1549380232-Automorphic_Forms_and_Applications__Sarnak_

134 J.W. COGDELL, £-FUNCTIONS FOR GLn

Proposition 3.6. Let n be a generic representation of GLn(k). Then there is a
finite set of finite functions X(n) = {xi} on An(k), depending only on n, so that

for every WE W(n, '¢;) there are Schwartz functions </Ji E S(kn-l x Kn) such that

for all a E A n (k) with an= 1 we have

W(nak) = 'lf;(n) L Xi(a)¢i(a1 (a), ... , O:n-1 (a), k).

X(7r)

Now the finite functions are r elated to the exponents of the representation n and

through the Langlands classification to the representation T(n) of wk. We retain

the same convergence statements as in the non-archimedean case [45, I, Proposition

3.17; II, Proposition 2.6], [47, Proposition 5.3].

Proposition 3.7. The integrals Wj(s; W, W') and w(s; W, W', <I>) converge

absolutely in the half plane Re( s) ~ 1 under the unitarity assumption and for

Re( s) > 0 if 7r and n' are tempered.

The meromorphic continuation and "bounded denominator" statement in the

case of a non-archimedean local field is now replaced by the following. Define

M ( n x n') to be the space of all meromorphic functions ¢( s) with the property

that if P(s) is a polynomial function such that P(s)L(s, n x n') is holomorphic in

a vertical strip S[a, b] = { s I a :::; Re( s) :::; b} then P( s )¢( s) is bounded in S[a, b].

Note in particular that if¢ E M(n x n') then the quotient ¢(s)L(s,n x n')-^1 is

entire.

Theorem 3.5. The integrals Wj(s; W, W') or w(s; W, W', <I>) extend to meromor-

phic functions of s which lie in M(n x n'). In particular, the ratios

e (s; W, W') = Wj(s; W, W') or e(s; W, W', <I>)= w(s; W, W', <I>)

(^1) L(s, n x n') L(s, n x n')
are entire.
This statement has more content than just the continuation and "bounded
denominator" statements in the non-archimedean case. Since it prescribes the
"denominator" to be the L factor L(s, n x n')-^1 it is bound up with the actual
computation of the poles of the local integrals. In fact, a significant part of the
paper of Jacquet and Shalika. [47] is taken up with the simultaneous proof of this
and the local functional equations:
Theorem 3.6. We have the local functional equations
Wn-m-j-l (1 - s; p(wn,m)W, W') = w' (-l)n-^1 1(s, 7f x n', '¢;)11' j (s ; W, W')
or
11'(1 - s; W, W', ) = w^1 (-1)n-^1 1(s, n x n', 'lf;)w(s; W, W', ).
The one fact that we are missing is the statement of "minimality" of the L-
factor. That is, we know that L(s, n x n') is a standard archimedean Euler factor