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ON THE SL(2) PERIOD INTEGRAL By U. K. ANANDAVARDHANAN and DIPENDRA PRASAD
 

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 SL2-invariant 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 L-packet 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,

  

Source: Anandavardhanan, U. K. - Department of Mathematics, Indian Institute of Technology Bombay
Prasad, Dipendra - School of Mathematics, Tata Institute of Fundamental Research

 

Collections: Mathematics