1549380232-Automorphic_Forms_and_Applications__Sarnak_

118 J.W. COGDELL, £-FUNCTIONS FOR GLn

where the unfolded global integral is

with the h integral over Nm(A)\ GLm(A) and the x integral over Mn-m-1,m(A), the

space of ( n -m - 1) x m matrices, p denoting right translation, and Wn,m the Weyl

element Wn,m = (Im ) with Wn-m = ( · ·

1

) the standard long Weyl

Wn-m l

element in GLn-m· Also, for WE W(7r,7/i) we set W(g) = W(wng") E W(?r,?ji-^1 ).

The extra unipotent integration is the remnant of ifD::::i. As before, W(s; W, W') is

absolutely convergent for Re( s) > > 0. For <p and <p^1 factorizable as before, these

integrals W ( s; W 'P, W~,) will factor as well. Hence we have

v

where

where now with the hv integral is over Nm(kv)\ GLm(kv) and the Xv integral is over

the matrix space Mn-m-l,m(kv)· Thus, coming back to our functional equation,

we find that the right hand side is Eulerian and factors as

v

2.3. Eulerian integrals for GLn x GLn

The paradigm for integral representations of £-functions for GLn x GLn is not

Hecke but rather the classical papers of Rankin [71] and Selberg [73]. These were

first interpreted in the framework of automorphic representations by Jacquet for

GL2 x GL 2 [37] and then Jacquet and Shalika in general [45].

Let (7r, V1f) and (7r^1 , V1f') be two cuspidal representations of GLn(A). Let <p E V7r

and <p^1 E V7r' be two cusp forms. The analogue of the construction above would be

simply

f i.p(g)i.p'(g)I det(g)ls dg.

JGLn(k)\ GLn(A)

This integral is essentially the £^2 -inner product of <p and <p^1 and is not suitable

for defining an £-function, although it will occur as a residue of our integral at a

pole. Instead, following Rankin and Selberg, we use an integral representation that

involves a third function: an Eisenstein series on G Ln (A). This family of Eisenstein

series is constructed using the mirabolic subgroup once again.