Strong density of spherical characters attached to unipotent subgroups

• Published in: arXiv.org 2022-02
• Main Authors: Aizenbud, Avraham, Bernstein, Joseph, Sayag, Eitan
• Format: Article
• Language: English
• Subjects: Modules; Subgroups
• ISSN: 2331-8422

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.