On the SL(2) period integral

Abstract

Let E/F be a quadratic extension of number fields. For a cuspidal representation π of SL₂(A_E), we study in this paper the integral of functions in π on SL₂(F)\SL₂(A_F). We characterize the nonvanishing of these integrals, called period integrals, in terms of π having a Whittaker model with respect to characters of E\A_E which are trivial on A_F. 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(A_E) whose period integral vanishes identically while each local constituent admits an SL₂-invariant linear functional. Finally, we construct an automorphic representation π on SL₂(A_E) which is abstractly SL₂(A_F) distinguished but for which none of the elements in the global L-packet determined by it is distinguished by SL₂(A_F).