128 J.W. COGDELL, £-FUNCTIONS FOR GLnLet us say a few words about the proof of this proposition, because it is
another application of the analysis of the restriction of representations of GLn
to the mirabolic subgroup P n [42, Sections 2.10 and 2.11 J. In the case where
m < n the local integrals involve the restriction of the Whittaker functions in
W(7r,'lj!) to GLm(k) C Pn, that is , the Kirillov model K(7r,'lj!) of 71". In thecase m = none notes that S 0 (kn) = { E S(kn) I (O) = 0}, which has co-
dimension one in S(kn), is isomorphic to the compactly induced representation
ind~~Uc~k)(5:P~^12 ) so that by Frobenius reciprocity a GLn(k) quasi-invariant trilin-
ear form on W(7r,'lj!) x W(7r','lj!-^1 ) x S 0 (kn) reduces to a Pn(k)-quasi-invariant
bilinear form on K ( 7r, 'lj!) x K ( 11"^1 , 'lj!-^1 ). So in both cases we a re naturally working in
the restriction to Pn(k). The restrictions of irreducible representations of GLn(k)
to P n ( k) are no longer irreducible, but do have composition series of finite length.
One of the tools for analyzing the restrictions of representations of G Ln to P n, or
an alyzing the irreducible representations of P n, are the derivatives of Bernstein and
Zelevinsky [2, 15]. These derivatives 7r(n-r) are naturally representations of GLr(k)
for r ::; n. 7r(o) = 7r and since 7r is generic the highest derivative 7r(n) corresponds to
the irreducible common submodule ( T, VT) of all Kirillov models, and is hence the
non-zero irreducible representation of the trivial group GL 0 (k). The poles of our
local integrals can be interpreted as giving quasi-invariant pairings between deriv-
atives of 7r and 11"^1 [15]. The s for which such pairings exist for all but the highest
derivatives are the exceptional s of the proposition. There is always a unique p air-
ing b etween the highest derivatives 7r(n) and 7r^1 (m), which are necessarily non-zero
since they since these correspond to the common irreducible subspace ( T, VT) of any
Kirillov model, and this is the unique Bs or Ts of the proposition.
As a consequence of this Proposition, we can define the local -y-factor which
gives the local functional equation for our integrals.Theorem 3.2. There is a rational function -y(s,11" x 11"^1 ,'lj!) E C(q-s) such that we
have
~(1 - s; p(wn,m)W, W') = w'(-l)n-^1 -y(s, 7r x 11"^1 , 'lj!)il!(s; W, W') if m < n
or
il!(l - s; W, W', ) = w'(-l)n-^1 -y(s, 7r x 11"^1 , 'lj!)iI!(s; W, W', ) if m = n
for all WE W(7r, 'lj!), W' E W(7r', 'lj!-^1 ), and if necessary all E S(kn).
Again, if 11"^1 = 1 is the trivial representation of GL 1 (k) we write -y(s,11",'lj!) =
-y(s,11" x 1,'lj!). The fact that -y(s,11" x 11"^1 ,'lj!) is rational follows from the fact that it
is a ratio of local integrals.
An equally important local factor, which occurs in the current formulations of
the local Langlands correspondence [32, 35], is the local €-factor.
Definition. The local factor € ( s, 7r x 11"^1 , 'lj!) is defined as the ratio
(
, ·'·) _ -y(s,11" x 11"^1 ,'lj!)L(s,11" x 11"^1 )
€ s, 71" x 71" , '// - L ( 1 - s, 7r - x 11"- 1 ) •With the local €-factor the local functional equation can be written in the form~(l-s;p(wn,m)W,W') _ '(-l)n-l ( / 'lj!)il!(s;W,W')
L(l -s,11" x 7r') -w € s,11" x 7r' L(s,11" x 7r') ifm < n