Uniqueness of Shalika functionals (the Archimedean case)
arXiv:0904.0922
Abstract
Let F be either R or C. Let be an irreducible admissible smooth \Fre representation of GL(2n,F). A Shalika functional $ϕ:V \to \C$ is a continuous linear functional such that for any $g\in GL_n(F), A \in \Mat_{n \times n}(F)$ and we have $$ ϕ[πg & A 0 & g)v] = \exp(2πi \re(\tr (g^{-1}A))) ϕ(v).$$ 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].
9 pages. v2:corrected version, to appear in Pacific Journal of Mathematics