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 $G={\bf G}\left(\mathcal{O}/\mathfrak{p}^n\right)$ be the adjoint Chevalley group. Let $m_f(G)$ denote the smallest possible dimension of a faithful representation of $G$. Using the Stone-von Neumann theorem, we determine a lower bound for $m_f(G)$ 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