124 J.W. COGDELL, £-FUNCTIONS FOR GLn

in the case m < n, where the h integration is over Nm(k)\ GLm(k) in both integrals
and the x integration is over the matrix space Mn-m-1,m(k), and in the case n = m

w(s; W, W', q>) = f W(g)W'(g)gj(eng)I det(g)ls dg

}Nn(k)\ GLn(k)

all integrals being convergent for Re(s) >> 0. To make the notation more con-
venient for what follows, in the case m < n for any 0 :::; j :::; n - m - 1 let us
set

'11;('' W, W') ~ j j W (Z I; In-m-J dx W'(h)I det(h)l•-(n-m)/' dh,

where the h integral is still over Nm(k)\ GLm(k) and now the x integral is over
the matrix space Mj,m(k), so that w(s; W, W') = '11 0 (s; W, W') and '1i(s; W, W') =
Wn-m- 1 (s; W, W'), which is still absolutely convergent for Re(s) >> 0.
We need to understand what type of functions of s these local integrals are. To
this end, we need to understand the local Whittaker functions. So let WE W(7r, 't/J).
Since W is smooth, there is a compact open subgroup K', of finite index in the
maximal compact subgroup K n = GLn(o), so that W(gk) = W(g) for all k EK'. If
we let { ki} be a set of coset representatives of G Ln ( o) / K', using that W transforms
on the left under Nn(k) via 't/J and the Iwasawa decomposition on GLn(k) we see that
W(g) is completely determined by the values of W(aki ) = Wi(a) for a E An(k), the
maximal split (diagonal) torus of GLn(k). So it suffices to understand a general
Whittaker function on the torus. Let cxi, i = 1,... , n - 1, denote the standard

simple mots of GL., so that ifa ~ (a, ·.

0

J E A 0 (k) then a,(a) ~ a,/a;+<·

By a finite function on An(k) we mean a continuous function whose translates span
a finite dimensional vector space [39, 40 , Section 2.2]. (For the field k x itself the
finite functions are spanned by products of characters and powers of the valuation
map.) The fundamental result on the asymptotics of Whittaker functions is then
the following [40, Prop. 2.2].

Proposition 3.1. Let 7r be a generic representation of GLn(k). Then there is a
finite set of finite functions X(7r) = {xi} on An(k), depending only on 7r, so that
for every WE W(7r,'t/J) there are Schwartz -Bruhat functions </>i E S(kn-l) such
that for all a E An ( k) with an = 1 we have

W(a) = L Xi(a)</>i(cx1 (a), ... , CXn-1 (a)).
X(7r)
The finite set of finite functions X ( 7r) which occur in the asymptotics near 0 of
the Whittaker functions come from analyzing the J acquet module of 7r in the form
W(7r, 't/J)/(7r(n)W - Win E Nn) which is naturally an An(k)- module. Note that
due to the Schwartz-Bruhat functions, the Whittaker functions vanish whenever
any simple root cxi(a) becomes large. The gauge estimates alluded to in Lecture 2
are a consequence of this expansion and the one in Proposition 3.6.
Several nice consequences follow from inserting these formulas for W and W'
into the local integrals wj(s; W, W') or w(s; W, W', q>) [40, 42].