1549380232-Automorphic_Forms_and_Applications__Sarnak_

LECTURE 3. LOCAL L-FUNCTIONS 133

More generally, if in addition 1f^1 is an irreducible generic representation of GLm(k)

then for all sufficiently highly ramified characters 'T/ of GL 1 ( k) we have

1(s, (7r1 © ry) x 1f^1 , '¢) = 1(s, (7r2 © ry) x 1f^1 , '¢)

and

L(s, (7r1 © ry) x 7r^1 ) = L(s, (7r2 © ry) x 7r^1 ) = l.

3.2. The archimedean local factors
We now take k to be an archimedean local field, i.e., k =JR. or C. We take (7r, V1r)
to be the space of smooth vectors in an irreducible admissible unitary generic rep-
resentation of GLn(k) and similarly for the representation (7r', V1r') of GLm(k). We
take 'ljJ a non-trivial continuous additive character of k.
The treatment of the archimedean local factors parallels that of the non-
archimedean in many ways, but there a re some significant differences. The major
work on these factors is that of Jacquet and Shalika in [47], which we follow for
the most part without further reference, and in the archimedean parts of [45].
One significant difference in the development of the archimedean theory is that
the local Langlands correspondence was already in place when the theory was de-
veloped [62]. The correspondence is very explicit in terms of the usual Langlands
classification. Thus to 7f is associated an n dimensional semi-simple representation
r = r(7r) of the Weil group Wk of k and to 1f^1 an m-dimensional semi-simple repre-
sentation r' = r(7r') of Wk. Then r(7r) © r(7r') is an nm dimensional representation
of Wk and to this representation of the Weil group is attached Artin-Weil L-and
£"- factors [92], denoted L(s, r © r') and c(s, r © r', '¢). In essence, Jacquet and
Shalika define

L(s, 7f x 7r^1 ) = L(s, r(7r) © r(7r')) and c(s, 7f x 7f^1 , '¢) = c(s, r(7r) © r(7r^1 ), '¢)

and then set

(

, ·'·) _ c(s,7f x 1f^1 ,'l/!)L(l-s,ir x if')

'Y s, 7f x 7f , 'f/ - L( s , 7f x 1f 1 ).

For example, if 7f is unramified, and hence of the form 7f e::::: Ind(μ 1 © · · · © μn)
with unramified characters of the form μi(x) = lxlr' then

n

L(s,7r) = L(s,r(7r)) = IJrv(s+ri)

i=l

is a standard archimedean Euler factor of degree n, where

if kv =JR.

if kv = C

They then proceed to show that these functions have the expected relation to
the local integrals. Their methods of analyzing the local integrals '1t j ( s; W, W')
and w(s;W, W',\T?), defined as in the non-archimedean case for WE W(7r, '¢),
W' E W(7r^1 , 'lj.;-^1 ), and \T? E S(kn), are direct analogues of those used in [42] for the
non-archimedean case. Once again, a most important fact is [47, Proposition 2.2]