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Let  be the commuting variety of the Lie algebra  of a connected noncommutative reductive algebraic group G over an algebraically closed field of characteristic zero. Let  be the singular locus of  and let  be the locus of points whose G-stabilizers have dimension > rk G. We prove that: (a)  is a nonempty subset of ; (b)  where the maximum is taken over all simple ideals  of  and  is the “lacety” of ; and (c) if  is a Cartan subalgebra of  and  root of  with respect to , then  is an irreducible component of  of codimension 4 in . This yields the bound  and, in particular, . The latter may be regarded as an evidence in favor of the known longstanding conjecture that  is always normal. We also prove that the algebraic variety  is rational.

Irregular and Singular Loci of Commuting Varieties