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

Abstract

Let ${\sigma }_1$ and ${\sigma }_2$ be commuting involutions of a connected reductive algebraic group $G$ with $\mathfrak{g}=Lie\left(G\right)$. Let

$$\mathfrak{g}=\underset{i,j=0,1}{\oplus }{\mathfrak{g}}_{@ij}$$

be the corresponding ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$-grading. If $\left\{\alpha ,\beta ,\gamma \right\}=\left\{01,10,11\right\}$, then $\left[,\right]$ maps ${\mathfrak{g}}_{@\alpha }\times {\mathfrak{g}}_{\beta }$ into ${\mathfrak{g}}_{\gamma }$, and the zero fiber of this bracket is called a $\overrightarrow{\sigma }$-commuting variety. The commuting variety of $\mathfrak{g}$ 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 $\overrightarrow{\sigma }$-commuting variety is isomorphic to the commuting variety of the simple Jordan algebra associated with ${\sigma }_1$. As an application, we show that if $\mathcal{J}$ is the Jordan algebra of symmetric matrices, then the product map $\mathcal{J}\times \mathcal{J}\to \mathcal{J}$ is equidimensional, while for all other simple Jordan algebras equidimensionality fails.