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 $k$?



Cohen-Macaulayness of the scheme of centralizer
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](http://link.springer.com/article/10.1007%2Fs00031-008-9018-9)*. 
Transform. Groups 13 (2008), no. 3-4, 819–837 ([MR](http://www.ams.org/mathscinet-getitem?mr=2452617)).


