Smooth representations of unit groups of split basic algebras over non-Archimedean local fields

TL;DR

This paper studies smooth representations of unit groups of split basic algebras over non-Archimedean local fields, proving a conjecture that all irreducible representations are induced from one-dimensional subalgebra representations.

Contribution

It proves Gutkin's conjecture for these groups, showing all irreducible smooth representations are compactly induced from one-dimensional subalgebra representations.

Findings

Every irreducible smooth representation is compactly induced from a one-dimensional subalgebra representation.

Discussion on admissibility and unitarisability of these representations.

Establishment of a structural understanding of representations of unit groups of split basic algebras.

Abstract

We consider smooth representations of the unit group $G = \mathcal{A}^{\times}$ of a finite-dimensional split basic algebra $\mathcal{A}$ over a non-Archimedean local field. In particular, we prove a version of Gutkin's conjecture, namely, we prove that every irreducible smooth representation of $G$ is compactly induced by a one-dimensional representation of the unit group of some subalgebra of $\mathcal{A}$. We also discuss admissibility and unitarisability of smooth representations of G.