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I want to understand centralizers of semisimple elements in unitary groups.

Let us begin with example of $GL_n(k)$. Centralizers of semisimple elememts are a product of smaller $GL_m(k)$ thus connected. Steinberg proved that for a connected, semisimple, simply connected algebraic group centralizers of semisimple elements are connected. However this need not be true in general. For example $-1$ in orthogonal group.

I believe this is true that for unitary groups centralizers of semisimple elements are connected. I would also like to determine structure. Any help or reference will be helpful.

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    $\begingroup$ What do you mean by unitary group? Connectedness of centralizers has more to do with the fundamental group of the algebraic group than anything else. Just consider the centralizer in $\textbf{PGL}_2$ of the image of the diagonal matrix with entries $+1$ and $-1$. Are your unitary groups simply connected? $\endgroup$
    – Jason Starr
    Commented Feb 29, 2016 at 13:31
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    $\begingroup$ Your statement about centralizers of semisimple elements in GL$_n(k)$ is only for diagonalizable elements; with $k$ not algebraically closed, one cannot really describe them in a uniform manner. Of course, the notion of "unitary group" involves a quadratic Galois extension $k'$of the ground field $k$, so for that you can't assume $k$ to be algebraically closed. Since SU$(h)$ for hermitian spaces $(V',h)$ of dimension $n \ge 2$ over $k'$ are Galois-twisted forms of SL$_n$, they're also simply connected. Hence, Steinberg's theorem gives connectedness for U$(h)$ too. $\endgroup$
    – nfdc23
    Commented Mar 1, 2016 at 1:43
  • $\begingroup$ @nfdc23. This is what confused me: Is the OP asking about geometric connectedness of the scheme-theoretic centralizer (which should follow from Steinberg's theorem, after base change to an algebraic closure), or is the OP asking about connectedness for some other topology (p-adic topology, real Euclidean topology)? $\endgroup$
    – Jason Starr
    Commented Mar 1, 2016 at 11:54

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