Representations of reductive groups over finite local rings of length two

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 $\mathbb{C}[\mathbb{G}({\mathbb{F}}_q[t]/t^2)]\cong \mathbb{C}[\mathbb{G}(W_2({\mathbb{F}}_q))]$.

16 pages, typos fixed, final version