Strong density of spherical characters attached to unipotent subgroups

Abstract

We prove the following result in relative representation theory of a reductive p-adic group $G$: Let $U$ be the unipotent radical of a minimal parabolic subgroup of $G$, and let $\psi$ be an arbitrary smooth character of $U$. Let $S\subset Irr(G)$ be a Zariski dense collection of irreducible representations of $G$. Then the span of the Bessel distributions $B_{\pi }$ attached to representations $\pi$ from $S$ is dense in the space ${\mathcal{S}}^{\ast }(G{)}^{U\times U,\psi \times \psi }$ of all $(U\times U,\psi \times \psi )$-equivariant distributions on $G.$ We base our proof on the following results: 1. The category of smooth representations $\mathcal{M}(G)$ is Cohen-Macaulay. 2. The module $in{d}_U^G(\psi )$ is a projective module.

17 pages