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
17
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