1549380232-Automorphic_Forms_and_Applications__Sarnak_

130 J .W. COGDELL, £-FUNCTIONS FOR GLn

of unramified representations associates to each of these representation the semi-

simple conjugacy classes [A71"] E GLn (C) and [A71"'] E GLm(C) given by

(Recall that w is a uniformizing parameter fork, that is, a generator of p.)

In the Whittaker models there will be unique normalized K = GL(o)- fixed

Whittaker functions, W 0 E W(7r,'¢') and W~ E W(7r','¢'-^1 ), normalized by W 0 (e) =

W~(e) = 1. Let us concentrate on W 0 for the moment. Since this function is right

Kn-invariant and transforms on the left by'¢' under Nn we have that its values are

completely determined by its values on diagonal matrices of the form

for J = (j1,... ,Jn) E zn. There is an explicit formula for W 0 (w^1 ) in terms of

the Satake parameter A11" due to Shintani [87] for GLn and generali zed to arbitrary

reductive groups by Casselman and Shalika [4].

Let T+(n) be the set of n-tuples J = (J 1 , ... ,Jn) E zn with J 1 ~···~Jn· Let

PJ be the rational representation of GLn(C) with dominant weight A 1 defined by

Then the formula of Shintani says that

= tjl 1... tJn n ·

if J tJ. r+(n)

if J E T+(n)

under the assumption that '¢' is unramified. This is proved by analyzing the recur-

sion relations coming from the action of the unramified Hecke algebra on W 0.

We have a similar formula for W~(w^1 ) for J E zm.

If we use these formulas in our local integrals, we find [45, I, Prop. 2.3]

w(s; Wo, W~) = L Wo ( WJ In-m) W~(w^1 )1 det(w^1 w-(n~m) 68! (w^1 )

JET+(m), j,,,;:::o

L tr(P(J,O) (A11" )) tr(p 1 (A11"' ))q-IJls

JET+(m), j,,,;:::o

L tr(P(J,O)(A11") ©p1(A11"1))q-IJls

JET+(m), j,,,;:::o

where we let IJI =Ji+ ... + Jm and we embed zm '-...-4 zn by J = (J 1 , ... ,Jm) 1--7
(J, 0) = (J1, · · · ,Jm, 0, · · · , 0). We now use the invariant theory facts that

L tr(P(J,o)(A11") 0 P1(A11"' )) = tr(Sr(A71" 0 A11"' )),

JET+(m), j,,,;:::o, IJl=r