Representations of ${\mathrm{SL}}_n$ 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 integer $n$ such that $p$ divides $n$, the groups ${\mathrm{SL}}_n({\mathbb{F}}_q[t]/t^2)$ and ${\mathrm{SL}}_n(W_2({\mathbb{F}}_q))$ have the same number of irreducible representations of dimension $d$, for each $d$.

14 pages