Submission: 2008, Jan 20
We prove that the singular locus of the commuting variety of a noncommutative reductive Lie algebra is contained in the irregular locus and we compute the codimension of the latter. We prove that one of the irreducible components of the irregular locus has codimension 4. This yields the lower bound of the codimension of the singular locus, in particular, implies that it is at least 2. We also prove that the commuting variety is rational.
2000 Mathematics Subject Classification: 14M99, 14L30, 14R20, 14L24, 17B45
Keywords and Phrases: reductive algebraic group, reductive Lie algebra, commuting variety, irregular element, singular point, semisimple and nilpotent elements
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