Real characters and real classes of $\mathrm{GL}_2$ and $\mathrm{GU}_2$ over discrete valuation rings

TL;DR

This paper classifies real and strongly real classes and characters of the groups $ ext{GL}_2( ext{o}_ ext{ell})$ and $ ext{GU}_2( ext{o}_ ext{ell})$ over discrete valuation rings, revealing differences in realizability over the real numbers.

Contribution

It provides a complete classification of real classes and characters for these groups and shows that all real-valued characters of $ ext{GL}_2$ are realizable over $ ext{R}$, unlike $ ext{GU}_2$.

Findings

All real-valued irreducible characters of $ ext{GL}_2( ext{o}_ ext{ell})$ are realizable over $ ext{R}.

Some real-valued irreducible characters of $ ext{GU}_2( ext{o}_ ext{ell})$ are not realizable over $ ext{R}.

The results extend known phenomena from finite groups over finite fields.

Abstract

Let $\mathfrak{o}$ be the ring of integers of a non-archimedean local field with residue field of odd characteristic, $\mathfrak{p}$ be its maximal ideal and let $\mathfrak{o}_\ell = \mathfrak{o}/\mathfrak{p}^\ell$ for $\ell\ge 2$. In this article, we study real-valued characters and real representations of the finite groups $\mathrm{GL}_2(\mathfrak{o}_\ell)$ and $\mathrm{GU}_2(\mathfrak{o}_\ell)$. We give a complete classification of real and strongly real classes of these groups and characterize the real-valued irreducible complex characters. We prove that every real-valued irreducible complex character of $\mathrm{GL}_2(\mathfrak{o}_\ell)$ is afforded by a representation over $\mathbb{R}$. In contrast, we show that $\mathrm{GU}_2(\mathfrak{o}_\ell)$ admits real-valued irreducible characters that are not realizable over $\mathbb{R}$. These results extend the parallel known phenomena for the finite groups $\mathrm{GL}_n(\mathbb{F}_q)$ and $\mathrm{GU}_n(\mathbb{F}_q)$.