Characters of the Norm-One Units of Local Division Algebras of Prime Degree

TL;DR

This paper explicitly constructs all complex irreducible characters of the norm-one units in local division algebras of prime degree and computes their representation zeta functions, revealing their induction structure from compact subgroups.

Contribution

It provides an explicit construction of all irreducible characters of ${ m SL}_1(D)$ for prime degree division algebras over local fields and derives their zeta functions.

Findings

All characters are induced from linear characters of compact-open subgroups.

Explicit formulas for the representation zeta functions are obtained.

Characters are classified for division algebras of prime degree.

Abstract

We give an explicit construction of all complex continuous irreducible characters of the group ${\rm SL}_1(D)$, where $D$ is a division algebra of prime degree $\ell$ over a local field of odd residual characteristic different than $\ell$. For $\ell$ odd, we show that all such characters of ${\rm SL}_1(D)$ are induced from linear characters of compact-open subgroups of ${\rm SL}_1(D)$. We also compute an explicit formula for the representation zeta function of ${\rm SL}_1(D)$.