840 research outputs found
On commuting varieties of parabolic subalgebras
Let be a connected reductive algebraic group over an algebraically closed
field , and assume that the characteristic of is zero or a pretty good
prime for . Let be a parabolic subgroup of and let be
the Lie algebra of . We consider the commuting variety . Our main
theorem gives a necessary and sufficient condition for irreducibility of
in terms of the modality of the adjoint action of
on the nilpotent variety of . As a consequence, for the case a Borel subgroup of , we give a classification of when is irreducible; this builds on a partial classification given
by Keeton. Further, in cases where is irreducible, we
consider whether is a normal variety. In particular,
this leads to a classification of when is normal.Comment: 19 pages; minor update
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