On Uniform Admissibility of Unitary and Smooth Representations

TL;DR

This paper establishes an equivalence between uniform admissibility of irreducible unitary and smooth representations of certain topological groups, and shows this property is preserved under subgroup and overgroup relations.

Contribution

It proves the equivalence of uniform admissibility for unitary and smooth representations and demonstrates inheritance of this property by finite-index subgroups and overgroups.

Findings

Uniform admissibility of unitary and smooth representations are equivalent.

Inheritance of uniform admissibility by finite-index subgroups and overgroups.

Necessary mild assumption for the equivalence to hold.

Abstract

Let $G$ be a locally compact totally disconnected topological group. Under a necessary mild assumption, we show that the irreducible unitary representations of $G$ are uniformly admissible if and only if the irreducible smooth representations of $G$ are uniformly admissible. We also show that the latter property is inherited by finite-index subgroups and overgroups of $G$.