Commuting involutions of Lie algebras, commuting varieties, and simple Jordan algebras

Abstract: We study certain "\sigma-commuting varieties" associated with a pair of commuting involutions of a semisimple Lie algebra $\g$. The usual commuting variety of $\g$ and commuting varieties related to one involution are particular cases of our construction. We develop a general theory of \sigma-commuting varieties and point out some cases, when they have especially good properties. We show that, for a special choice of commuting involutions, the \sigma-commuting variety is isomorphic to the commuting variety of a simple Jordan algebra. As a by-product of our theory, we show that if $J$ is the Jordan algebra of symmetric matrices, then the product map $J \times J\to J$ is equidimensional; while for all other simple Jordan algebras equidimensionality fails.