A Polynomial Result for Dimensions of Irreducible Representations of Smooth Affine Group Schemes Over Principal Ideal Local Rings

Abstract

Denote by $\mathfrak{o}$ the valuation ring of a non-Archimedean local field with prime ideal $\mathfrak{p}$ and finite residue field, and let $r\geq 1$ be an integer. We prove that for every smooth affine group scheme $G$ over $\mathbb{Z}$, the dimension of each irreducible representation of $G(\mathfrak{o}/\mathfrak{p}^r)$ is given by one of finitely many polynomials with coefficients in $\mathbb{Q}$ evaluated at $q=|\mathfrak{o}/\mathfrak{p}|$, provided that the residue characteristic $p=\mathrm{char} \mathfrak{o}/\mathfrak{p}$ is large and fixed.