Algebra Number Theory 7 (6), 1505-1534, (2013) DOI: 10.2140/ant.2013.7.1505
KEYWORDS: semisimple Lie algebra, commuting variety, Cartan subspace, quaternionic decomposition, nilpotent orbit, Jordan algebra, 14L30, 17B08, 17B40, 17C20, 22E46
Let and be commuting involutions of a connected reductive algebraic group with . Let
be the corresponding -grading. If , then maps into , and the zero fiber of this bracket is called a -commuting variety. The commuting variety of and commuting varieties related to one involution are particular cases of this construction. We develop a general theory of such varieties and point out some cases, when they have especially good properties. If is a Hermitian symmetric space of tube type, then one can find three conjugate pairwise commuting involutions , , and . In this case, any -commuting variety is isomorphic to the commuting variety of the simple Jordan algebra associated with . As an application, we show that if is the Jordan algebra of symmetric matrices, then the product map is equidimensional, while for all other simple Jordan algebras equidimensionality fails.