Faithful representations of Chevalley groups over quotient rings of non-Archimedean local fields

Abstract

Let $F$ be a non-Archimedean local field with the ring of integers $\mathcal{O}$ and the prime ideal $\mathfrak{p}$ and let $G={\mathbf{G}}_{\mathrm{ad}}(\mathcal{O}/{\mathfrak{p}}^n)$ be the adjoint Chevalley group. Let $m_{\mathsf{f}}(G)$ denote the smallest possible dimension of a faithful representation of $G$. Using the Stone–von Neumann theorem, we determine a lower bound for $m_{\mathsf{f}}(G)$ which is asymptotically the same as the results of Landazuri, Seitz and Zalesskii for split Chevalley groups over ${\mathbb{F}}_q$. Our result yields a conceptual explanation of the exponents that appear in the aforementioned results.