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

Abstract

Let ${\mathcal{O}}_2$ and ${\mathcal{O}}_2^{\prime }$ be two distinct finite local rings of length two with residue field of characteristic $p$. Let $\mathbb{G}({\mathcal{O}}_2)$ and $\mathbb{G}({\mathcal{O}}_2^{\prime })$, 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 $K[\mathbb{G}({\mathcal{O}}_2)]\cong K[\mathbb{G}({\mathcal{O}}_2^{\prime })]$, where $K$ is a sufficiently large field of characteristic different from $p$.

9 pages, all comments are welcome!

References in corpus (1)

• On Representations of General Linear Groups over Principal Ideal Local Rings of Length Two