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

Let ψ: U → ℂ^x be 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 π of G was expressed by Rodier in terms of the limit of values of the trace of π at certain measures concentrated near the origin. An analogous statement holds in the twisted case. This twisted analogue is used in [F, p. 47] to provide a local proof of the multiplicity one theorem for U(3). This asserts that each discrete spectrum automorphic representation 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. It is pointed out in [F, p. 47] that a proof of the twisted analogue of Rodier's theorem does not appear in print. It is then given below. Detailing this proof is necessitated in particular by the fact that the attempt in [F, p. 48] at a global proof of the multiplicity one theorem for U(3), although widely quoted, is incomplete, as we point out here.