Alexander Jackson | Searcharxiv

arXiv2025

The Representations of Automorphism Groups of $\mathfrak{o}$-modules of type $(\ell,1^n)$

Let $\mathfrak{o}$ be the valuation ring of a non-Archimedean local field with finite residue field. We give a procedure to find the representation zeta polynomial of $\mathrm{Aut}_\mathfrak{o}(\mathfrak{o}_\ell\oplus\mathfrak{o}_1^{\oplus n})$ by induction on $n$. In particular, we show that the dimensions of the representations are given by evaluating finitely many polynomials at $q=|\mathfrak{o}_1|$.

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arXiv2024

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 $r\geq 1$ 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.

math.RT↗