# Centralizers of Nilpotent Elements in Semisimple Lie Algebras

Let $G$ be a connected, simply-connected, complex, semisimple Lie group with Lie algebra $\frak{g}$, and let $\xi\in\frak{g}$ be a nilpotent element. I am interested in understanding the structure of $C_{\frak{g}}(\xi)=\{\eta\in\frak{g}:[\xi,\eta]=$0$\}$, $C_G(\xi)=\{g\in G:Ad_g(\xi)=\xi\}$, and $\pi_0(C_G(\xi))=C_G(\xi)/C_G(\xi)_0$. I would appreciate any references you suspect would give useful structural information. Also, I would welcome any advice and suggestions.

Thanks!

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P.S. Concerning structural information on the centralizers, you can also consult Roger Carter's 1985 book on characters of finite groups of Lie type. There he includes a lot of details about the classes and centralizers in your question over an algebraically closed field. Since there are only finitely many unipotent classes or nilpotent orbits (the same number in good characteristic), his tables provide a clear overview. There is less detail about exceptional types in the book by Collingwood-McGovern on nilpotent orbits, but it provides the full Dynkin-Kostant theory over $\mathbb{C}$. Fine points of structure are also treated extensively in the newer AMS book by Martin Liebeck and Gary Seitz, in arbitrary characteristic (including good and bad primes).