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In this book, the authors establish global Rankin Selberg integrals which determine the standard $L$ function for the group $GL_r\times G'$, where $G'$ is an isometry group of a nondegenerate symmetric form. The class of automorphic representations considered here is for any pair $\prod_1\otimes\prod_2$ where $\prod_1$ is generic cuspidal for $GL_r(A)$ and $\prod_2$ is cuspidal for $G'(A)$. The construction of these $L$ functions involves the use of certain new'models'of local representations; these models generalize the usual generic models. The authors also compute local unramified factors in a new way using geometric ideas.

Automorphic L-functions, introduced by Robert Langlands in the 1960s, are natural extensions of such classical L-functions as the Riemann zeta function, Hecke L-functions, etc. They form an important part of the Langlands Program, which seeks to establish connections among number theory, representation theory, and geometry. This book offers, via the Rankin-Selberg method, a thorough and comprehensive examination of the degree 16 standard L-function of the product of two rank two symplectic similitude groups, which includes the study of the global integral of Rankin-Selberg type and local integrals, analytic properties of certain Eisenstein series of symplectic groups, and the relevant residue representations.