Uniqueness of Shalika functionals: the archimedean case

Abstract

Let F be either ‫ޒ‬ or ‫.ރ‬Let (π, V ) be an irreducible admissible smooth Fréchet representation of GL 2n (F).A Shalika functional φ : V → ‫ރ‬ is a continuous linear functional such that for any g ∈ GL n (F), A ∈ Mat n×n (F) and v ∈ V we haveIn this paper we prove that the space of Shalika functionals on V is at most one-dimensional.For nonarchimedean F (of characteristic zero) this theorem was proved by Jacquet and Rallis.