136 J.W. COGDELL, £-FUNCTIONS FOR GLn
As a final result, let us note that in [16] it is established that the linear function-
als e(s; W) = w(s; W)L(s, n x n')-^1 and e(s; W , )= w(s; W , )L(s, n x n')-^1 are
continuous on W(n®n', 'l/J), uniformly for sin compact sets. Since there is a choice
of WE W(n®n',1/J) such that e(s;W) = 1 or Wi E W(n®n',1/J) and i E S(kn)
such that I: e(s; W i, i ) = 1, as a result of this continuity and the fact that the
algebraic tensor product W(n, 'l/J) ® W(n', 1/J-^1 ) is dense in W(n®n', 'lfJ) we have the
following result [16].
Proposition 3.8. For any s 0 E C there are choices of W E W(n, 'l/J), W' E
W(n',1/J-^1 ) and if necessary such that e(s 0 ;W, W') i=- 0 or e(so;W, W',) ;j=. O.
The continuity of the local integrals plays a role in proving the following result
of Sta de [89, 90] and Jacquet and Shalika (unpublished).
Theorem 3.8. In the cases m = n and m = n - 1 there exist a finite collection of
K- finite functions Wi E W(n,1/J), Wf E W(n',1/J-^1 ), and i E S(kn) if necessary
such that
L(s, 7r x n') = L w(s; w i, Wf) or L(s, 7r x n') = L w(s; wi, w:, i)·
In the case where both 7r and n' are unramified, Stade shows that one obtains
the £-function exactly with the K- invariant Whittaker functions (and Schwartz
function if necessary). In the general case, Jacquet has provided us with a sketch
of his argument with Shalika. First one proves that the integrals involving K-finite
functions are equal to the product of a polynomial and the £-factor. It suffices to
prove this for principal series, since the other representations embed into principal
series. For principal series one proceeds by an induction argument on n, however
one must prove the m = n and m = n - 1 cases simultaneously. The (essentially
formal) arguments needed are to be found in the published papers of Jacquet and
Shalika. The polynomials in question then form an ideal and the point now is to
show this ideal is the full polynomial ring. This is then implied by Proposition 3.8
above.