L-FUNCTIONS FOR THE ORTHOGONAL GROUP

In this Memoir we establish global Rankin Selberg integrals which dete rmine the standard L function for the group GL(r) x G' where G' is an isometry group of a nondegenerate symmetric form. The class of automor phic representations considered here is for any pair Pi(1) circle time s Pi(2) where Pi(1) is generic cuspidal for GL(r)(A) and Pi(2) is cusp idal for G'(A). The constructon of these L functions involves the use of certain new ''models'' of local representations; these models gener alize the usual generic models. Also we compute local unramified facto rs in a new way using geometric ideas.