A NULLSTELLENSATZ FOR AMOEBAS

466 DAVID GINZBURG
cuspidality of the lift. Since the computations are quite similar in all five cases, we
concentrate only on the first case.
Let M = Sp 2m(2n+1) .Letσ denote an irreducible cuspidal representation of
Sp 2n (A). In this case, the lift is given by the integrals
∫ ∫
( )
f (h) =
ϕ σ (g)θ ɛ u(g, h) ψU (u) dudg,
H 1 (F )\H 1 (A) U(F )\U(A)
where H 1 = Sp 2n and h ∈ H = Sp 2(m+n) . (The group U and the character ψ U were
described in Section 2.)
As often happens in the constructions of liftings using small representations, the
image of the lift is not always cuspidal. Usually, there is an obstruction for the lift to
be cuspidal. To understand when this can happen, assume, for example, that n ≥ m.
Assume that σ itself is an endoscopic lift from two cuspidal representations, from a
cuspidal representation σ ′ on Sp 2(n−m) and from a cuspidal representation ɛ ′ on SO 2m .
For example, if ɛ ′ = ɛ, then it is not expected that the representation π is cuspidal. In
this case, there is an obstruction for the lift to be cuspidal, which is basically expressed
in terms of lifts to groups of smaller rank.
To simplify notation, we prove Theorem 2 only when n ≤ m. Whenn>m,
the formal computations of the constant terms are similar. Since we assumed that the
cuspidal representation ɛ lifts to a cuspidal representation τ(ɛ) on GL 2m , the image
of the lift can fail to be cuspidal only in the case where n = m. Thus, in this case, for
the image to be cuspidal we have to assume that a certain integral is zero. In our case,
the integral we need to assume to be zero is given by integral (7), defined in the proof
of Theorem 2. One can interpret this integral as a lift to a group of a lower rank.
The proof of the cuspidality of the lift requires a manipulation of Fourier expansions
performed on the automorphic functions θ ɛ . Here the function θ ɛ liesinthe
space of the residue representation ɛ . At each step, one has to check that the integrals
converge absolutely. These justifications are now quite standard; the main reference
required is [MW, I.2.10].
THEOREM 2
With the above notation, let π denote the automorphic representation of Sp 2(m+n) (A)
generated by the space of functions f (h) defined above. Assume that n ≤ m. Inthe
case where n = m, assume also that the integral (7) is zero for every choice of data.
Then π is a cuspidal representation.
Proof
For 1 ≤ j ≤ m + n, letV j denote the standard unipotent radical of the maximal
parabolic subgroup of H whose Levi part is GL j × Sp 2(m+n−j) . Thus, we prove that