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

TL;DR

This paper proves that the dimensions of irreducible representations of certain smooth affine group schemes over local rings are given by finitely many polynomials in the size of the residue field, for large enough residue characteristic.

Contribution

It establishes a polynomial formula for the dimensions of irreducible representations of these group schemes over principal ideal local rings, extending prior understanding.

Findings

Dimensions are given by finitely many polynomials in q.

Results hold for large fixed residue characteristic p.

Provides a uniform polynomial description across representations.

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.