1549380232-Automorphic_Forms_and_Applications__Sarnak_

LECTURE 2. EULERIAN INTEGRALS FOR GLn 121

will be bounded in vertical strips away from the poles and satisfies the functional

equation

I(s; cp, cp', 'P) = I(l - s; cp, cp', <i>),

coming from the functional equation of the Eisenstein series, where we still have

cp(g) = cp(g') = cp( Wn9') E V;r and similarly for cp'.

These integrals will be entire unless we have ry(a) = w(a)w'(a) = laliM is

unramified. In that case, the residue at s = -ia will be

Res I( s; cp, cp', ip) = - c'P(O) r cp(g )cp' (g) I det(g) 1- ia dg

s=-ia lzn(A) GLn(A)\ GLn(A)

and at s = 1 - ia we can write the residue as

Res. I(s; cp, cp', 'P) = c<i>(O) { cp(g)cp'(g)I det(g)li" dg.

s=l-ia lzn (A) GLn(k)\ GLn(A)

Therefore these residues define GLn(A) invariant pairings between 7r and rr' ®

I <let 1-ia or equivalently between if and if' ® I <let Ii". Hence a residues can be

non-zero only if 7r '.:::::'. if' ® I <let lia and in this case we can find cp, cp', and 'P such

that indeed the residue does not vanish.

We have yet to check that our integrals are Eulerian. To this end we take the

integral, replace the Eisenstein series by its definition, and unfold:

I(s; cp, cp', 'P) = { cp(g)cp'(g)E(g, 'P; s, 77) dg

lzn(A) GLn(k)\ GLn(A)

= f cp(g)cp'(g)F(g,'P;s,ry)dg

lzn(A) P~(k)\ GLn (A)

= f cp(g)cp'(g)I <let(g)ls f 'P(aeng)lalnsry(a) da dg

lzn(A) Pn(k)\ GLn(A) J Ax

= f cp(g)cp'(g)'P(eng)I <let(g)ls dg.

}Pn(k)\ GLn(A)

We next replace cp by its Fourier expansion in the form

cp(g) =

"YENn(k)\ Pn(k)

and unfold to find

I(s; cp, cp','P) = { Wcp(g)cp'(g)'P(eng)I det(g)ls dg

}Nn(k)\ GLn(A)

= f Wcp(g) f cp'(ng)'ifJ(n) dn 'P(eng)I det(g)ls dg

}Nn(A)\ GLn(A) }Nn(k)\ Nn(A)

= f Wcp(g)W~,(g)'P(eng)I det(g)l5 dg
}Nn(A)\ GLn(A)
= w(s; Wcp, w~,, 'P).
This expression converges for Re( s) > > 0 by the gauge estimates as before.
To continue, we assume that cp, cp' and 'P are decomposable tensors under
the isomorphisms 7r '.:::::'. ®' 7r v, H^1 '.:::::'. ®' 7r~, and S (An) '.:::::'. ®' S ( k;J) so that we have
Wcp(g) = ITv WcpJ9v), w~, (g) = ITv w~~ (gv) and 'P(g) = ITv 'Pv(9v)· Then, since