Lecture 2. Eisenstein series and £-functions
2.1. Eisenstein Series and Intertwining Operators; The Constant
Term
Let 7f = 0v1fv be a cusp form on M. Given a KM-finite function cp in the space
of w, we extend cp to a function rp on G as follows. The representation w is a sub-
representation of L6(ZtM(F)\M, p), where zt is the Ap-points of the connected
component Z~ of the center of M and p is a character of ZR,x ( F) \ zt. The function
cp is then in this L^2 -space and being KM-finite, its right translations by elements
in KM = II vKM,v, KM,v = Kv n Mv, generate a finite dimensional representation
T of KM (cf. [Car]). We may assume cp is so chosen that Tis irreducible and write
T = @vTv , where for almost all v, Tv is trivial. Next we will choose irreducible (finite
dimensional) representations iv of each K v, containing Tv· Moreover, we assume ivis the trivial representation for almost all Kv. Seti= ®viv (cf. [Sh6]).
Let Pr be the projection on the space of T and fix measures dkv on each KM,v
whose total mass is 1. Let dk be the product measure on KM. Set(2.1) rj;(m) = { cp(mk)i(k)dk ·Pr.
}KM
Observe that rp(mk-^1 ) = i(k)rp(m). This is a 7- function on Min Harish-Chandra's
terminology [HC]. We extend::; to all of G by rp(nmk) = i(k-^1 )rp(m). It is easily
checked to be a well- defined operator valued function on G ([Sh6]).
Next, set
(2.2) <i>s(g) = rp(g) exp(sa +pp, Hp(g))
and let <I> s be a matrix coefficient of this operator valued function. (See (2. 7) below.)
The corresponding Eisenstein series is then defined by(2.3) E(s,<I> 5 , g,P) =
1EP(F)\G(F)We will also define the operator valued Eisenstein series by
(2.4) E(s, <l>s, g, P) =
1EP(F)\G(F)
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