Nilpotent commuting varieties of reductive Lie algebras

By A. Premet. 2003

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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 ℋn⊂Hilbn(ℙ2) is irreducible over any algebraically closed field.

Subfields: Geometry and Topology, Mathematical Physics

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