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 $$C_{\mathfrak{g}}(\xi)=\{\eta\in\mathfrak{g}:[\xi,\eta]=0\}, \quad C_G(\xi)=\{g\in G:\mathrm{Ad}_\mathfrak{g}(\xi)=\xi\},$$ 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!
Best Answer
This determination of component groups goes back to Elashvili and Alexeevskii, but has been improved somewhat in a 1998 IMRN paper by Eric Sommers and a later joint paper by him and George McNinch here. Your set-up is essentially equivalent to studying the same problem for a semisimple algebraic group and its Lie algebra in arbitrary characteristic, but good characteristic (including 0) is essential for getting uniform results.
In particular, the situation for nilpotent elements of the Lie algebra and unipotent elements of the group is essentially the same, by Springer's equivariant isomorphism between the two settings The classes/orbits and centralizers correspond nicely in good characteristic.
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).