$L$ Functions for the Orthogonal Group

Book excerpt: In this book, the authors establish global Rankin Selberg integrals which determine the standard [italic capital]L function for the group [italic capitals]GL[subscript italic]r x [italic capital]Gʹ, where [italic capital]Gʹ is an isometry group of a nondegenerate symmetric form. The class of automorphic representations considered here is for any pair [capital Greek]Pi1 [otimes/dyadic/Kronecker/tensor product symbol] [capital Greek]Pi2 where [capital Greek]Pi1 is generic cuspidal for [italic capitals]GL[subscript italic]r([italic capital]A) and [capital Greek]Pi2 is cuspidal for [italic capital]Gʹ([italic capital]A). The construction of these [italic capital]L functions involves the use of certain new "models" of local representations; these models generalize the usual generic models. The authors also computer local unramified factors in a new way using geometric ideas.