CUSP FORMS ON GL(2N) WITH GL(N)XGL(N) PERIODS, AND SIMPLE ALGEBRAS

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