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 $d$.