Theta Functions on Covers of Symplectic Groups

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by  Solomon Friedberg

THETA FUNCTIONS ON COVERS OF SYMPLECTIC GROUPS
SOLOMON FRIEDBERG AND DAVID GINZBURG
To Freydoon Shahidi on his 70th birthday
6
1 Abstract. We study the automorphic theta representation Θ2(rn) on the r-fold cover of
0 the symplectic group Sp2n. This representation is obtained from the residues of Eisenstein
2 series on this group. If r is odd, n ≤ r < 2n, then under a natural hypothesis on the
n theta representations,we show that Θ(r) may be used to construct a generic representation
2n
Ja σ2(2nr−)r+1 onthe2r-foldcoverofSp2n−r+1. Moreover,whenr =ntheWhittakerfunctionsof
this representation attached to factorizable data are factorizable, and the unramified local
9
factors may be computed in terms of n-th order Gauss sums. If n = 3 we prove these
1
] trhesautltins,fwachticthheinrethparetsceansteatpioenrtcaoinnsttorutchteedsixh-efroel,dσc(o2vr)er of,Sispp4,reucnisceolnydΘit(i2orn)ally.; tWhaeteixsp,ewcet
T 2n−r+1 2n−r+1
conjecture relations between theta representations on different covering groups.
N
.
h
t
a
m 1. Introduction
[
Theclassical metaplectic groupisadoublecover ofasymplectic group, andmaybedefined

1
over a local field or the ring of adeles of a number field. This group arises in the study of
v
0 theta functions, which may be constructed directly as sums over the global rational points
7 of an isotropic subspace. This construction of theta functions appears to be special to the
9
double cover case.
4
0 To generalize, let r > 1 be any integer, and let F be a number field with a full set of
1. r-th roots of unity. Then there is an r-fold cover of the symplectic group Sp2n(A), where
0 A is the adeles of F, denoted Sp(r)(A). When r > 2 one may construct theta functions
6 2n
1 for such a group, but only indirectly – theta functions are obtained as the multi-residues
v: of the minimal parabolic (Borel) Eisenstein series on Sp(r)(A) at the “right-most” (suitably
2n
i normalized) pole. (This construction obtains when r = 2 as well.) As such, these theta
X
(r)
r functions are automorphic forms and give an automorphic representation Θ , called the
a 2n
theta representation.
Since for a general cover the theta functions may only be defined via residues, it is con-
siderably more challenging to determine basic information such as their Whittaker or other
Fourier coefficients in this situation. Indeed, the Fourier coefficients of the theta function
have not been determined for covers of SL of any degree higher than 4. And when n > 1
2
very little is known. Moreover, except in the double cover case there are few examples of any
automorphic functions on such a group, there is no information about when their Whittaker
2010 Mathematics Subject Classification. Primary 11F70; Secondary 11F27,11F55.
Key words and phrases. Symplectic group, metaplectic cover, theta representation, descent integral,
unipotent orbit, generic representation, Whittaker function.
This work was supported by the BSF, grant number 2012019, and by the NSF, grant number 1500977
(Friedberg).
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coefficients might be factorizable (one expects rarely), and it is not clear to what extent
one might expect relations between theta functions or other automorphic forms on different
covers of different groups.
In a first example of such information, the authors [6] have established relations between
the Whittaker coefficients of certain automorphic functions on different covering groups in
two specific cases, related to the conjectures of Patterson and Chinta-Friedberg-Hoffstein
concerning the Fourier coefficients of theta functions on covers of GL . The key to doing so
2
was to adapt descent methods, originally used by Ginzburg-Rallis-Soudry [9] in the context
of algebraic groups or double covers, to higher degree covering groups. In particular, the
treatment of the adelic version of the conjecture of Chinta-Friedberg-Hoffstein relied on the
study of the theta function on the three-fold cover of Sp .
4
In this work we investigate theta functions and descent integrals on covers of symplectic
groups of arbitrary rank. As we shall explain, for r odd a descent construction allows one
(r) (2r)
to pass from the theta representation Θ to a representation σ realized inside the
2n 2n−r+1
space of automorphic square-integrable functions on Sp(2r) (F)\Sp(2r) (A). We expect
2n−r+1 2n−r+1
(2r)
that this representation is in fact the theta representation Θ (Conjecture 2), and we
2n−r+1
(2r)
show that σ has non-zero projection to this representation. In Theorem 1 below we
2n−r+1
(2r)
establish that if r is odd, n ≤ r < 2n, then the representation σ is generic. Moreover,
2n−r+1
(2n)
if r = n, we show that the Whittaker coefficients of σ arising from factorizable inputs
n+1
are factorizable. As a first new case, this is true of the Whittaker coefficients of the descent
representation on the six-fold cover of Sp . Then in Theorem 2 we show that the unramified
4
local contributions to the Whittaker coefficients in the factorizable case may be expressed as
sums of Whittaker coefficients of the theta representation on the n-fold cover of GL . These
n
coefficients are n-th order Gauss sums by the work of Kazhdan and Patterson [12]. Thus we
exhibit a new class of automorphic forms on covering groups whose Whittaker coefficients
are factorizable with algebraic factors at good places.
These results are proved on two conditions. One concerns the specific unipotent orbit
(r)
attached to Θ (Conjecture 1). This is an object of on-going study [4]; in the last Section
2n
we give a sketch of the proof of this Conjecture in the first new case, n = r = 3. A second
concerns the precise characters within a given unipotent orbit which support a nonzero
Fourier coefficient (Assumption 1). This is not needed in all cases, and in particular is not
(2n)
needed in the factorizable case σ . Our results concerning Conjecture 1 are thus sufficient
n+1
to establish unconditionally that the Whittaker function of the descent to the six-fold cover
of Sp is Eulerian for factorizable data, and to compute its unramified local factor in terms
4
of cubic Gauss sums. We expect that this descent is in fact the theta function on the six-fold
cover of Sp .
4
2. Definition and Properties of the Theta Representation
Let n and r be two natural numbers. Let F be a number field containing a full set
µ of r-th roots of unity, and let A be the adeles of F. Let Sp denote the symplectic
r 2n
group consisting of the 2n × 2n matrices leaving invariant the standard symplectic form
< x,y >= n (x y − x y ). Denote by Sp(r)(A) the metaplectic r-fold cover
i=1 i 2n−i+1 n+i n−i+1 2n
of the symplectic group Sp (A). This group consists of pairs (g,ζ) with g ∈ Sp (A) and
2n 2n
P
ζ ∈ µ ; it may be obtained in the standard way from covers of the local groups Sp (F ) as ν
r 2n ν
2
rangesover theplaces of F, identifying thecopies of µ . The multiplication inthe localgroup
r
is determined by a cocyle σ. If σ is the 2-cocycle of Banks-Levy-Sepanski [1] for SL (F )
BLS 2n ν
and w is the permutation such that the conjugate of Sp by w preserves the symplectic form
2n
< x,y >′= n x y −y x , then we take σ(g ,g ) = σ (wg w−1,wg w−1).
i=1 i n+i i n+i 1 2 BLS 1 2
WeshallbeconcernedwiththethetarepresentationdefinedonthegroupSp(r)(A),denoted
P 2n
(r)
Θ . This representation is defined as the space spanned by the residues of the Borel
2n
Eisenstein series on Sp(r)(A), similarly to the definition for the general linear group in [12],
2n
p. 118. Our first task is to give a brief account of this construction.
Let B denote the standard Borel subgroup of Sp , and let T denote the maximal torus
2n 2n 2n
of Sp which is a subgroup of B . Working either locally at a completion of F or globally
2n 2n
over A, if H is any algebraic subgroup of Sp , we let H(r) denote its inverse image in Sp(r).
2n 2n
(r)
We will call this inverse image parabolic or unipotent if H has this property. Let Z(T )
2n
(r)
denote the center of T . Suppose first that r is odd. Given a character µ of T , we may
2n 2n
(r)
use it to define a genuine character, again denoted µ, of Z(T ). (The notion of genuine
2n
depends on an embedding of the group of r-th roots of unity µ into C×. We will fix this
r
and omit it from the notation.) Extending it trivially to any maximal abelian subgroup of
(r) (r) Sp(r)
T and then inducing up, we obtain a representation of Sp which we denote by Ind 2nµ.
2n 2n B(r)
2n
This representation is determined uniquely by the choice of µ and this procedure may be
carried out both locally and globally. Here we consider unnormalized induction, and shall
1/2
include the modular character δ when we require normalized induction.
B2n
Let s be complex variables, and let µ be the character of T given by
i 2n
µ(diag(a ,...,a ,a−1,...,a−1)) = |a |s1···|a |sn.
1 n n 1 1 n
If this construction is carried out over a local field, then |a| denotes the normalized local
absolute value, while if it is carried out over A, then |a| denotes the product of these over all
Sp(r)(A) 1/2
places of F. One may form the induced representation Ind 2n µδ , and for each vector
B(r)(A) B2n
2n
in this space, one may construct the multi-variable Eisenstein series E(h,s ,...,s ) defined
1 n
on the group Sp(r)(A). Computing the constant term as in [12], Prop. II.1.2, we deduce that
2n
the poles of the partial intertwining operator associated with the long Weyl element of Sp
2n
are determined by
[ζ (r(s −s ))ζ (r(s +s ))] ζ (rs )
i<j S i j S i j i S i
(1) .
[ζ (r(s −s )+1)ζ (r(s +s )+1)] ζ (rs +1)
i<j QS i j S i j Q i S i
Here S is a finiteQset of all places including the archimedeQan places and all finite places µ
such that |r| 6= 1 and ζ (s) is the partial global zeta function.
ν S
The expression (1) has a multi-residue at the point
1
s = , r(s −s ) = 1.
n i i+1
r
From this we deduce that the Eisenstein series E(h,s ,...,s ) has a multi-residue at that
1 n
(r)
point, and we denote the residue representation by Θ . If µ denotes the character of T
2n 0 2n
defined by
µ0(diag(a1,...,an,a−n1,...,a−11)) = |a1|nr ···|an−1|2r|an|r1,
3
(r) Sp(r)(A) 1/2
then it follows that Θ is a subquotient of the induced representation Ind 2n µδ .
2n B(r)(A) B2n
2n
(r)
From this we deduce that Θ is also a subrepresentation of the induced representation
2n
Sp(r)(A)
Ind 2n χ where
B2(rn)(A) Sp(2rn),Θ
χSp(r),Θ(diag(a1,...,an,a−n1,...,a−11)) = |a1|n(rr−1) ···|an−1|2(rr−1)|an|r−r1.
2n
We turn to even degree coverings. We shall only be concerned with the covers of degree
(2r)
twiceanoddinteger. Ifr isodd, thedefinitionofthethetarepresentation forSp issimilar.
2m
There is a small difference since the maximal parabolic subgroup of Sp whose Levi part
2m
is GL splits under the double cover. Because of this, when computing the intertwining
m
operator corresponding to the long Weyl element, one finds that its poles are determined by
[ζ (r(s −s ))ζ (r(s +s ))] ζ (2rs )
i<j S i j S i j i S i
.
[ζ (r(s −s )+1)ζ (r(s +s )+1)] ζ (2rs +1)
i<j QS i j S i j Q i S i
Accordingly we dQefine Θ(2r) to be the multi-residue of the QEisenstein series at the point
2m
1
s = , r(s −s ) = 1.
m i i+1
2r
(2r) Sp(2r)(A) 1/2
ThentherepresentationΘ isasubquotientoftheinducedrepresentationInd 2m µ δ
2m B(r)(A) e B2m
2m
where µ denotes the character of T defined by
e 2m
µe(diag(a1,...,am,a−m1,...,a−11)) = |a1|2m2r−1 ···|am−1|23r|am|21r.
(2r) Sp(2r)(A)
ItfollowsthatΘ isalsoasubrepresentationoftheinducedrepresentationInd 2m χ
2m B2(rm)(A) Sp(22mr),Θ
where
χSp(2r),Θ(diag(a1,...,am,a−m1,...,a−11)) = |a1|2mr−2(r2m−1) ···|am−2|6r2−r5|am−1|4r2−r3|am|2r2−r1.
2n
We now develop some properties of the theta representations that follow from induction
in stages, or, equivalently, by taking an (n− 1)-fold residue of E(h,s ,...,s ) to obtain a
1 n
maximal parabolic Eisenstein series attached to theta representations of lower rank groups.
Let 1 ≤ a ≤ n. Denote by P the maximal parabolic subgroup of Sp whose Levi part is
2n,a 2n
GL × Sp , and let L denote its unipotent radical. We write i for the inclusion of
a 2(n−a) 2n,a
GL in P in this paragraph but we suppress i afterwards. From the block compatibility
a 2n,a
of the cocycle σ (Banks-Levy-Sepanski [1]) and a short computation it follows that if
BLS
g ,g ∈ GL over a local field then σ(i(g ),i(g )) = σ (g ,g )2(detg ,detg )−1, where
1 2 a 1 2 GLa 1 2 1 2
σ is the metaplectic 2-cocycle of [1] for GL . Thus for both the r and 2r-fold covers,
GLa a
(r)
i determines an r-fold cover of GL . We write this cover GL (which case we are in will
a a
always be clear from context).
(r)
Suppose first that r is odd. Let Θ denote the theta representation of the group
GLa
GL(r)(A), as constructed in [12] (or the corresponding character if a = 1). Then it follows
a
Sp(r)(A) 1/2 Sp(r)(A) (r) (r) r+1
from induction in stages that Ind 2n µ δ is equal to Ind 2n (Θ ⊗Θ )δ 2r .
B(r)(A) 0 B2n P(r) (A) GLa 2(n−a) P2n,a
2n 2n,a
From this we deduce the following. Let E (g,s) denote the Eisenstein series of Sp(r)(A)
a 2n
4
associated with the induced representation IndSp(2rn)(A)(Θ(r) ⊗ Θ(r) )δs . (For the con-
P(r) (A) GLa 2(n−a) P2n,a
2n,a
struction of such Eisenstein series more generally, see Brubaker and Friedberg [2].) Then the
(r)
representation Θ is a residue of E (g,s) at the point s = (r +1)/2r. Of course, this can
2n a
also be verified directly by studying the corresponding intertwining operators.
In the case of a cover of degree 2r, r odd, the situation is roughly similar. In this case, the
representation IndSp(22mr)(A)µ δ1/2 is equal to IndSp(22mr)(A)(Θ(r) ⊗Θ(2r) )δ(r2+r1(2)(m2m−a−+a1)+)r. Hence
B(r)(A) e B P(r) (A) GLa 2(m−a) P2n,a
2m 2m,a
if we let E (g,s) denote the Eisenstein series associated with the induced representation
a
IndSp(22mr)(A)(Θ(r) ⊗Θ(2r) )δs , then we deduce that Θ(2r) is the residue of E (g,s) at the
P(r) (A) GLa 2(m−a) P2n,a 2m a
2m,a
point s = (r+1)(2m−a)+r.
2r(2m−a+1)
From these observations, we will deduce the following proposition. Here and below, ma-
trices are embedded in metaplectic groups by the trivial section without additional notation,
and we call metaplectic elements diagonal or unipotent when their projections to the linear
group are.
Proposition 1. Suppose that r is odd. Let θ(r) be in the space of Θ(r). Then there exist
2n 2n
functions θ(r) ∈ Θ(r) , θ(r) ∈ Θ(r) such that for all diagonal g in GL(r)(A) which
GLa GLa 2(n−a) 2(n−a) a
lies in the center of the Levi part of the parabolic group P(r) (A) and for all unipotent h ∈
2n,a
Sp(r) (A), v ∈ GL(r)(A), one has
2(n−a) a
(r) (r) (r)
(2) θ (u(gv,h))du = χ (g)θ (v)θ (h).
2n Sp(2rn),Θ GLa 2(n−a)
Z
L2n,a(F)\L2n,a(A)
A similar identity holds in the even-degree cover case.
The requirement that h,v be unipotent could be dropped at the expense of a cocyle; being
unipotent guarantees that it is 1.
Proof. Since Θ(r) is the residue of the Eisenstein series E (·,s), we first consider the constant
2n a
term
(3) E (u(gv,h),s)du.
a
Z
L2n,a(F)\L2n,a(A)
For Re(s) large we can unfold the Eisenstein series and deduce that (3) is equal to
(4)
f (wγu(gv,h))du.
s
w∈P2n,a(F)\SXp2n(F)/P2n,a(F)Lw2n,a(F)Z\L2n,a(A) γ∈(w−1P2n,a(F)wX∩P2n,a(F))\P2n,a(F)
Here Lw = w−1L w ∩ L− where L− is the conjugate of L by the long Weyl
2n,a 2n,a 2n,a 2n,a 2n,a
element. Also, we have f ∈ IndSp(2rn)(A)(Θ(r) ⊗ Θ(r) )δs . Notice that all elements in
s P(r) (A) GLa 2(n−a) P2n,a
2n,a
P (F)\Sp (F)/P (F) can be chosen to be Weyl elements. Similarly to [13], Section
2n,a 2n 2n,a
1.2, one can check that for all Weyl elements in P (F)\Sp (F)/P (F) which are not
2n,a 2n 2n,a
the long Weyl element, the inner summation is a certain Eisenstein series or product of
5
such series. Moreover, one can also check that the Eisenstein series in the corresponding
summand in (4) is holomorphic at s = (r+1)/2r. Hence, taking the residue in (3) and (4) at
s = (r+1)/2r, we are left only with the long Weyl element, and for that element we obtain
the identity
(r)
θ (u(gv,h))du = Res M f (gv,h)
2n s=(r+1)/2r w s
Z
L2n,a(F)\L2n,a(A)
where M is the intertwining operator attached to w. From this identity (2) follows. (cid:3)
w
Similar statements for other groups are established in [3] Proposition 3.4 and [6] Proposi-
tion 1.
We end this section with a general conjecture about the unipotent orbit attached to the
(r)
representation Θ , and give a consequence of this conjecture. Recall that if π denotes an
2n
automorphic representation of a reductive group, then the set O(π) was defined in [7]. It is
the largest unipotent orbit that supports a nonzero coefficient for this representation. The
extension of the definition of this set to the metaplectic groups is clear. In this paper we are
(r)
interested in the set O(Θ ). The conjecture regarding this set is given as follows. Assume
2n
that r < 2n. Write 2n = a(n,r)r + b(n,r) where a(n,r) and b(n,r) are both nonnegative
numbers such that 0 ≤ b(n,r) ≤ r −1. We recall that a partition is a symplectic partition
if any odd number in the partition occurs with even multiplicity. As defined in [5], given
a partition λ of 2n, we define the Sp collapse of λ to be the greatest symplectic partition
which is smaller than λ. We have
Conjecture 1. Let r be an odd number. If r < 2n, then the set O(Θ(r)) consists of the
2n
partition which is the Sp collapse of the partition (ra(n,r)b(n,r)). If r > 2n, then the repre-
sentation Θ(r) is generic.
2n
The conjecture is studied in [4]. We shall write Conjecture 1 for this statement for
r,n
a specific pair (r,n). We remark that an extension to even degree covers has not been
formulated at this time.
To give a consequence of this conjecture, let r be odd, let a be an integer with 1 ≤ a ≤
n − (r − 1)/2, and let b = n − (r − 1)/2 − a. Let U be the unipotent radical of the
2n,n−b
standard parabolic subgroup of Sp whose Levi part is GLn−b×Sp . (In particular, U is
2n 1 2b 2n,n
the standard maximal unipotent subgroup of Sp .) Let U be the subgroup of U
2n 2n,n−b,1 2n,n−b
which consists ofallmatricesu = (u ) ∈ U suchthat u = 0foralln−b+1 ≤ i ≤ n.
i,j 2n,n−b n−b,i
Let α = (α ,...,α ) where for all i we have α ∈ {0,1}. Let ψ be a nontrivial character of
1 a−1 i
F\A, which will be fixed throughout this paper. Let ψ be the character of U
U2n,n−b,1,α 2n,n−b,1
given by
a−1 n−b
ψ (u) = ψ α u + u .
U2n,n−b,1,α i i,i+1 j,j+1
!
i=1 j=a
X X
Then we have the following result, which will be used later.
Lemma 1. Suppose r is odd, r < 2n, and Conjecture 1 holds for all 0 ≤ i ≤ a−1.
r,n−i0 0
Then the integral
(r)
(5) θ (ug)ψ (u)du
2n U2n,n−b,1,α
Z
U2n,n−b,1(F)\U2n,n−b,1(A)
6
is zero for all choices of data, that is, for all θ(r) ∈ Θ(r).
2n 2n
Proof. Consider first the case where α = 1 for all i. Then the Fourier coefficient (5) is the
i
Fourier coefficient which corresponds to the unipotent orbit ((2n − 2b)12b). From Conjec-
(r)
ture1 wehavethatO(Θ )consistsofthepartitionwhichistheSpcollapseofthepartition
r,n 2n
(ra(n,r)b(n,r)). Hence ((2n−2b)12b)isgreater thanornot relatedtoO(Θ(r)). Indeed, thisfol-
2n
lowsfromtherelation2b = 2n−2a−r+1 which implies that((2n−2b)12b) = ((r+2a−1)12b).
Thus, if α = 1 for all i, then the integral (5) is zero for all choices of data.
i
Next assume that at least one of the scalars α is zero. Let i ≤ a−1 be the largest index
i 0
such that α = 0. Then the integral (5) is equal to
i0
(6)
Z Z Z
H(F)\H(A)L2n,i0(F)\L2n,i0(A)U2(n−i0),n−i0−b,1(F)\U2(n−i0),n−i0−b,1(A)
(r)
θ (luhg)ψ (u)ψ (h)dudldh.
2n U2(n−i0),n−i0−b,1 H,α
Here H is a certain unipotent subgroup of GL which will not be important to us. No-
i0
tice that the integration over L (F)\L (A) is the constant term along this unipotent
2n,i0 2n,i0
group. Therefore, it follows from Proposition 1 above that the integration along the quotient
space U (F)\U (A) is the coefficient of the representation Θ(r)
2(n−i0),n−i0−b,1 2(n−i0),n−i0−b,1 2(n−i0)
corresponding to the unipotent orbit ((2(n−i −b))12b). Thus the vanishing of the integral
0
(r)
(6) will follow if this unipotent orbit is greater than or not related to the orbit O(Θ ).
2(n−i0)
From Conjecture 1 , this is the case if 2(n − i − b) > r. Since this inequality follows
r,n−i0 0
from the relations 2(n−a−b) = r −1 and a > i , the result is proved. (cid:3)
0
3. The Descent Construction
RecallthatU denotestheunipotent radicalofthestandardparabolicsubgroupofSp
2m,k 2m
whose Levi part is GLk×Sp . The quotient group U \U may be identified with
1 2(m−k) 2m,k−1 2m,k
the Heisenberg group in 2(m−k)+1 variables, H . Indeed, there is a homomorphism
2(m−k)+1
l : U 7→ H which is onto, and whose kernel is the group U .
2m,k 2(m−k)+1 2m,k−1
For the rest of this section we suppose that r > 1 is odd. Set r′ = (r −1)/2. For α ∈ F∗
let ψU,α be the character of Ur−1,r′(F)\Ur−1,r′(A) given by
ψU,α(u) = ψ(u1,2 +u2,3 +···+ur′−1,r′ +αur′,r′+1) u = (ui,j).
From now on we shall suppose that Conjecture 1r,r′ holds, that is, that the representation
Θ(r) is generic. This is known if r = 3 [12]. Thus there is an α ∈ F∗ and a choice of data,
r−1
(r) (r)
i.e. an automorphic function θ in the space of Θ , such that the integral
r−1 r−1
(r)
(7) θ (ug)ψ (u)du
r−1 U,α
Z
Ur−1,r′(F)\Ur−1,r′(A)
7
is not zero. Replacing ψ by ψα−1 and changing the function, we may assume that α = 1. In
(r)
other words, there is a choice of data θ such that the integral
r−1
(r)
(8) θ (ug)ψ (u)du
r−1 U
Z
Ur−1,r′(F)\Ur−1,r′(A)
is not zero, where ψ is defined as ψ with α = 1.
U U,α
Remark: In fact we expect that for all α ∈ F∗, the integral (7) will be nonzero for some
choice of data.
Let Θψ,φ denote the theta representation attached to the Weil representation and defined
2m
on the double cover of Sp (A). Here φ is a Schwartz function on Am. Since r is odd, using
2m
the isomorphism µ ∼= µ × µ we may map g ∈ Sp(2r) (A) to its image in the double
2r 2 r 2n−r+1
and r-fold covers; we will not introduce separate notation for this. To define the descent
construction, consider the function of g ∈ Sp(2r) (A) given by
2n−r+1
(9) f(g) = θψ,φ (l(u)g)θ(r)(ug)ψ (u)du.
2n−r+1 2n U2n,r′
Z
U2n,r′(F)\U2n,r′(A)
Hereθψ,φ andθ(r) arevectors inthespacesoftherepresentations Θψ,φ andΘ(r) (resp.),
2n−r+1 2n 2n−r+1 2n
ψU2n,r′ is the character of U2n,r′(A) given by ψU2n,r′(ui,j) = ψ(u1,2+u2,3+···+ur′−1,r′), and
g ∈ Sp is embedded in Sp as
2n−r+1 2n
Ir′
g 7→ g .
 
Ir′
 
(One could also consider descent integrals similar to (9) if r is even but we shall not do so
here.) Then f(g) is a genuine automorphic function defined on Sp(2r) (A). Let σ(2r)
2n−r+1 2n−r+1
denote the representation of Sp(2r) (A) generated by all the functions f(g).
2n−r+1
4. Computation of the Constant Term of the Descent Integral
Let V denote any standard unipotent subgroup of Sp , and let ψ be a character of
2n−r+1 V
V(F)\V(A), possibly trivial. In this paper we will compute integrals of the type
(10) f(vg)ψ (v)dv,
V
Z
V(F)\V(A)
where f(g) is given by (9).
Using (9), we arrive at an iterated integral. We first unfold the theta function θψ,φ .
2n−r+1
Collapsing summation and integration, and then using the formulas for the action of the
Weil representation ω , the integral (10) is equal to
ψ
(r)
(11) ω (g)φ(x)θ (uvj(x)g)ψ (u)ψ (v)dudvdx.
ψ 2n U2n,r′,1 V
AnZ−r′V(F)Z\V(A)U2n,r′,1(F)Z\U2n,r′,1(A)
Here x = (x1,...,xn−r′) ∈ An−r′ is embedded in Sp2n via the map j(x) = I2n +x1e∗r′,r′+1 +
x2e∗r′,r′+2 + ··· + xn−r′e∗r′,n, where ei,j denotes the (i,j)-th elementary matrix and e∗i,j =
8
ei,j − e2n−j+1,2n−i. Recall that the group U2n,r′,1 is the subgroup of U2n,r′ consisting of all
matrices u = (ui,j) ∈ U2n,r′ such that ur′,k = 0 for all r′ + 1 ≤ k ≤ n. The character
ψU2n,r′,1(u) is defined as the product of ψU2n,r′(u) and the character ψU02n,r′,1 of U2n,r′,1 given
by ψU02n,r′,1(u) = ψ(ur′,2n−r′+1).
At this point we consider the case when V = L where 1 ≤ a ≤ n − r′ and
2n−r+1,a
ψ is the trivial character. Thus, V is the unipotent radical of the standard maximal
V
parabolic subgroup of Sp whose Levi part is GL × Sp . We recall that
2n−r+1 a 2n−2a−r+1
2b = 2n−2a−r +1.
Let w denote the Weyl group element of Sp defined by
a 2n
I
a
I
n−a−b
 
w = I .
a 2b
 I 
n−a−b
 
 I 
a
 
 
(r)
Since θ is left-invariant under rational points, we may conjugate by w from left to right.
2n a
Doing so, we see that the integral (11) is equal to
(12)
AnZ−r′Mat(n−a−b−1)×a(F)Z\Mat(n−a−b−1)×a(A)V0(F)Z\V0(A)U0(F)Z\U0(A)
(r)
ω (g)φ(x)θ (u v k(y)w j(x)g)ψ (v )du dv dydx.
ψ 2n 0 0 a V0 0 0 o
The notations are defined as follows. Recall that L denotes the unipotent radical of the
2n,a
standard maximal parabolic subgroup of Sp whose Levi part is GL ×Sp . The group
2n a 2(n−a)
U is defined to be the subgroup of L consisting of all matrices in Sp of the form
0 2n,a 2n
I 0 ∗ ∗ ∗
a
I 0 0 ∗
n−a−b
 
I 0 ∗ .
2b
 I 0
n−a−b
 
 I 
a
 
 
The embedding of Mat into Sp is given by
(n−a−b−1)×a 2n
I
a
y I
n−a−b
 
k(y) = I ,
2b
 I 
n−a−b
 
 y∗ I 
a
 
where y∗ is chosen to make the matrix symplectic. Here the embedding of y is such that it
consistsofallmatricesofsize(n−a−b)×a suchthatitsbottomrowiszero. FinallyV isequal
0
tothegroupU , definedsimilarlytothedefinitionofU reviewedafter(11)(in
2(n−a),n−a−b,1 2n,a,1
particular, thegroupV isa subgroupof Sp ), andthecharacter ψ := ψ as
0 2(n−a) V0 U2(n−a),n−a−b,1
defined there. In coordinates, ψ (v) = ψ(v +v +···+v +v ).
V0 a+1,a+2 a+2,a+3 n−b−1,n−b n−b,n+b+1
The next step is to carry out some root exchanges. The notion of root exchange was
described in detail in the authors’ paper [6], Section 2.2; a more abstract formulation may
9
be found in Ginzburg, Rallis and Soudry [11], Section 7.1. In our context, we perform the
following root exchange. For z ∈ Mat , let m(z) in Sp be given by
a×(n−a−b) 2n
I z
a
I
n−a−b
 
m(z) = I .
2b
 I z∗
n−a−b
 
 I 
a
 
 
We embed the group Mat (A) inside Sp (A) by considering all matrices m(z)
a×(n−a−b−1) 2n
as above such that the first column of z is zero. Expand the integral (12) along the group
so obtained. Performing the root exchange with the group of all matrices k(y) such that
y ∈ Mat (A), one sees that the integral (12) is equal to
(n−a−b−1)×a
(13)
AnZ−r′Mat(n−a−Zb−1)×a(A)V0(F)Z\V0(A)L02n,a(F)Z\L02n,a(A)
(r)
ω (g)φ(x)θ (uv k(y)w j(x)g)ψ (v )dudv dydx.
ψ 2n 0 a V0 0 0
Here L0 is the subgroup of L which consists of all matrices u = (u ) ∈ L such that
2n,a 2n,a i,j 2n,a
u = 0 for all 1 ≤ i ≤ a.
i,a+1
Consider the quotient space of L0 \L . This quotient may be naturally identified with
2n,a 2n,a
the column vectors of size a. This group embeds in Sp as the group of all matrices of the
2n
form I +z e∗ +z e∗ +···+z e∗ . Expand the integral (13) along this group. The
2n 1 1,a+1 2 2,a+1 a a,a+1
group GL (F) acts on this expansion with two orbits.
a
The contribution of the nontrivial orbit to the integral (13) is the expression
(14)
AnZ−r′Mat(n−a−Zb−1)×a(A)V0(F)Z\V0(A)L2n,a(F)Z\L2n,a(A)
(r)
ω (g)φ(x)θ (uv k(y)w j(x)g)ψ (v )ψ (u)dudv dydx,
ψ 2n 0 a V0 0 L2n,a 0
where for u = (u ) ∈ L (F)\L (A), ψ (u) = ψ(u ). We claim that the integral
i,j 2n,a 2n,a L2n,a a,a+1
(14) is zero for all choices of data.
To prove this, we carry out similar expansions repeatedly. To start, consider the quotient
space L \L . This space may be identified with the column vectors of size a − 1,
2n,a−1 2n,a
embedded inSp asthegroupofallmatrices oftheformI +z e∗ +z e∗ +···+z e∗ .
2n 2n 1 1,a 2 2,a a−1 a−1,a
Expand the integral (14) along this group. The group GL (F) acts on this expansion with
a−1
two orbits. If a−1 > 1 we continue in this way. Then the vanishing of the integral (14) for
all choices of data then follows from Lemma 1 above.
We conclude that the integral (10) with V = U and ψ = 1 is equal to
2n−r+1,a V
(15)
AnZ−r′Mat(n−a−Zb−1)×a(A)V0(F)Z\V0(A)L2n,a(F)Z\L2n,a(A)
(r)
ω (g)φ(x)θ (uv k(y)w j(x)g)ψ (v )dudv dydx.
ψ 2n 0 a V0 0 0
10