Uniqueness of Shalika functionals (the Archimedean case)

TL;DR

This paper proves that the space of Shalika functionals for irreducible admissible smooth representations of GL(2n,F) over Archimedean fields is at most one-dimensional, extending known results from the non-Archimedean case.

Contribution

It establishes the uniqueness of Shalika functionals in the Archimedean setting, filling a gap in the representation theory of GL(2n,F).

Findings

Shalika functional space is at most one-dimensional.

Extension of non-Archimedean results to Archimedean fields.

Provides foundational result for automorphic forms and representation theory.

Abstract

Let F be either R or C. Let $(\pi,V)$ be an irreducible admissible smooth \Fre representation of GL(2n,F). A Shalika functional $\phi:V \to \C$ is a continuous linear functional such that for any $g\in GL_n(F), A \in \Mat_{n \times n}(F)$ and $v\in V$ we have $$ \phi[\pi g & A 0 & g)v] = \exp(2\pi i \re(\tr (g^{-1}A))) \phi(v).$$ In this paper we prove that the space of Shalika functionals on V is at most one dimensional. For non-Archimedean F (of characteristic zero) this theorem was proven in [JR].