Cusp forms on GL(2n) with GL(n)×GL(n) periods, and simple algebras

The notion of a period of a cusp form on GL(2,D(double-struck)), with respect to the diagonal subgroup D(double-struck A)^× × D(double-struck A)^×, is defined. Here D is a simple algebra over a global field F with a ring A of adeles. For D^× =GL(1), the period is the value at 1/2 of the L-function of the cusp form on GL(2, double-struck). A cuspidal representation is called cyclic if it contains a cusp form with a non zero period. It is investigated whether the notion of cyclicity is preserved under the Deligne-Kazhdan correspondence, relating cuspidal representations on the group and its split form, where D is a matrix algebra. A local analogue is studied too, using the global technique. The method is based on a new bi-period summation formula. Local multiplicity one statements for spherical distributions, and non-vanishing properties of bi-characters, known only in a few cases, play a key role.