Real characters and real classes of ${\mathrm{G}\mathrm{L}}_2$ and ${\mathrm{G}\mathrm{U}}_2$ over discrete valuation rings

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 ${\mathfrak{o}}_{\mathrm{\ell }}=\mathfrak{o}\mathrm{/}{\mathfrak{p}}^{\mathrm{\ell }}$ for $\mathrm{\ell }\ge 2$. In this article, we study real-valued characters and real representations of the finite groups ${\mathrm{G}\mathrm{L}}_2({\mathfrak{o}}_{\mathrm{\ell }})$ and ${\mathrm{G}\mathrm{U}}_2({\mathfrak{o}}_{\mathrm{\ell }})$. 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 ${\mathrm{G}\mathrm{L}}_2({\mathfrak{o}}_{\mathrm{\ell }})$ is afforded by a representation over $\mathbb{R}$. In contrast, we show that ${\mathrm{G}\mathrm{U}}_2({\mathfrak{o}}_{\mathrm{\ell }})$ admits real-valued irreducible characters that are not realizable over $\mathbb{R}$. These results extend the parallel known phenomena for the finite groups ${\mathrm{G}\mathrm{L}}_n({\mathbb{F}}_q)$ and ${\mathrm{G}\mathrm{U}}_n({\mathbb{F}}_q)$.