A NULLSTELLENSATZ FOR AMOEBAS

460 DAVID GINZBURG
check that up to a product of local intertwining operators, the poles of the Eisenstein
series are determined by
∏n−1
i=1
L S (τ × τ,s i − s i+1 ) L S (τ × µ(ɛ),s n )
L S (τ × τ,s i − s i+1 + 1) L S (τ × µ(ɛ),s n + 1) L τ (¯s),
where L τ (¯s) is a product of partial L-functions that are holomorphic and do not vanish
at the point s i = n − i + 1.Let¯s 0 = (n, n − 1,...,1). It follows from the above that
the Eisenstein series has a pole at that point.
If we denote by ɛ the residue of this Eisenstein series at the point ¯s 0 ,then
O G ( ɛ ) = ((2m) 2n+1 ). Thus, the difference between this Eisenstein and the series that
we studied at the beginning Section 2.2 is the use of the representation µ(ɛ) defined
on Sp 2m (A). This difference is technical in its nature, and the statements proved above
follow easily using similar arguments.
(2) We have a similar situation on the double cover of the symplectic group. Denote
G = ˜Sp 2m(2n+1) .Letɛ denote a generic cuspidal representation of ˜Sp 2m (A), andlet
τ = τ(ɛ) denote the functorial lift to GL 2m from ɛ. By that we mean the following. It is
well known (see, e.g., [GRS2]) that every generic cuspidal representation of ˜Sp 2m (A)
has a functorial lift to a generic cuspidal representation of SO 2m+1 (A) or to a generic
cuspidal representation of SO 2m−1 (A). Using the result of [CKPS], we can thus deduce
that ɛ has a functorial lift to a cuspidal representation of GL 2m (A) or GL 2(m−1) (A).
Henceforth, we assume that the given cuspidal representation ɛ has a functorial lift
to a cuspidal representation τ = τ(ɛ) of GL 2m (A). In this context, we can therefore
view the group Sp 2m (C) as the “L group” of ˜Sp 2m (see also [S]).
We can form the Eisenstein series Ẽ τ(ɛ),ɛ (g, ¯s) as in case (1) and prove similar
results.
(3) Let G denote the split orthogonal group SO 2m(2n+1) .Letɛ denote a generic
irreducible cuspidal representation of SO 2m (A), andletτ = τ(ɛ) denote its lift
to GL 2m . We assume that τ is cuspidal. Let Q denote the standard parabolic subgroup
of G whose Levi part is GL 2m × ··· × GL 2m × SO 2m .LetE τ(ɛ),ɛ (g, ¯s)
denote the Eisenstein series defined on G and associated with the induced representation
Ind G(A)
Q(A) (τ ⊗···⊗τ ⊗ 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
cases (1), (2), it follows that up to a product of local intertwining operators, the poles
of the Eisenstein series are determined by
∏n−1
i=1
L S (τ × τ,s i − s i+1 ) L S (τ × ɛ, s n )
L S (τ × τ,s i − s i+1 + 1) L S (τ × ɛ, s n + 1) L τ (¯s),
where L τ (¯s) is a product of partial L-functions that are holomorphic and do not vanish
at the point s i = n − i + 1.Let¯s 0 = (n, n − 1,...,1). It follows from the above that
the Eisenstein series has a pole at that point.