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

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 $\mathrm{GU}_n(\mathbb{F}_q)$.