Let $G$ be a simply connected group over an algebraically closed field $k$, and
$I:=\{(g,\gamma)\in G\times G\vert~ g\gamma=\gamma g\}$
the scheme of centralizer.
Is $I$ a Cohen-Macaulay scheme over $k$?
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The answer is not known. In fact, the analogous result for the "commuting variety" $\mathcal{C}(\mathfrak{g})$ of a reductive Lie algebra $\mathfrak{g}$ ($\mathcal{C}(\mathfrak{g})=\{(x,y)\in \mathfrak{g}\times \mathfrak{g}\ |\ [x,y]=0\}$) is a classical conjecture -- even the normality of $\mathcal{C}(\mathfrak{g})$ is not known. See V. Popov, Irregular and singular loci of commuting varieties. Transform. Groups 13 (2008), no. 3-4, 819–837 (MR).
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$\begingroup$ A fairly recent arXiv paper of Charbonnel claims to prove Cohen-Macaulayness for ${\mathcal C}({\mathfrak g})$ - see Thm. 4.3(iii) in arxiv.org/abs/1206.5592. I guess Charbonnel's paper must be under review. $\endgroup$ Commented Nov 18, 2013 at 21:56
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$\begingroup$ @PaulLevy, there is a recent J. Algebra paper with a similar title. However, unlike the arXiv paper you linked, it has a co-author, and at least the introduction is completely different, so who knows what happened. $\endgroup$– LSpiceCommented Jul 11, 2016 at 20:09
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1$\begingroup$ I think that is not directly related - it is arxiv.org/abs/1204.0377, which appeared before 1206.5592. $\endgroup$ Commented Jul 13, 2016 at 15:40