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 $\mathrm{ GU}_n(\mathbb{ F}_q)$.
Categories: math.RT math.GR math.RA