Characters, genericity, and multiplicity one for U(3)

Abstract

Publications on automorphic representations of the group U(3) assumed the validity of multiplicity one theorem since I claimed it in 1982. But the argument, published 1988, was based on a misinterpretation of a claim of Gelbart and Piatetski-Shapiro, SLN 1041 (1984), Prop. 2.4(i): ``L^2_{0,1} has multiplicity 1'', as meaning that each irreducible in the space L^2_{0,1} of generic cusp forms has multiplicity one in the space L^2_0 of cusp forms. The statement meant in SLN 1041, ``each irreducible in L^2_{0,1} has multiplicity 1 in L^2_{0,1}'' is too weak to be useful for nongeneric representations, as the present article points out. To remedy the situation, we detail in this paper the local method we sketched in the paper of 1988. Let ψbe a generic character of the unipotent radical U of a Borel subgroup of a quasisplit p-adic group G. The number (0 or 1) of ψ-Whittaker models on an admissible irreducible representation πof G was expressed by Rodier in terms of the limit of values of the trace of πat certain measures concentrated near the origin. An analogous statement holds in the twisted case. This article proves this twisted analogue for an involution when G=U(3) and p is not 2, and uses it to provide a local proof of the multiplicity one theorem for U(3). This asserts that each discrete spectrum automorphic representation (with induced components at the dyadic places) of the quasisplit unitary group U(3) associated with a quadratic extension E/F of number fields occurs in the discrete spectrum with multiplicity one.