A NULLSTELLENSATZ FOR AMOEBAS

ENDOSCOPIC LIFTING 459
⎧⎛
⎞
⎫
⎨ I 2m Y Z
⎬
N 1 m = ⎝ I
⎩
4m(n−1) Y ∗ ⎠,Y ∈ Mat 2m×4m(n−1) : Y 2m,i = 0,Z ∈ Mat 0 2m×2m⎭ .
I 2m
Next, we expand the above integral along the unipotent group N m /Nm 1 with points in A
modulo points in F . This is an abelian group, and Sp 4m(n−1) (F ) acts on this expansion
with two orbits. Here Sp 4m(n−1) is embedded in Sp 4mn as g ↦→ diag(I 2m ,g,I 2m ).The
trivial orbit produces the integral on the right-hand side of the identity at the statement
of the lemma. Thus we need to prove that the nontrivial orbit contributes zero. In other
words, we need to prove that the integral
∫ ∫
θ τ (y˜X)ψ V (GL2m )(X)ψ m (y) dy dX (1)
N m (F )\N m (A)
is zero for every choice of data. Here
⎛⎛
⎞⎞
I 2m Y Z
ψ m (y) = ψ m
⎝⎝
I 4m(n−1) Y ∗ ⎠⎠ = ψ(Y 2m,1 ).
I 2m
This is done in a way similar to [GRS9, Lemma 3.3]. Indeed, if not zero, the above
integral induces a local functional that is nonzero on each of the constituents of τ .
However, as in the above reference, one can show that the local unramified constituent
of τ cannot support such a functional. We omit the details.
For the construction of the lifting defined in Section 2.3, we need to consider residues
of other Eisenstein series which are defined on other classical groups. As we did
above, we need to study their properties. Since the arguments are exactly the same,
we define only the residues and indicate the corresponding unipotent orbit attached to
these representations. We start with the following step.
(1) Let G = Sp 2m(2n+1) .Letɛ denote a cuspidal generic irreducible representation
of the split orthogonal group SO 2m (A). Letτ = τ(ɛ) denote the functorial lift of ɛ to
GL 2m (A), aswasprovedin[CKPS]. We assume that τ is cuspidal. Let µ(ɛ) denote
the lift of ɛ to Sp 2m given by the theta representation. It follows from [GRS2] that
µ(ɛ) is generic. Let Q denote the standard parabolic subgroup of G whose Levi part
is GL 2m ×···×GL 2m × Sp 2m . Here GL 2m occurs n times. Let E τ(ɛ),µ(ɛ) (g, ¯s) denote
the Eisenstein series defined on G and associated with the induced representation
Ind G(A)
Q(A) (τ(ɛ) ⊗ ··· ⊗ τ(ɛ) ⊗ µ(ɛ))δ¯s Q . Write an element in the Levi part of Q as
g = diag(g 1 ,...,g n ,h,gn ∗,...,g∗ 1 ). Then we define δ¯s Q = ∏ n
i=1 |g i| s i δ1/2 Q (g). Asin
the case of the Eisenstein series which we considered in Section 2.2, it is not hard to