Nilpotent commuting varieties of reductive Lie algebras

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) Hubert scheme Hn ⊂ Hilbnn (ℙ 2) is irreducible over any algebraically closed field.