A NULLSTELLENSATZ FOR AMOEBAS

ENDOSCOPIC LIFTING 489
(1) For general values of m and n, this case was studied in [GRS3]. The case where
m = n was studied in [GP]. In this case, let Ẽ τ (h, s) denote the Eisenstein series defined
on the group ˜Sp 4m (A) associated with the induced representation Ind ˜Sp 4m (A)
˜P m
τδ s (A) P m
.
Here P m is the standard parabolic subgroup of Sp 4m whose Levi part is GL 2m .Let
θ ψ Sp 2(n+m)
denote the theta representation defined on the group ˜Sp 2(n+m) (A). The global
integral is then given by (see [GRS3, page 210])
∫
∫
ϕ π (h)θ ψ (
Sp 2(n+m)
l(u)h
)Ẽτ (uh, s)ψ Um−n (u) dudg.
Sp 2(n+m) (F )\Sp 2(n+m) (A) U m−n (F )\U m−n (A)
(28)
Here U m−n is a unipotent group (denoted as H n · V 2k,k−n−1 in [GRS3]), defined as
follows. Using the notation of Section 2.2, we consider the unipotent orbit of Sp 2(n+m)
defined by O = ((2m − 2n)1 2(m+n) ).Asexplainedin[G3], to this unipotent orbit we
can associate a unipotent group, denoted by U m−n and a character ψ Um−n . These are
the unipotent group and the character that we use in the above integral. The function
l is the projection from the group U m−n onto the Heisenberg group of 2(n + m) + 1
variables. Arguing as in [GRS6], this Eisenstein series can have at most a simple pole
at s 0 = (k + 2)/(2(k + 1)). We denote by Ẽ τ (h) a vector in the residue representation
at that point. Taking the residue at s 0 in (28), we denote by P(π, τ(ɛ)) the family of
integrals
∫
Sp 2(n+m) (F )\Sp 2(n+m) (A) U m−n (F )\U m−n (A)
∫
ϕ π (h)θ ψ Sp 2(n+m)
(
l(u)h
)Ẽτ (uh)ψ Um−n (u) dudg.
(2) This case is similar to case (1). The only difference is that now, π is an
irreducible generic cuspidal representation defined on ˜Sp 2(n+m) (A), and the Eisenstein
series is defined on the symplectic group Sp 4m (A). The corresponding global integral
wasdefinedin[GRS3, page 210]. Denoting by E τ (h) a vector in the corresponding
residue representation, we denote by P(π, τ(ɛ)) the family of integrals
∫
∫
˜ϕ π (h)θ ψ ( )
Sp 2(n+m)
l(u)h Eτ (uh)ψ Um−n (u) dudh.
Sp 2(n+m) (F )\Sp 2(n+m) (A) U m−n (F )\U m−n (A)
(3) This case was considered in [G1] and also in [So]. The case when m = n was
studied in [GP]. We consider the Eisenstein series E τ (h, s) defined on SO 4m+1 (A),
which corresponds to the induced representation Ind SO 4m+1(A)
P m (A)
τδm s . Here P m is the
parabolic subgroup of SO 4m+1 whose Levi part is GL 2m .LetO = ( (2(m − n) +
1)1 2(m+n)) .In[G3], we attached to this unipotent orbit a unipotent group U m−n and a