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Nilpotent commuting varieties of reductive Lie algebras

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Abstract

Let G be a connected reductive algebraic group over an algebraically closed field k of characteristic p≥0, and 𝔤=Lie G. In positive characteristic, suppose in addition that p is good for G and the derived subgroup of G is simply connected. Let 𝒩=𝒩(𝔤) denote the nilpotent variety of 𝔤, and ℭnil(𝔤):={(x,y)∈𝒩×𝒩 |  [x,y]=0}, the nilpotent commuting variety of 𝔤. Our main goal in this paper is to show that the variety ℭnil(𝔤) is equidimensional. In characteristic 0, this confirms a conjecture of Vladimir Baranovsky; see [2]. When applied to GL(n), our result in conjunction with an observation in [2] shows that the punctual (local) Hilbert scheme ℋ n ⊂Hilbn(ℙ2) is irreducible over any algebraically closed field.

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  1. Department of Mathematics, University of Manchester, Oxford Road, M13 9PL, UK

    Alexander Premet

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20G05

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Premet, A. Nilpotent commuting varieties of reductive Lie algebras. Invent. math. 154, 653–683 (2003). https://doi.org/10.1007/s00222-003-0315-6

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