A NULLSTELLENSATZ FOR AMOEBAS

ENDOSCOPIC LIFTING 499
from which it follows that
(η m+4 |ϖ m ) ↓ Spin7 ×Spin 2m+1
= (µ 3 |0) ⊕ (µ 3 |2ϖ m ) ⊕ m−1
i=1 (µ 3|ϖ i ). (33)
The representations on the right-hand side of this equality are representations of the
group Spin 7 (C)×Spin 2m+1 (C).Let(01) denote the second fundamental representation
of G 2 (C). Its degree is 7. Then µ 3 restricted to G 2 (C) gives us 1 ⊕ (01). From this,
we obtain the fact that
(η m+4 |ϖ m ) ↓ G2 ×Spin 2m+1
= (0|0) ⊕ (01|0) ⊕ (0|2ϖ m ) ⊕ (01|2ϖ m )
⊕ m−1
i=1 [(0|ϖ i) ⊕ (01|ϖ i )],
where the representations of the right-hand side are representations of the group
G 2 (C) × Spin 2m+1 (C).
Assume that Conjecture 2(3) holds. Then the above branching decomposition
induces a factorization of the partial L-function L S (π × ɛ, Spin 2(m+4) × Spin 2m+1 ,s),
which contains at least one zeta factor. One expects that for generic representations,
the other partial L-function must be nonzero at s = 1. Moreover, if the representations
ν and ɛ are in general position, one would expect that these L-functions also must be
holomorphic at s = 1. This would then imply that at s = 1, the pole of the above
L-function would actually be a simple pole. This then motivates the implication that
(3) implies (1) in Conjecture 2.
Next, we introduce the period integral Q(π, ɛ). To simplify things, we introduce
the period not in the similitude groups but on the orthogonal and symplectic groups
themselves. The adoption to the similitude groups is quite simple but requires some
more notation. Let P denote the standard parabolic subgroup of M ′ = SO 2(3m+2)
whose Levi part is GL m−1
2 × SO 2(m+4) ,andletU ′ denote its unipotent radical. In terms
of matrices, this unipotent group is described in (2) with r = m − 2,p = 1, and
q = m + 4. As explained in Section 2.2(c), one can define a projection l from the
group U ′ onto the Heisenberg group H 4(m+4)+1 .LetK = SO 2(m+4) × SL 2 .Wedefine
Q(π, ɛ) to be the period integral
∫ ∫
ϕ π (k 1 )θ ψ (
Sp l(u)(k1
4(m+4)
,k 2 ) ) θ ɛ( ′ u(k1 ,k 2 ) ) ψ U ′(u) dudk 1 dk 2 . (34)
K(F )\K(A) U ′ (F )\U ′ (A)
Here k 1 ∈ SO 2(m+4) ,andk 2 ∈ SL 2 . The function θ
ɛ ′ is a vector in the space of the
representation ′ ɛ defined on the group M′ (A) as follows. Recall the representation
ɛ defined on the group M = SO 14m+8 in Section 2.2(4), with n = 3. This representation
is a residue representation that was attached to the induced representation
from the parabolic subgroup Q whose Levi part is GL 3 2m+1 × SO 2(m+1).Todefinethe
representation ′ ɛ , we start with the parabolic subgroup Q′ of M ′ whose Levi part