1549380232-Automorphic_Forms_and_Applications__Sarnak_

LECTURE 2. EULERIAN INTEGRALS FOR GLn 119

2.3.1. The mirabolic Eisenstein series

To construct our Eisenstein series we return to the observation that P n \ GLn '.:::'.

kn - {O}. If we let S(An) denote the Schwartz-Bruhat functions on An, then

each !fl E S defines a smooth function on GLn(A), left invariant by Pn(A), by

g 1--+ !f>((O, ... , 0, l)g) = !i>(eng). Let rJ be a unitary idele class character. (For our

application rJ will be determined by the central characters of 7r and 7r^1 .) Consider

the function

If we let P~ = Zn P n be the parabolic of GLn associated to the partition ( n - 1, 1)

then one checks that for p' = (~ ~) E P~(A) with h E GLn-i(A) and d E Ax we

have,

F(p'g, !I>; s, TJ) =I det(h)l81dl-(n-l}s'T](d)-^1 F(g, !I>; s, rJ)

=bf,~ (p')rJ-^1 (d)F(g, !fl; s, rJ),

with the integral absolutely convergent for Re(s) > 1/n, so that if we extend rJ to
a character of P~ by rJ(p') = rJ(d) in the above notation we have that F(g, !I>; s, rJ)
is a smooth section of the normalized induced representation Ind~t(~:) ( b~~^112 rJ).
Since the inducing character b~-;^112 'T] of P~(A) is invariant under P~(k) we may
form Eisenstein series from this family of sections by

E(g, !I>; s, rJ) = F('yg, !I>; s, rJ).

If we replace F in this sum by its definition we can rewrite this Eisenstein series as

E(g, !I>; s, ?J) =I det(g)ls 1 L !f>(a~g)lalnsrJ(a) dxa

kX\AX ~Ekn-{o}

=I det(g)ls f e~(a,g)lalnsrJ(a) dxa

JkX\AX

and this first expression is convergent absolutely for Re( s) > 1 [45].
The second expression essentially gives the Eisenstein series as the Mellin trans-
form of the Theta series

e<J?(a,g) = 'L !f>(a~g),

~Ekn

where in the above we have written

e~(a,g) = L !f>(a~g) = e<J?(a,g) - !I>(O).

~Ekn-{o}