1549380232-Automorphic_Forms_and_Applications__Sarnak_

L ( l - s,7r x 7r^1 ) - w € s, 7r x 7r ' 'V L(s,7r x 7r')

ifm=n.

This can also be expressed in terms of the e(s; W, W'), etc .. In fact, since we know

we can choose a finite set of Wi, Wf, and if necessary <I>i so that

'°"' w(s,; wi, Wi) = '°"' (. W· W') = 1

L L(s,7r x 7r') Les, " i

t t

or

'°"' w(s; wi, Wf, <I>i) = '°"' (. W· W' n-..) =

L. L( S,7r X 7r ') Le. s, ,, i' 'l'i 1

t t

we see that we can write either

or

c:(s, 7r x 7r^1 , 'I/;)= w'(-l)n-l L e(l - s; p(wn,m)Wi, Wi)

c(s, 7r x 7r^1 , 'I/;)= w'(-1t-^1 L e(l - s; wi, Wf, ~i)

i

and hence c:(s,7r x 7r^1 ,'l/;) E <C[q^8 ,q-^8 ]. On the other hand, applying the functional

equation twice we get

c:(s,7r x 7r^1 ,'l/;)c:(l - s,?r x ?r','l/;-^1 ) = 1

so that c( s, 7r x 7r^1 , 'I/;) is a unit in <C[q^8 , q-s]. This can be restated as:

Proposition 3.5. c:(s, 7r x 7r^1 , 'I/;) is a monomial function of the form cq-fs.

Let us make a few remarks on the meaning of the number f occurring in the

c:-factor in the case of a single representation. Assume that 'I/; is unramified. In

this case write c(s,7r,'I/;) = c:(0,7r,'l/;)q-f(7r)s. In [43] it is shown that f(7r) is a non-

negative integer, f(7r) = 0 iff 7r is unramified, that in general the space of vectors

in v7r which is fixed by the compact open subgroup

K, (pf<•l) ~ { g E GL,(o) lg= ( O * O ;) (mod pf(•l)}

has dimension exactly 1, and that if t < f(7r) then the dimension of the space of

fixed vectors for K 1 (pt) is 0. Depending on the context, either the integer f(7r)

or the ideal f(7r) = pf(7r) is called the conductor of 7r. Note that the analytically

defined c:-factor carries structural information about 7r.

3.1.3. The unramified calculation
Let us now turn to the calculation of the local £-functions. The first case to consider
is that where both 7r and 7r^1 are unramified. Since they are assumed generic, they are
both full induced representations from unramified characters of the Borel subgroup
[97]. So let us write 7r c:::: Ind~~n (μ1 0 · · · 0 μn) and 7r^1 c:::: Ind~:=(μ~ 0 · · · 0 μ'm)
with the μi and μj unramified characters of kx. The Satake parameterization