Representations of reductive groups over finite local rings of length two

Abstract

Let Fq be a finite field of characteristic p, and let W2(Fq) be the ring of Witt vectors of length two over Fq. We prove that for any reductive group scheme G over Z such that p is very good for G×Fq, the groups G(Fq[t]/t2) and G(W2(Fq)) have the same number of irreducible representations of dimension d, for each d. Equivalently, there exists an isomorphism of group algebras C[G(Fq[t]/t2)]≅C[G(W2(Fq))].