1549380232-Automorphic_Forms_and_Applications__Sarnak_

LECTURE 3. LOCAL £-FUNCTIONS 127

Corollary. The functions e(s; W, W') and e(s; W, W', <f?) are entire functions,

bounded in vertical strips, and for each s 0 E C there is a choice of W, W', and if

necessary <f? such that e(so; W, W')-/=- 0 or e(s 0 ; W, W', <f?)-/=-0.

3.1.2. The local functional equation

Either by analogy with Tate's thesis or from the corresponding global statement,

we would expect our local integrals to satisfy a local functional equation. From the

functional equations for our global integrals, we would expect these to relate the

integrals \Il(s; W, W') and ~(1-s; p(wn,m)W, W') when m < n and \Il(s; W, W', <f?)

and \Il(l - s; W, W', 4>) when m = n. This will indeed be the case. These functional

equations will come from interpreting the local integrals as families (ins) of quasi-

invariant bilinear forms on W(7r,'¢') x W(7r','¢'-^1 ) or trilinear forms on W(7r,'¢') x

W(7r','¢'-^1 ) x S(kn) depending on the case.

First, consider the case when m < n. In this case we have seen that

\II (s;7r e In-m) W,7r

1

(h)W') = ldet(h)l-s+(n-m)/^2 \If(s;W, W')

and one checks that ~(1 - s; p(wn,m)W, W') has the same quasi-invariance as a

bilinear form on W(7r,'¢') x W(7r','¢'-^1 ). In addition, if we let Yn,m denote the

unipotent radical of the standard parabolic subgroup associated to the partition

(m + 1, 1, ... , 1) as before then we have the quasi-invariance

\Il(s; 7r(y)W, W') = '¢'(y)\Il(s; W, W')

for ally E Y n,m· One again checks that ~(1 - s; p(wn,m)W, W') satisfies the same

quasi-invariance as a bilinear form on W(7r,'¢') x W(7r','¢'-^1 ).

For n = m we have seen that

\Il(s; 7r(h)W, 7r^1 (h)W',p(h)<f?) =I det(h)l-s\Il(s; W, W', <f?)

and it is elementary to check that \Il(l - s; W, W', 4>) satisfies the same quasi-

invariance as a trilinear form on W(7r, '¢') x W(7r^1 , '¢1-^1 ) x S(kn). Our local functional

equations will now follow from the following result [42, Propositions 2. 10 and 2.11].

Proposition 3.4. (i) If m < n, then except for a finite number of exceptional

values of q-s there is a unique bilinear form Bs on W(7r, '¢') x W(7r^1 , '¢1-^1 ) satisfying

Bs ( 7r e In-m) W, 7r

1

(h)W') =I det(h)l-s+(n-ml/^2 Bs(W, W')

and Bs(7r(y)W, W') = '¢'(y)Bs(W, W')

for all h E GLm(k) and y E Yn,m(k).

(ii) If n = m, then except for a finite number of exceptional values of q-s there

is a unique trilinear form Ts on W(7r,'¢') x W(7r','¢'-^1 ) x S(kn) satisfying

T 8 (7r(h)W, 7r^1 (h)W', p(h)<f?) = I det(h) 1-srs(W, W', <f?)

for all h E GLn(k).