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On the Irreducibility of Commuting Varieties Associated with Involutions of Simple Lie Algebras

Authors:	D I Panyushev

Institution:	(1) Independent University of Moscow, Russia

Abstract:	Let $\mathfrak{g}$ be a reductive Lie algebra over an algebraically closed field of characteristic zero and $\mathfrak{g} = \mathfrak{g}_0\oplus \mathfrak{g}_1$ an arbitrary $\mathbb{Z}_2$ -grading. We consider the variety $\mathfrak{C}_1= \{ (x,y)\|x,y] = 0\}\subset \mathfrak{g}_1\times \mathfrak{g}_1$ , which is called the commuting variety associated with the $\mathbb{Z}_2$ -grading. Earlier it was proved by the author that $\mathfrak{C}_1$ is irreducible, if the $\mathbb{Z}_2$ -grading is of maximal rank. Now we show that $\mathfrak{C}_1$ is irreducible for $(\mathfrak{g},\mathfrak{g}_0 ) = (\mathfrak{s}\mathfrak{l}_{2n} ,\mathfrak{s}\mathfrak{p}_{2n} )$ and (E₆,F₄). In the case of symmetric pairs of rank one, we show that the number of irreducible components of $\mathfrak{C}_1$ is equal to that of nonzero non--regular nilpotent G ₀-orbits in $\mathfrak{g}_1$ . We also discuss a general problem of the irreducibility of commuting varieties.

Keywords:	semisimple Lie algebra ₂-grading" target="_blank">gif" alt="Zopf" align="BASELINE" BORDER="0">₂-grading commuting variety
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