Theta lifting of cusp forms on the unitary group \(U(d, d)\) (Q1196403)

This paper studies some fundamental problems for the theta lifting for unitary groups over a totally imaginary quadratic extension of a totally real number field. After a discussion of the structure of quasi-split unitary groups, the author computes the Fourier coefficients of the theta lifting. A main result of the paper is the following theorem: If an irreducible nontrivial cuspidal representation of \(U(d,d)\) admits a Whittaker model, its theta lifting to \(U(n,n)\) is nonzero when \(n > d\); further, the theta lifting is not cuspidal when \(n > d+1\).