A NULLSTELLENSATZ FOR AMOEBAS

ENDOSCOPIC LIFTING 453
Definition 1
Let π denote an automorphic representation of G = Sp 4mn . We denote by O G (π)
the set of all unipotent classes of H with the following property. A unipotent class
O ∈ O G (π) if, for all unipotent classes Õ > O, the representation π does not have a
nonzero Fourier coefficient corresponding to Õ. When there is no confusion, we write
O(π) for O G (π).
The main result of Section 2 is the following theorem.
THEOREM 1
We have O( τ ) = ((2m) 2n ).
Proof
To prove this theorem, we need to prove two things. First, we need to prove that given a
unipotent orbit O of Sp 4mn which is greater or not related to ((2m) 2n ), the representation
τ has no nonzero Fourier coefficient that corresponds to O. Then we need to show
that τ has a nonzero Fourier coefficient that corresponds to the unipotent orbit
((2m) 2n ). The first part follows as in [GRS9, Sections 2, 3], but by replacing n with
m and k with n. Indeed, the proof is local. We show that the unramified constituents
of the residue cannot support any functional that is induced from a global Fourier
coefficient corresponding to a unipotent orbit that is bigger than or not related to
((2m) 2n ). It follows from [GRS9, Definition 2.1, Lemma 3.1] that the local unramified
constituents are the same as in our case defined above. By applying [GRS9, Lemma 3.3,
Proposition 3.6], we may deduce that a similar result holds for τ defined here.
Thus we need only prove the second part, namely, that τ has a nonzero Fourier
coefficient that corresponds to the unipotent orbit ((2m) 2n ).Let̂Q denote the standard
parabolic subgroup of Sp 4mn whose Levi part is GL 2n ×···×GL 2n .LetU denote its
unipotent radical. In terms of matrices, we have
⎧⎛
⎞
⎫
I 2n X 1 ∗
I 2n X 2 ∗
. ..
⎪⎨
Xm−1 ∗
U =
I 2n Y
⎪⎬
I 2n Xm−1
∗ : X i ∈ Mat 2n×2n ,Y∈ Mat 0 2n×2n .
.
⎜
.. ⎟
⎝
⎪⎩
I 2n X1
∗ ⎠
⎪⎭
I 2n
On U(F )\U(A), we define a character ψ U as follows. For Y = (Y i,j ),defineψ U (u) =
ψ(tr(X 1 +···+X m−1 ) + tr ′ (Y )), where tr ′ (Y ) = Y 1,1 +···+Y n,n .
For a vector θ τ in the space of τ , consider the Fourier coefficient θ U,ψ U
τ
(g).
Following [G3], it is not hard to check that this Fourier coefficient does indeed