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