Spectral equivalence of smooth group schemes over principal ideal local rings

2022/07/12 by Itamar Hadas, Hadas, Itamar
Mathematics · #11E72 #11U07 #20C15 #20G25 #20J06 #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #Finite Group Theory Research #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2207.05830

openalex publication_date 2022/07/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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

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