126 J.W. COGDELL, £-FUNCTIONS FOR GLnFina lly, in the case m < n one can show by a rather elementary although
somewhat involved manipulation of the integrals that all of the ideals Ij ( 7r , -rr') are
the same [42, Section 2.7]. We will write this ideal as I(-rr, -rr') and its generator as
P11:,11:'(q-s )-1.
This gives us the definition of our local L-function.
Definition. Let -rr and -rr' be as above. Then L(s, 7r x -rr') = P11:,11:'(q-s)-^1 is
the normalized generator of the fractional ideal I(-rr, -rr') formed by the family of
local integrals. If -rr' = 1 is the trivial representation of GL 1 (k) then we write
L(s,-rr) = L(s,-rr x 1).
One can show easily that the ideal I( -rr, -rr') is independent of the character
'lj; used in defining the Whittaker models, so that L(s, -rr x -rr') is independent
of the choice of 'lj;. So it is not included in the notation. Also, note that for
-rr' = x an automorphic representation (character) of GL 1 (A) we have the identity
L(s, -rr xx) = L(s, -rr ® x) where -rr ®xis the representation of GLn(A) on Vn given
by -rr ® x(g)~ = x(det(g))-rr(g)f
We summarize the above in the following theorem.
Theorem 3.1. Let -rr and -rr' be as above. The family of lo cal integrals form a
C[q^8 ,q-^8 ]- fractional ideal I(-rr,-rr') in qq-^8 ) with generator the local L-function
L(s, -rr x -rr').
Another useful way of thinking of the local L-function is the following. The
function L(s, 7r x -rr') is the minimal (in terms of degree) function of the form
P(q-s)-^1 , with P(X) a polynomial satisfying P(O) = 1, such that the ratios
-W(s; W, W') w(s; W, W', )
or
L(s, -rr x -rr') L(s, -rr x -rr')
are entire for all WE W(-rr,'l/;) and W' E W(-rr','l/;-^1 ), and if necessary E S(kn).
That is , L(s, 7r x -rr') is the standard Euler factor determined by the poles of the
functions in I(-rr, -rr').
One should note that since the L-factor is a generator of the ideal I ( 7r, -rr'), then
in particular it lies in I(-rr, -rr'). Since this ideal is spanned by our local integrals,
we have the following useful Corollary.
Corollary. There are a finite collection of Wi E W(-rr,'l/;), Wf E W(-rr','lj;-^1 ), and
if necessary i E S(kn) such that
L(s, 7r x 7r^1 ) = L w(s; wi, wn or L(s, 7r x -rr') = L w(s; wi, w:, i)·
For future reference, let us sete(s; W, W') = W~s; W, W'?' ·(. WW')= -WJ(s; W , W')
L s,-rr x -rr' eJ
8
' ' L(s,-rr x -rr')'_(. WW')= ~(s; W, W') d (. WW' "") = -W(s; W, W', <I>)
e s, , L( s, -rr x -rr '), an e s, , , '±' L ( s, -rr x -rr' ) •Then all of these functions are Laurent polynomials in q±s, that is, elements of
C[q^8 , q-^8 ]. As such they are entire and bounded in vertical strips. As above,
there are choices of Wi, Wf, and if necessary i such that I:: e(s; Wi, Wf) = 1 or
L::e(s;Wi, Wf,i) = l. In particular we have the following result.