The $\ell$-modular representation of reductive groups over finite local rings of length two

TL;DR

This paper proves that for reductive groups over certain finite local rings of length two, their group algebras are isomorphic over sufficiently large fields, revealing a form of structural invariance.

Contribution

It establishes an isomorphism between the group algebras of reductive groups over distinct finite local rings of length two, extending understanding of modular representations.

Findings

Group algebras of reductive groups over different finite local rings are isomorphic.

The isomorphism holds over sufficiently large fields of characteristic not equal to p.

Results apply to rings with residue fields of characteristic p and p very good for the group.

Abstract

Let $\mathcal{O}_2$ and $\mathcal{O}'_2$ be two distinct finite local rings of length two with residue field of characteristic $p$. Let $\mathbb{G}(\mathcal{O}_2)$ and $\mathbb{G}(\mathcal{O}'_2)$, be the group of points of any reductive group scheme $\mathbb{G}$ over $\mathbb{Z}$ such that $p$ is very good for $\mathbb{G} \times \mathbb{F}_q$. We prove that there exists an isomorphism of group algebra $K[\mathbb{G}(\mathcal{O}_2)] \cong K[\mathbb{G}(\mathcal{O}'_2)]$, where $K$ is a sufficiently large field of characteristic different from $p$.