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

Let sigma(1) and sigma(2) be commuting involutions of a connected reductive algebraic group G with g = L i e (G). Let g = circle plus(gij)(i,j=0,1) be the corresponding Z(2) x Z(2)-grading. If {alpha, beta, gamma} = {01; 10; 11}, then T; U maps g(alpha) x g(beta) into g gamma, and the zero fiber of this bracket is called a (sigma) over right arrow- commuting variety. The commuting variety of g 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 G/G(sigma 1) is a Hermitian symmetric space of tube type, then one can find three conjugate pairwise commuting involutions sigma(1), sigma(2), and sigma(3) = sigma(1)sigma(2). In this case, any E-commuting variety is isomorphic to the commuting variety of the simple Jordan algebra associated with sigma(1). As an application, we show that if J is the Jordan algebra of symmetric matrices, then the product map J x J -> J is equidimensional, while for all other simple Jordan algebras equidimensionality fails.