20 2. L-FUNCTION OF GSp(2) x GSp(2)
£(o,o) = F(e'l,e'l) 0 F(e'2,e2) © F(e
2
,e
2
) or L(0,0) = £o7o where
r ° o \
I ° °
2
I
7 o = o ° i 1 ° (17)
0 1 1 1
\ 1 0 - 1 ° /
\ 1 - 1 0 0 /
It is easy to see that £(o,o) is the diagonal embedding into W of a three-dimensional
subspace L* Fe[ 0 Fe
2
0 Fe
2
of V. Let Q = Stab# (L(0,o))- Then we deduce that
Q = {(&,&) G ff : ?I|L* - &|L*, L*9i = L \ i = 1, 2}. (18)
Let Px2 = StabG5P(2)(£*)- Then Px2 is a maximal parabolic subgroup of GSp(2), which
is of form, under the chosen basis {ei,e2, e'1? e
2
},
(
a x 2 s \
2 0l d X0 \eGSP(2) ad =
Xl
x4-x2x3^0}.
0 x3 x' x\ )
If g G GSp(2) such that ^|L* 1, then e ^ = ei + zei, i.e. # = X2ei(z) where
\2£l is the one-parameter subgroup of GSp(2) associated to the root 2s\. Let Z2 =
{X2ei(£) t G F } . Then we obtain that
Q = Stab„(L
o 7 o
) = P?' A (Z
2
x J4) (19)
where
Pj2'
is the diagonal embedding of P\ into Q, and P3J0H corresponds to
£(0,0) = L(oyo)H = Lo^oH.
2. Global Integral of Rankin-Selberg Type
We assume from now on that F be a number field and A its ring of adeles. Let
Fv be the local field of F associated to the place v. When v is finite, we denote
by Ov the ring of local integers in Fv. By automorphic representations of adelic
groups we mean that in the sense of Borel and Jacquet's Corvallis paper [BoJ a].
We shall establish a global zeta integral via the doubling method, which will be a
Rankin-Selberg convolution of two cusp forms of GSp(2) against an Eisenstein series
of GSp(A). The location and the order of poles of such a family of Eisenstein series are
explicitly determined in Chapter III. Before doing so, we shall first study Whittaker
functions on GSp(2,A).
2.1. T h e Whittaker functions on GSp(2). The properties of Whittaker func-
tions studied here will play a critical role in our proof of the eulerian properties of
our global zeta integral in the next subsection. For general description of Whittaker
models, see, J. Shalika [Shi] or S. Gelbart and F. Shahidi [GeSh].
Let 7r be irreducible admissible automorphic cuspidal representation of GSp(2, A)
with trivial central character. Let tp be a generic character of the standard maximal
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