Representations of reductive groups over finite local rings of length two

TL;DR

This paper establishes an isomorphism between the representation theories of reductive groups over certain finite local rings of length two, showing they have identical counts of irreducible representations of each dimension.

Contribution

It proves that for reductive group schemes over integers with good characteristic, the groups over _q[t]/t^2 and Witt vectors W_2(_q) have equivalent representation theories.

Findings

Same number of irreducible representations of each dimension for both groups.

Existence of an isomorphism between their group algebras.

Results hold under the condition that p is very good for the group.

Abstract

Let $\mathbb{F}_{q}$ be a finite field of characteristic $p$, and let $W_{2}(\mathbb{F}_{q})$ be the ring of Witt vectors of length two over $\mathbb{F}_{q}$. We prove that for any reductive group scheme $\mathbb{G}$ over $\mathbb{Z}$ such that $p$ is very good for $\mathbb{G}\times\mathbb{F}_{q}$, the groups $\mathbb{G}(\mathbb{F}_{q}[t]/t^{2})$ and $\mathbb{G}(W_{2}(\mathbb{F}_{q}))$ have the same number of irreducible representations of dimension $d$, for each $d$. Equivalently, there exists an isomorphism of group algebras $\mathbb{C}[\mathbb{G}(\mathbb{F}_{q}[t]/t^{2})]\cong\mathbb{C}[\mathbb{G}(W_{2}(\mathbb{F}_{q}))]$.