Uniqueness of Shalika functionals: the archimedean case

Avraham Aizenbud, Dmitry Gourevitch and Hervé Jacquet

Vol. 243 (2009), No. 2, 201–212

DOI: 10.2140/pjm.2009.243.201

Abstract

Let F be either ℝ or ℂ. Let (π,V ) be an irreducible admissible smooth Fréchet representation of GL_2n(F). A Shalika functional ϕ : V → ℂ is a continuous linear functional such that for any g ∈ GL_n(F), A ∈ Mat_n×n(F) and v ∈ V we have

In 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.

Keywords

multiplicity one, Gelfand pairs, Shalika functionals, uniqueness of linear periods

Mathematical Subject Classification 2000

Primary: 22E45

Milestones

Received: 10 April 2009

Accepted: 18 May 2009

Published: 1 December 2009