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()), with respect to the diagonal subgroup D() X × D() X , is defined. Here D is a simple algebra over a global field F with a ring of adeles. For D x = GL(1), the period is the value at 1/2 of the L‐ function of the cusp form on GL(2, ). 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.