Spectral equivalence of smooth group schemes over principal ideal local rings

TL;DR

This paper demonstrates spectral equivalence between group algebras of smooth group schemes over certain local rings and establishes uniform bounds on the dimensions of irreducible representations over finite fields.

Contribution

It proves the isomorphism of group algebras for group schemes over Witt vector rings and finite fields, revealing spectral equivalences and uniform bounds in representation theory.

Findings

Group algebras of $ ext{G}( ext{F}[t]/(t^k))$ and $ ext{G}(W_k( ext{F}))$ are isomorphic for large enough characteristic.

The set of irreducible representation dimensions over finite fields is uniformly bounded.

Spectral equivalence holds for smooth linear group schemes over principal ideal local rings.

Abstract

Let $\mathcal{G}$ be a smooth linear group scheme of finite type. For any positive integer $k$ and a finite field $\mathbb{F}$, let $W_k(\mathbb{F})$ be the ring of Witt vectors of length $k$ over $\mathbb{F}$. We show that the group algebras of $\mathcal{G}(\mathbb{F}[t]/(t^k))$ and $\mathcal{G}(W_k(\mathbb{F}))$ are isomorphic (i.e. the multi-sets of the dimensions of the irreducible representations are equal) for any positive integer $k$ and finite field $\mathbb{F}$ with large enough characteristic. We also prove that if $\mathrm{char}\mathbb{F}$ is large enough, then the cardinality of the set $\{\dim\rho\big|\rho\in \mathrm{irr}(\mathcal{G}(\mathbb{F}))\}$ is bounded uniformly in $\mathbb{F}$.