L functions for the orthogonal group

In this Memoir we establish global Rankin Seiberg integrals which determine the standard L function for the group GL_r × G′ where G′ is an isometry group of a nondegenerate symmetric form. The class of automorphic representations considered here is for any pair Π₁ ⊗ Π₂ where Π₁ is generic cuspidal for GL_r(double-struck A) and Π₂ is cuspidal for G′(double-struck A). The constructon of these L functions involves the use of certain new "models" of local representations; these models generalize the usual generic models. Also we compute local unramified factors in a new way using geometric ideas.