Nilpotent commuting varieties of reductive Lie algebras

Let G be a connected reductive algebraic group over an algebraically closed field k of characteristic p0, and =LieG. In positive characteristic, suppose in addition that p is good for G and the derived subgroup of G is simply connected. Let =() denote the nilpotent variety of , and ^nil():={(x,y)×|x,y]=0}, the nilpotent commuting variety of . Our main goal in this paper is to show that the variety ^nil() 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 _n Hilb ⁿ (²) is irreducible over any algebraically closed field. Mathematics Subject Classification (2000) 20G05