Spectral equivalence of smooth group schemes over principal ideal local rings

Abstract

Let G be a smooth linear group scheme of finite type. For any positive integer k and a finite field F, let Wk(F) be the ring of Witt vectors of length k over F. We show that the group algebras of G(F[t]/(tk)) and G(Wk(F)) are isomorphic (i.e. the multi-sets of the dimensions of the irreducible representations are equal) for any positive integer k and finite field F with large enough characteristic. We also prove that if charF is large enough, then the cardinality of the set {dim⁡ρ|ρ∈irr(G(F))} is bounded uniformly in F.