CHAPTER 2

Degree 16 Standard L-function of GSp(2) x GSp(2)

In this Chapter, we are going to establish, via the doubling method, a global integral of

Rankin-Selberg type,, which represents the degree 16 standard L-function of GSp(2) x

GSp(2).

1. Preliminaries

We shall first recall some basic facts on representation theory of algebraic groups

for our later use. For the general theory, we prefer T. A. Springer [Spr]. We only

consider two algebraic groups H and G of symplectic similitudes, which relates to

each other via the doubling method. We will study the //-orbit decomposition of

some flag variety constructed from G and the negligibility of those //-orbits. General

discussion of the doubling method can be found in S. Rallis' IMC paper [Ral].

1.1. The Group of Symplectic Similitudes. Let (V, ( , )) be a 4-dimensional

non-degenerate symplectic vector space over a field F with characteristic zero and

GSp(V) the group of similitudes of (V, ( , )), i.e., GSp(V) = {g e GL(4) : (gu, gv) =

v(g)(u,v) for u,v E V and s(g) is a scalar in F x } . Let (W, ( , )) be the doubling

symplectic space of (V, ( , )), i.e. W = V+ © V~, where V+ = {(t,0) v € V} and

V~ — {(0,v) v eV}, and the symplectic form is defined by

((uuu2),(vuV2)) = (uuvi) - (u2,v2) for Ui,u2,vuv2 e V.

Then (W, ( , )) is an 8-dimensional non-degenerate symplectic vector space over the

field F. We denote by G = GSp(W) be the group of similitudes of (W, ( , )).

In (V, ( , )), choose a symplectic basis {ei,e2, e[,e2} so that the underlying vector

space V is identified with F4 (row vectors) and the form ( , ) corresponds to the

matrix J2 =( _r

Q

2 1. Then in (W, ( , )), we have a typical symplectic basis

{(e

1

,0),(e

2

,0),(0,-e

1

),(0,-e

2

),(e ,

1

,0),(e ,

2

,0),(0,ei),(0,e ,

2

)} (13)

under which the underlying vector space W is identified with F8 (row vectors) and the

form ( , ) of W corresponds to the matrix J4 = ( _7

Q

4 j . Under the chosen basis,

the group of symplectic similitudes can be embedded into a general linear group, i.e.,

GSp(V) = GSp(2) = {ge GL(4) : gj^g = s(g)J2} and GSp(W) = GSp(4) = {g €

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