1549380232-Automorphic_Forms_and_Applications__Sarnak_

LECTURE 3. LOCAL £-FUNCTIONS 125

Proposition 3.2. The local integrals '1i 1 (s; W, W') or '1i(s; W , W', <I?) satisfy the

following properties.

(1) Each integral converges for Re(s) >> 0. For n and n' unitary, as we have

assumed, they converge absolutely for Re( s) ;::: 1. For n and n' tempered,

we have absolute convergence for Re(s) > 0.

(2) Each integral defines a rational function in q-s and hence meromorphically

extends to all of C.

(3) Each such rational function can be written with a common denominator

which depends only on the finite functions X ( n) and X ( n') and hence

only on n and n'.

In deriving these when m < n - 1 note that one has that

W (~ I; In-m-;_J "0

implies that x lies in a compact set independent of h E GLm(k) [42].

Let Ij ( n , n') denote the complex linear span of the local integrals 1]i j ( s; W, W')

if m < n and I(n, n') the complex linear span of the '1i(s; W, W', <I?) if m = n.

These are then all subspaces of C(q-s) which have "bounded denominators" in the

sense of (3). In fact, these subspaces have more structure - they are modules for

C[q^8 , q-^8 ] c qq-^8 ). To see this, note that for any h E GLm(k) we have

]ij ( s; 7f e In-m) w, n'(h)W') =I det(h)1-s-j+(n-m)/21]ij(s; w, W')

and

'1i(s; n(h)W, n'(h)W', p(h)<I?) =I det(h)l-^8 '1i(s; W , W', <I?).

So by varying hand multiplying by scalars, we see that each Ij(n, n') and I(n, n')

is closed under multiplication by C[q^8 , q-s]. Since we have bounded denominators,

we can conclude:

Proposition 3.3. Each I 1 (n, n') and I(n, n') is a fractional C[q^8 , q-^8 ]-ideal of

qq-s).

Note that C[q^8 , q-s] is a principal ideal domain, so that each fractional ideal
I 1 (n,n') has a single generator, which we call Q 1 ,-ir,-rr'(q-^8 ), as does I(n,n'), which
we call Q-rr,-rr'(q-^8 ). However, we can say more. In the case m < n recall that
from what we have said about the Kirillov model that when we restrict Whittaker
functions in W(n,¢ ) to the embedded GLm(k) C Pn(k) we get all functions of
compact support on GLm(k) transforming by¢. Using this freedom for our choice
of W E W( n, ¢) one can show that in fact the constant function 1 lies in I 1 ( n, n').
In the case m = n one can reduce to a sum of integrals over P n ( k) and then use the
freedom one has in the Kirillov model, plus the complete freedom in the choice of <I?
to show that once again 1 E I(n, n'). The consequence of this is that our generator

can be taken to be of the form Q 1 ,-rr,-rr'(q-s) = Pj,-rr,-rr'(q^8 ,q-s)-^1 form< nor

Q-rr,-rr'(q-^8 ) = P-rr,-rr'(q^8 ,q-s)-^1 for appropriate polynomials in C[q^8 ,q-^8 ]. Moreover,
since q^8 and q-s are units in C[q^8 , q-s] we can always normalize our generator to
be of the form Pj,-rr,-rr'(q-s)-^1 or P-rr,-rr'(q-s)-^1 where the polynomial P(X) satisfies
P(O) = 1.