ζ p b ∗ **δ −1 b ∗
*
Bn
b
b
*
* *
In
b
*
φ(ξ0 )**
*ωψ (x, 0, z)
∗
y In
b
*

*

*
*
Ik−n *
*
In
b
* dA dB db dy.
*f˜σ,s γ(A, 0, B)


∗
*
*
y In
b
*
*
I
k−n
(Recall that Bk−n,k−n = z, Ak−n = x). Here b is integrated over the
diagonal subgroup of GL(n, F ). We have
ϕσ,ζ
b
b
∗
1
= | det b|−(n+1)(ζ+ 2 ) Wσ (b),
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where Wσ ∈ W (σ , ψ). Thus (6.26) has the form
(6.27)
* *
*
*
*
*Wσ (b)* · | det b|s −ζ χ0 (b)**ωψ In
φ(ξ0 b + x)*
y In
*


*
*
In
*
*
* *fσ,s  Ik−n
; b
* dA dB db dy,
*
*
A B Ik−n
Ik−n
*
*
y A
In
where ζ = (n+ 1) Re(ζ + 12 ), s = (k + 1) Re(s) and χ0 is a certain positive
is the right
quasi-character (obtained from δ and change of variable) fσ,s
γ-translate of f˜σ,s . Write the Iwasawa decomposition


(6.28)

In
Ik−n

A B Ik−n
y A
In

 t1



= v


..



r,


.
tk
t−1
k
..
.
t−1
1
where v ∈ V2n,n , k ∈ K(Sp2k ). As in [S1, p. 81], we have

* t *

 [z]−2j ≤ * j * ≤ [z]2j ,
tj+1

 −k
[z] ≤ | det t| ≤ [z]−1 ,
(6.29)
j = 1, . . . , k − 1,
A B
and [z] = max{1, z}, where z is the sup-norm of
y A
the coordinates of z. As in (6.9), we have a majorization
where z =
(6.30)
*
*
Ik
b
* *
;
*fσ,s
*
z Ik
Ik−n
≤ | det t|s
αj ηj
b
Ik−n
t ,
where t = diag(t1 , . . . , tn ) and ηj are positive quasi-characters and,
b
t lies in the support of a gauge on GLk (F ). Assume
moreover,
Ik−n
that s ≥ 0. Then, by (6.29)
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(6.31)
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(if s < 0, then, by (6.29), | det |s ≤ [z]−ks ). Inequalities (6.29) also
implies that there is an integer K1 , such that
ηj (t) ≤ [z]K1 ,
(6.32)
Since
b
Ik−n
∀j.
t lies in the support of a gauge, then, writing
b = diag(b1 b2 · · · bn , b2 · · · bn , . . . , bn−1 bn , bn ),
there are positive constants cj , such that
*b t *
* j j*
*
* ≤ cj ,
tj+1
j = 1, . . . , k − 1,
and hence
* tj+1 *
* ≤ cj [z]2j
|bj | ≤ cj *
tj
(6.33)
(put bn+1 = · · · = bk = 1). Using (6.30)–(6.33) (for s > 0) we get a
majorization
(6.34)
*
*
Ik
b
*
* ;
* ≤ c [z]s +L1 .
*fσ,s
z Ik
Ik−n
Let ξσ be a gauge on GLn (F ), majorizing Wσ . Thus (6.27) is majorized
by a sum of constant multiples of integrals of the form
(6.35)
ξσ (b)| det b|s −ζ χ
(b)
|bn |≤cn [z]2n
*
* In
*
*
φ(ξ0 b + x)*[z]−s +L1 db dz.
* ωψ
y In
A B
, and Ak−n = x). χ
is of the form χ0 ηj . We
y A
used (6.33) and (6.34). Now write the Iwasawa decomposition
(Recall that z =
In
y In
=
In T
In
my
m∗y
ry ,
where ry ∈ K(Sp2n ) and my ∈ GLn (F ). As in (6.29), we find
(6.36)
[y]−n ≤ | det my | ≤ [y]−1 ≤ 1.
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From (1.3) and (1.4),
*
* *
* *
*
1
In
*
*
φ(u)* = | det my | 2 · * ωψ (ry )φ(u · my )* ≤ * ωψ (ry )φ(umy )*.
*ωψ
y In
* *
In
*
*
Thus there is a bound cφ , such that *ωψ
φ(u)* ≤ cφ , for all y
y In
and u, and hence (6.35) is majorized by a constant multiple of
(6.37)
ξσ (b)| det b|s −ζ χ(b)[z]−s +L1 db dz.
|bn |≤cn [z]2n
This integral is considered in [S1, p. 86] and converges in the indicated
domain.
PROPOSITION 6.9. — The integral.
(6.38)
ϕσ,ζ (g, m) ωψ ((x, 0, z)g)φ(e)
0 1 0 I t e
n
˜
1
fσ,s γ0 (A, 0, B)g; 0 0 Ik−n−1
m0
y
(k+1)s−(n+1)(ζ+ 12 )−k
| det m|
0
0
Ik−n−1
de dy dm d(A, B) dg.
converges absolutely in a domain of the form

1

 (1 − ε1 )(n + 1) Re(ζ + 2 ) + E < (k + 1) Re(s),
(6.39)
(k + 1) Re(s) < (n + 1) Re(ζ + 12 ) + D,


M < Re(ζ),
where D, E, M, ε1 are constants which depend on (σ, σ , k, n) and ε1 is
positive. In (6.38), g is integrated over GLn \ Sp2n , where GLn is identified
with the Levi part of P2n,n . The variable m is integrated over Nn \ GLn .
Proof. — Using the Iwasawa decomposition for g, in (6.38), it is enough
In
to take g of the form
. Also, using the Iwasawa decomposition
u In
for m, in (6.38), it is enough to take m of the form b ·r, where b is diagonal
and r ∈ K(GLn ). As before, let ξσ be a gauge majorizing ϕσ ,ζ (I; m).
Thus, it is enough to consider
(6.40)
*
* I
*
*
φ(e)*β0 (b)| det b|s −ζ
ξσ (b)* ωψ (x, 0, z)g n
u In
*
*


*
0 1 0 I t e 0 *
In
n
*
* *f  Ik−n
* d(. . .).
; 0 0 Ik−n−1
1
0
*
* σ,s
A B Ik−n
Ik−n−1
br 0
y
*
*
u A
In
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L-FUNCTIONS FOR SYMPLECTIC GROUPS
is the right
Here β0 is a certain fixed positive quasi-character and fσ,s
˜
γ0 -translate of fσ,s . We have
I
I
φ(e) = ψ(z − 2x · t e) ωψ n
φ(e)
ωψ (x, 0, z g n
u In
u In
and hence
* * *
* In
I
* *
*
*
φ(e)* = * ωψ n
φ(e)*
*ωψ (x, 0, z)g
u In
u In
We have
* *
In
*
*
φ(e)*
* ωψ
u In
* *
Jn
In
*
*
= *ωψ
φ(2e)*
−Jn
u In
* *
In −Jn uJn
Jn
*
*
= * ωψ
φ(2e)*
In
−Jn
*
* Jn
*
*
φ(2e)* = φ̃(e).
= *ωψ
−Jn
Here φ̃ is a positive Schwartz function. Thus (6.40) becomes
(6.41)
*
*
ξσ (b)β0 (b)| det b|s −ζ *φ̃(e)*
*
0 1 0 In t e 0 **
* Ik
*fσ,s
*
1
0
; 0 0 Ik−n−1
*
*
W Ik
br 0
y
Ik−n−1
db d(W, e, y) dr.
I
n
te
0
0
“Move” e =
Ik−n−1
Ik
. Denote
via
W Ik
1
e
e∗
and conjugate it
in (6.41) to the left in fσ,s
Ik
W Ik
−1
e
e
∗
=
Ik
W Ik
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and change variable W → W ( dW = dW ). Then (6.41) becomes
(6.42)
ξσ (b)β0 (b)| det b|s −ζ φ̃(e)
*
*
0 1
0
* Ik
*
e
*fσ,s
; 0 0 Ik−n−1 ** db d(W, e, y) dr.
*
W Ik
e∗
br 0
y
Since φ̃ is a linear combination of characteristic functions of small neighbourhoods, it is enough to consider, instead of (6.44), integrals of the
form
(6.43)
ξσ (b)β0 (b)| det b|s −ζ
*
0 1 0 *
* *
Ik
*fσ,s
; 0 0 Ik−n−1 ** db d(W, y) dr.
*
W Ik
br 0
y
Change variable y →
by (this changes β0 (b) to β1 (b), by a fixed power of
| det b|). We have
0 1 0 1
1
I
0 0 Ik−n−1
Ik−n−1
Ik−n−1 .
= k−n
b
br 0
by
y
In
r
Denote r =
1
r
Ik−n−1
, i.e.
. In (6.43), “move” r to the left in fσ,s
0 1 0 Ik
; 0 0 Ik−n−1
W Ik
br 0
by
1
Ik−n
Ik
r
Ik−n−1
;
= fσ,s
.
r∗ W r−1 Ik
r∗
b
y
In
(6.44) fσ,s
Change variable r∗ W r−1 → W . Now, it suffices to consider the integral
)
of the form (we use the K-finiteness of fσ,s
(6.45)
ξσ (b)β1 (b)| det b|s −ζ
*
*
1
*
*
Ik−n
*fσ,s In
* db d(W, y).
Ik−n−1
;
*
*
W Ik
b
y
In
Write the Iwasawa decomposition
1
Ik−n−1
(6.46)
y
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L-FUNCTIONS FOR SYMPLECTIC GROUPS
241
where v ∈ Nk , ty = diag(t1 , . . . , tk ), r ∈ K(GLk ). As before, by
“moving” r back, changing variable r∗ W r−1 → W , etc. It is enough
to consider
(6.47)
ξσ (b)β1 (b)| det b|s −ζ
*
*
Ik
I
*
*
t * db d(W, y).
; k−n
*fσ,s
W Ik
b y
As in (6.30) and (6.31) we have, for s > 0, the estimate
*
*
Ik
I
Ik−n
*
*
t y * ≤ [W ]−s ξσ
t ytw ,
; k−n
(6.48) *fσ,s
W Ik
b
b
where ξσ is a gauge on GLn (F ) (majorizing the Whittaker functions in
σ, m → fσ,s (ρ; m) for p ∈ K(Sp2k )). The matrix tw is obtained from the
Iwasawa decomposition
(6.49)
Ik
W Ik
=u
tw
t∗w
ρ,
u ∈ V2k,k , tw = diag(t1 , . . . , tk ), ρ ∈ K(Sp2k ). So now consider
(6.50)
ξσ (b)β1 (b)| det b|s −ζ [W ]−s ξσ
Ik−n
b
ty tw db d(W, y).
Clearly, in (6.46), we have t1 = 1. Now, we are at the situation of the proof
of Proposition 11.16 (11.16.1) of [S1]. We conclude that the integral (6.38)
converges absolutely in a domain of the form (6.39).
PROPOSITION 6.10. — The integral (6.38) is equal (in the domain (6.39))
to
ωσ (−1)k−2 γ σ × σ , (k + 1)s − (n + 1)ζ − 12 k, ψ −1 J1 (W, φ, f˜σ,s )
(and hence the analytic continuations of (6.38) and J1 are related by the
same functional equation.)
Proof. — By the remark following Proposition (6.7) we know that the
integral (6.38) and J1 (W, φ, f˜σ,s ) are proportional (in the domain (6.39)).
(The equivariance properties, mentioned in the remark, which guarantee
the proportionality are easily seen to be satisfied since (6.38) is obtained
from J1 by formal manipulations.)
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We will find the proportionality factor by a special substitution of data.
Choose ϕσ ,ζ to have support in P2n,n · Ω, where Ω is a small neighbourhood of I2n , and such that ϕσ ,ζ , φ and f˜σ,s are right Ω-invariant. Let
ϕσ ,ζ
m
−(n+1)(ζ+ 12 )
Wσ (m).
∗ ; In = | det m|
∗ m
Then, using the form (6.2) of J1 , we get (for this substitution)
I
φ(ξ0 m + x)
(6.51) J1 (W, φ, f˜σ,s ) = c(Ω) Wσ (m)ωψ n
u In



In

γ; m
ψ(z)f˜σ,s  A Ik−n
B Ik−n
Ik−n
u
A
In
(k+1)s−(n+1)ζ+ 12 n−k
| det m|
dm d(u, A, B).
Here m is integrated over Nn \ GLn , c(Ω) ≡ c is the measure of Ω. Recall
that Ak−n = x and Bk−n,1 = z. Next choose f˜σ,s so that its right σtranslate has support in P2k,k ·Ω1 , and is also right Ω1 -invariant, where Ω1
is a small neighbourhood of I2k . Denote f˜σ,s (γ; r) = Wσ (r). Take Ω1 so
small that
 I
n



ω
φ = φ,

ψ

In
u In

 Ik−n
 ∈ Ω1 =⇒
A B Ik−n
φ(e + x) = φ, ψ(z) = 1



u A
In

(x = Ak−n , z = Bk−n,1 ).
We get from (6.51) (and this substitution) that
˜
$
(6.52) J1 (W, φ, fσ,s ) = c1 cωσ (−1)
Wσ (m)σ(φ)
Wσ
Here Wσ = σ
Nn \GLn
m
Ik−n
1
| det m|(k+1)s−(n+1)ζ+ 2 n−k dm.
Wσ and c1 = c(Ω1 ) is the measure of Ω1 . Apply
−In
Ik−n
now the same substitutions to the integral (6.38). We get
(6.53)
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L-FUNCTIONS FOR SYMPLECTIC GROUPS
1
| det m|(k+1)s−(n+1)ζ−k+ 2 n


In
I
f˜σ,s  A k−n
B Ik−n
u
A
γ;
In
0 1
0
0 0 Ik−n−1
m0
y
I te
n
1
= cc1
Nn \GLn
−I
Ik−n−1
$ Wσ (m)σ(φ)W
σ
0 1
n
0
0 0 Ik−n−1
m0
y
1
d(. . .)
Ik−n−1
1
| det m|(k+1)s−(n+1)ζ+ 2 n−k dm.
The integrals in (6.52) and (6.53) are related by the local functional
equation of [JPSS] by the stated gamma factor.
BIBLIOGRAPHY
[BFG] BUMP (D.), FRIEDBERG (S.), GINZBURG (D.).— Whittaker-Orthogonal
models, functoriality, and the Rankin-Selberg method, Invent Math.,
t. 109, , p. 55–96.
[BFH] BUMP (D.), FRIEDBERG (S.), HOFFSTEIN (J.). — p-Adic Whittaker
functions on the metaplectic group, Duke Math. J., t. 63, no 2, ,
p. 379–397.
[C-S] CASSELMAN (W.), SHALIKA (J.). — The unramified principal series of
p-adic groups II : the Whittaker function, Compos. Math., t. 41,
, p. 207–231.
[DM] DIXMIER (J.), MALLIAVIN (P.). — Factorizations des fonctions et de
vecters indéfiniment différetiables, Bull. Sci. Math. II, t. 102, ,
p. 307–330.
[GPS] GELBART (S.), PIATETSKI-SHAPIRO (I.). — L-functions for G × GL(n),
Springer Lecture Notes in Math., t. 1254, .
[GS] GELBART (S.), SHAHIDI (F.). — Analytic Properties of Automorphic
L-Functions, Perspectives in Mathematics, t. 6, .
[G] GINZBURG (D.). — L-functions for SOn ×GLk , J. reine angew. Math.,
t. 405, , p. 156–180.
BULLETIN DE LA SOCIÉTÉ MATHÉMATIQUE DE FRANCE
244
D. GINZBURG, S. RALLIS, D. SOUDRY
[GRS1] GINZBURG (D.), RALLIS (S.), SOUDRY (D.). — Periods, poles of Lfunctions and symplectic-orthogonal theta lifts, J. reine angew. math.,
t. 487, , p. 85–114.
[GRS2] GINZBURG (D.), RALLIS (S.), SOUDRY (D.). — A new construction of
the inverse Shimura correspondence, IMRN, t. 7, , p. 349–357.
[GRS3] GINZBURG (D.), RALLIS (S.), SOUDRY (D.). — Self Dual GLn Automorphic modules, construction of a backward lifting from GLn to
classical group, IMRN, t. 14, , p. 687–701.
[JPSS] JACQUET (H.), PIATETSKI-SHAPIRO (I.), SHALIKA (J.). — Rankin-Selberg convolutions, Amer. J. Math. 105, , p. 367–464.
[JS] JACQUET (H.), SHALIKA (J.). — Exterior square L-functions, in
Automorphic forms, Shimura varieties and L-functions, L. Clozel
and J. Milne eds, vol. II, .
[P] PERRIN (P.). — Representations de Schrödinger, Indice de Maslov et
groupe metaplectique, in Non Commutative Harmonic Analysis and
Lie Groups, Proc. Marseille-Luming 1980, Springer Lecture Notes,
t. 880, , p. 370–407.
[P.S.] PIATETSKI-SHAPIRO (I.). — Euler Subgroups, in Lie groups and their
Representations, Halsted, New York, .
[S1] SOUDRY (D.). — Rankin-Selberg Convolutions for SO2+1 × GLn :
Local Theory, Memoirs of AMS, t. 500, , p. 1–100.
[S2] SOUDRY (D.). — On the Archimedean theory of Rankin-Selberg
convolutions for SO2+1 × GLn , Ann. Scient. Éc. Norm. Sup., t. 28,
, p. 161–224.
[T1] TON-THAT (T.). — On holomorphic representations of symplectic
groups, Bull. Amer. Math. Soc., t. 81, , p. 1069–1072.
[T2] TON-THAT (T.). — Lie groups representations and harmonic polynomials of a matrix variable, Trans. Amer. Math. Soc., t. 126, ,
p. 1–46.
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