An Integral Representation of Standard Automorphic L Functions for Unitary Groups

Let F be a number field, G the general linear group of degree n defined over F. Let π be an irreducible cuspidal automorphic representation of G(A). In [1–3], a Rankin-Selbergtype integral is constructed to represent the L function of π. That the integrals of Jacquet, Piatetski-Shapiro, and Shalika are Eulerian follows from the uniqueness of Whittaker models and the fact that cuspidal representations of GLn are always generic. For other reductive group whose cuspidal representations are not always generic, in [4], PiatetskiShapiro and Rallis construct a Rankin-Selberg integral for symplectic group G= Sp2n to represent the partial L function of a cuspidal representation π of G(A). In this paper, we apply similar method to the quasi-split unitary group of rank n. Let F be a number field, E a quadratic field extension of F. Let V be a 2n-dimensional vector space over E with an anti-Hermitian form