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

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