A proof of the first Kac-Weisfeiler conjecture in large characteristics

2018/10/30 by Martin, Benjamin, Stewart, David, Tikaradze, Akaki +1
#03C60 #17B35 #FOS: Mathematics #Primary: 17B50 #Representation Theory (math.RT) #Rings and Algebras (math.RA) #Secondary: 17B10

paper · doi:10.48550/arxiv.1810.12632

Abstract

In 1971, Kac and Weisfeiler made two influential conjectures describing the dimensions of simple modules of a restricted Lie algebra \mathfrakg. The first predicts the maximal dimension of simple \mathfrakg-modules and in this paper we apply the Lefschetz principle and classical techniques from Lie theory to prove this conjecture for all restricted Lie subalgebras of \mathfrakgl_n(k) whenever k is an algebraically closed field of characteristic p ≫ n. As a consequence we deduce that the conjecture holds for the the Lie algebra of a group scheme when specialised to an algebraically closed field of almost any characteristic. In the appendix to this paper, written by Akaki Tikaradze, a short proof of the first Kac--Weisfeiler conjecture is given for the Lie algebra of group scheme over a finitely generated ring R ⊆ ℂ, after base change to a field of large positive characteristic.

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