 
Summary: ON THE SL(2) PERIOD INTEGRAL
By U. K. ANANDAVARDHANAN and DIPENDRA PRASAD
Abstract. Let E/F be a quadratic extension of number fields. For a cuspidal representation of
SL2(AE), we study in this paper the integral of functions in on SL2(F)\SL2(AF). We characterize
the nonvanishing of these integrals, called period integrals, in terms of having a Whittaker model
with respect to characters of E\AE which are trivial on AF. We show that the period integral in
general is not a product of local invariant functionals, and find a necessary and sufficient condition
when it is. We exhibit cuspidal representations of SL2(AE) whose period integral vanishes identically
while each local constituent admits an SL2invariant linear functional. Finally, we construct an
automorphic representation on SL2(AE) which is abstractly SL2(AF) distinguished but for which
none of the elements in the global Lpacket determined by it is distinguished by SL2(AF).
1. Introduction. Let F be a number field and AF its ad`ele ring. Let G be
a reductive algebraic group over F and H a reductive subgroup of G over F.
Assume that the center ZH of H is contained in the center ZG of G, a condition
that holds in the cases we study in this paper. For an automorphic form on
G(AF) on which ZH(AF) acts trivially, the period integral of with respect to H
is defined to be the integral (when convergent, which is the case if is cuspidal)
P() =
H(F)ZH(AF)\H(AF)
(h) dh,
