  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.
