Timeline for Centralizers of regular elements are abelian

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Oct 31, 2016 at 22:07	comment	added	Will Sawin		@YCor Ah, good point.
Oct 31, 2016 at 6:39	comment	added	YCor		To summarize erased comments: the 1st sentence of the post is true, but is not the point: this is the centralizer in $PGL_2$ of the diagonal matrix $(1,-1)$ viewed as element of $PGL_2$. But one has to consider instead, the centralizer in $PGL_2$ of this matrix viewed as an element of the Lie algebra. This is the same as the image in $PGL_2$ of its genuine centralizer in $GL_2$, so is reduced to diagonal matrices. And thus this does not yield a counterexample.
Oct 31, 2016 at 5:54	comment	added	YCor		Actually, given an element of $\mathfrak{sl}_n$, its centralizer in the whole algebras of matrices is a subalgebra, so its centralizer in $GL_n$ is an Zariski open subset of this algebra and in particular is Zariski connected, and in turn the image of the latter in $PGL_n$ is Zariski connected. So the result is true in $PGL_n$ (and connectedness of the centralizer of any Lie algebra element does not make use of regularity).
Oct 31, 2016 at 5:28	comment	added	Francois Ziegler		No, a matrix $\left(\begin{smallmatrix}0&\\&0\end{smallmatrix}\right)$ won't centralize $\left(\begin{smallmatrix}1&0\\0&-1\end{smallmatrix}\right)$ unless it is zero.
Oct 30, 2016 at 17:09	comment	added	SHP		In the paper they consider $G$ simply-connected, which I forgot to add to the question.
Oct 30, 2016 at 17:02	history	answered	Will Sawin	CC BY-SA 3.0