1549380232-Automorphic_Forms_and_Applications__Sarnak_

LECTURE 3. LOCAL £-FUNCTIONS

where sr(A) is the rth_symmetric power of the matrix A, and

r=O

for any matrix A. Then we quickly arrive at

w(s; Wo, W~) = det(I - q-s A,,.® A,,.1 )-^1 =II (1 - μ i(w)μj(w)q-s)-^1

i,j

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a standard Euler factor of degree mn. Since the L-function cancels all poles of the

local integrals, we know at least that det(I - q-s Arr® A,,.1) divides L(s, n x n')-^1.

Either of the methods discussed below for the general calculation of local factors

then shows that in fact these are equal.

There is a similar calculation when n = m and 1> = 1> 0 is the characteristic

function of the lattice on C kn. Also, since n unramified implies that its con-

tragredient 7r is also unramified, with W 0 as its normalized unramified Whittaker

function, then from the functional equation we can conclude that in this situation

we have c:(s,n x n','l/J) = l.

Theorem 3.3. If n, n', and 'ljJ are all unramified, then

L(s, n x n') = det(I - q-s A,,.® A,,.1 )-^1 = {\ll(s; Wo, W~)

\l!(s; Wo, W~, 1>o)

and c:(s,n x n','l/J) = l.

m<n

m=n

For future use, let us recall a consequence of this calculation due to Jacquet
and Shalika [45].

Corollary. Suppose n is irreducible unitary generic admissible (our usual as-
sumptions on n) and unramified. The the eigenvalues μi ( w) of A,,. all satisfy
q-1;2 < Jμi(w)I < q112.

To see this, we apply the above calculation to the case where n' = it the complex
conjugate representation. Then A,,.1 = A,,., the complex conjugate matrix, and we
have from the above

det(I - q-s A,,.® A,,. )\l!(s; Wo, Wo, 1>o) = l.

The local integral in this case is absolutely convergent for Re( s) 2: 1 and so the
factor det(I - q-s A,,. ®A,,.) cannot vanish for Re( s) 2: l. If μi ( w) is an eigenvalue
of A,,. then we have 1 - q-aJμi (w)J^2 =/. 0 for O' 2: l. Hence Jμi (w)J < q^112. Note
that if we apply this to the contragredient representation 7r as well we conclude
that q-1; 2 < Jμi(w)I < q1/2.

3.1.4. The supercuspidal calculation

The other basic case is when both n and n' are supercuspidal. In this case the
restriction of W to P n or W' to Pm lies in the Kirillov model and is hence compactly
supported mod N. In the case of m < n we find that in our integral we have W
evaluated along GLm(k) C Pn(k). Since W is smooth, and hence stabilized by
some compact open subgroup, we find that the local integral always reduces to a
finite sum and and hence lies in C[q^8 , q-s]. In particular it is always entire. Thus in
this case L(s, n x n') = l. In the case n = m the calculation is a bit more involved