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

TL;DR

The work analyzes real representations of $GL_2(\mathfrak{o}_\ell)$ and $GU_2(\mathfrak{o}_\ell)$ over the rings of integers modulo powers of the maximal ideal, providing a complete classification of real and strongly real classes and characterizing real-valued irreducible characters. It develops explicit constructions of regular representations for both even and odd levels using Serre lifts, inertia groups, and Clifford theory, and it delineates split non-semisimple, split semisimple, and cuspidal types (including cuspidals via Heisenberg-type extensions). The paper proves that all real-valued irreducible characters of $GL_2(\mathfrak{o}_\ell)$ are realizable over $\mathbb{R}$ (orthogonal), while $GU_2(\mathfrak{o}_\ell)$ admits real-valued irreducibles not realizable over $\mathbb{R}$, mirroring the finite-field parallels for $GL_n$ and $GU_n$. Through involution counts and Frobenius–Schur indicators, it connects the reality of characters with the stronger property of orthogonality and highlights a dichotomy between the GL and GU cases that persists at higher congruence levels.

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

Theorems & Definitions (31)

• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Proposition 2.3
• proof
• Lemma 2.4
• Theorem 3.1
• Theorem 3.2
• proof