LECTURE 2. EISENSTEIN SERIES AND L--FUNCTIONS 309Here 8M,M' is the Kronecker 8-function.
Its analytic properties and therefore those of M(s, 7r) are exactly the same as
E(s, <I> 5 , -, P).In the split case the proof of the first part is in [Lal], as in elsewhere. But the
rest is among the main properties of Eisenstein series and included in several places
[La2,HC,MW1], as well as Bernstein's lectures.
To prove the first part, one just substitutes (2.3) in (2.15) and expands and
uses Bruhat decomposition and cuspidality of t.p.
We finally express the functional equation of Eisenstein series by (cf. [HC,La2]),
(2.17) E(-s, M(7r, s) 5 , g, P') = E(s, 5 , g, P).
2.2. Constant Term and Automorphic £-Functions
It follows immediately from the Constant Term Theorem that ®vA(s, 1fv, wo) is a
meromorphic function of s with a finite number of simple poles for Re(s) > 0, none
on Re(s) = 0.
Assume v is an unramified place for 7r, i.e., that 7rv is spherical. Take f~ E
V(s, 7rv) such that f~(k) is a fixed vector invariant under M(Ov) for all k E G(Ov)·
m
With notation as in Section 1.2, let r = EB ri be the adjoint action of L M on Ln.
i=l
We have
Lemma 2.1 [Lal]. Assume 1fv is unramified. Then
m
(2.18) A( s, 7r v) wo)f~ ( ev) = II L( is, 7r V> ri) I L(l +is, 1f v , ri)f~ ( ev).
i=l
With a clever induction [Lal,La2,Sh3] the problem reduces to that of SL 2 ,
which we will now explain.
Here G = SL 2 and M = T =Gm= F:. We need to calculate
(2.19)
since Rwf~ = f~, where w = ( ~ -~ ). We then have
(2.20)
We now write
(2.21) (^1 O) (x-
1
1) (0 -1)
x 1 = 0 x 1 x -^1 'and therefore for lxlv > 1, which implies x-^1 E Ov,
f~( (x~l !) )
77;1(x)lxl;1-s,