1549380232-Automorphic_Forms_and_Applications__Sarnak_

LECTURE 3. LOCAL £-FUNCTIONS 135

Towards this end, Jacquet and Shalika enlarge the allowable space of local inte-
grals. Let A and A' be the Whittaker functionals on V1r and V7r' associated with the

Whittaker models W(7r,'l/;) and W(7r','l/;-^1 ). Then A= A @A' defines a continuous

linear functional on the algebraic tensor product v1r 0 v1r, which extends continu-
ously to the topological tensor product v1r®1r' = v1r® v1r,' viewed as representations
of GLn(k) x GLm(k).
Before proceeding, let us make a few remarks on smooth representations. If
( 7r, V1r) is the space of smooth vectors in an irreducible admissible unitary repre-
sentation, then the underlying Harish-Chandra module is the space of Kn-finite
vectors V7r,K· V1r then corresponds to the (Casselman-Wallach) canonical com-
pletion of V1r,K [94]. The category of Harish-Chandra modules is appropriate for
the algebraic theory of representations, but it is useful to work in the category of
smooth admissible representations for automorphic forms. If in our context we take
the underlying Harish-Chandra modules v1r ,K and v7r, ,K then their algebraic tensor
product is an admissible Harish-Chandra module for GLn(k) x GLm(k). The asso-
ciated smooth admissible representation is the canonical completion of this tensor
product, which is in fact V7r@7r'' the topological tensor product of the smooth rep-
resentations 7r and 7r^1 • It is also the space of smooth vectors in the unitary tensor
product of the unitary representations associated to 7r and 7r^1 • So this completion
is a natural place to work in the category of smooth admissible representations.
Now let
W( 7rG7r', 'l/!) = {W(g, h) = A( 7r(g) 0 Jr' (h)~) I~ E v7r 0 7r,}.

Then W( 7rG7r', 'lj!) contains the algebraic tensor product W( 7r, 'lj!) 0 W(1f^1 , 'l/;-^1 ) and
is again equal to the topological tensor product. Now we can extend all out local
integrals to the space W ( 7rG7r^1 , '¢') by setting

Wj(s; W) = j j W ( (~ Ij .) , h) dx I det(h)ls-(n-m)/^2 dh

In-m-1

and

w(s;W,<P) = J W(g,g)<P(eng)ldet(g)ls dh

for WE W(7rG7r','¢'). Since the local integrals a re continuous with respect to the
topology on the topological tensor product, all of the above facts remain true, in
particular the convergence statements, the local functional equations, and the fact
that these integrals extend to functions in M(7r x 7r^1 ).
At this point, let 'Ij(7r,7r^1 ) = {wj(s;W)IW E W(7rG7r')} and let I(7r,7r^1 ) be
the span of the local integrals {w(s;W,<P)IW E W(7rG7r','l/;), <PE S(kn)}. Once
again, in the case m < n we have that the space Ij ( 7r, 1f^1 ) is independent of j and
we denote it also by I(7r, 7r^1 ). These are still independent of 'lj!. So we know from
above that 'I(7r, 7r^1 ) c M(7r x 7r^1 ). The remainder of [47] is then devoted to showing
the following.

Theorem 3.7. I(7r,7r^1 ) = M(7r x 7r^1 ).

As a consequence of this, we draw the following useful Corollary.

Corollary. There is a choice of Whittaker function W in W ( 7rG7r', 'lj!) such that
L(s, 7r x Jr')= w(s; W) if m <nor finite coll ection of functions Wi E W(7rG7r', '¢')

and <Pi E S(kn) such that L(s, 7r x 7r') = ,L:i w(s; Wi, <Pi) if m = n.