Representations of reductive groups over finite local rings of length two

2018/04/13 by Alexander Stasinski, Stasinski, Alexander, Andrea Vera-Gajardo +1
Mathematics · #FOS: Mathematics #Group Theory (math.GR) #Representation Theory (math.RT) #math.GR #math.RT

paper · pdf · doi:10.48550/arxiv.1804.05043

16 pages, typos fixed, final version

arxiv created 2019/02/19 · arxiv updated 2019/02/20

Abstract

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

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