Strong density of spherical characters attached to unipotent subgroups

TL;DR

This paper establishes that Bessel distributions associated with a Zariski dense set of irreducible representations of a p-adic group densely span a space of equivariant distributions, using properties of smooth representations.

Contribution

It proves the density of Bessel distributions in the space of equivariant distributions for a dense set of irreducible representations, leveraging Cohen-Macaulay and projectivity properties.

Findings

Bessel distributions form a dense subset in the space of equivariant distributions.

The category of smooth representations is Cohen-Macaulay.

The induced module from a unipotent character is projective.

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 $\mathcal S^*(G)^{U\times U,\psi \times \psi}$ 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 $ind_U^G(\psi)$ is a projective module.