1549380232-Automorphic_Forms_and_Applications__Sarnak_

LECTURE 2. EISENSTEIN SERIES AND £ -FUNCTIONS

Av E X*(T). Now take t such that A(t) = a.v, where

a.v E X*(Tv)(= Hom(X*(T), Z) = A(T))

defines a root of yv. Then

(2.27)

where Hr(t) = A(t) = a.v.

In the case of SL 2 , t v = 'T/v· Moreover, setting Av= a. E X*(T),

(

-1

Thus t = wO

(2.28)

and therefore

(2.29)

O) and

Wv

q~oY,a) ja.(t)j

= qv.^2

q~v = a. V ( 'T/v)

= r(ryv)·

311

Thus q;;μv = f(TJv)· It is therefore f which will appear if one uses the standard
definition of Hr.

SL 2 (JR). It is instructive to also compute the case of SL 2 (JR)(CL 2 (JR), respec-
tively). We again need to calculate

(2.30)

Here K = S0 2 (JR)(0 2 (JR), respectively) and we can write

(x^1 o) 1 = (a 0 y) b k(e) ,

where

k(B) = ( co~e sine).

-sme cose

We can then take tane = -x, b = a-^1 = v'x2+1 and y = x/)x^2 + l. We need
to calculate

J

oo ( 2 1 )s (^1) (Vx2+1)s2(x2+1)-1/2dx,
-()() v'x2+1
where 'TJ ( ( ~ ~)) = jaj^81 jbj82, or
2 fooo (x2+1)-(s1-s2+1)/2dx.
Let s = s 1 - s 2 and set x = tan e, we need to calculate
r 12
2 lo (cose)s-^1 de.