Vol. 7, No. 6, 2013

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

Dmitri I. Panyushev

Vol. 7 (2013), No. 6, 1505–1534
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 ${ℤ}_{2}×{ℤ}_{2}$-grading. If $\left\{\alpha ,\beta ,\gamma \right\}=\left\{01,10,11\right\}$, then $\left[\phantom{\rule{1em}{0ex}}\phantom{\rule{0.3em}{0ex}},\phantom{\rule{1em}{0ex}}\right]$ maps ${\mathfrak{g}}_{@\alpha }×{\mathfrak{g}}_{\beta }$ into ${\mathfrak{g}}_{\gamma }$, and the zero fiber of this bracket is called a $\stackrel{\to }{\sigma }$-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 $\stackrel{\to }{\sigma }$-commuting variety is isomorphic to the commuting variety of the simple Jordan algebra associated with ${\sigma }_{1}$. As an application, we show that if $\mathsc{J}$ is the Jordan algebra of symmetric matrices, then the product map $\mathsc{J}×\mathsc{J}\to \mathsc{J}$ is equidimensional, while for all other simple Jordan algebras equidimensionality fails.

Keywords
semisimple Lie algebra, commuting variety, Cartan subspace, quaternionic decomposition, nilpotent orbit, Jordan algebra
Mathematical Subject Classification 2010
Primary: 14L30
Secondary: 17B08, 17B40, 17C20, 22E46