Representations of reductive groups over finite local rings of length two (1804.05043v2)

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 $\mathbb{C}[\mathbb{G}(\mathbb{F}{q}[t]/t^{2})]\cong\mathbb{C}[\mathbb{G}(W_{2}(\mathbb{F}_{q}))]$.