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Let $G$ be a connected, simply-connected, complex, semisimple Lie group with Lie algebra $\frak{g}$. Suppose that $X\in\frak{g}$ is a nilpotent element (ie. that $ad_X:\frak{g}\rightarrow\frak{g}$ is a nilpotent endomorphism), and let $C_G(X)\subseteq G$ denote its stabilizer with respect to the adjoint representation of $G$. What is known about the topology of $C_G(X)$ (ex. topological invariants)? I would appreciate any and all references/suggestions, particularly those concerning the stabilizers of irregular nilpotent elements of $\frak{g}$.

Thanks!

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2 Answers 2

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To supplement what Francois Ziegler says, I'd point out that the structure of semisimple complex Lie groups has been developed piecemeal over a century or so. The basic results on nilpotent elements in the Lie algebra are by now fairly old, but refinements continue to be made. The "topology" involved in the group centralizers seems to be limited mainly to the study of the component groups: groups of connected components in the centralizer group of a given nilpotent element. These groups have a topological interpretation as fundamental groups, summarized in Chapter 6 of the cited book by Collingwood-McGovern. (Unfortunately, that slim 1993 book has become unaffordably expensive and doesn't exist in all math libraries.)

It should be kept in mind that what Collingwood-McGovern do is mostly not original, so there are older sources in the literature. Moreover, most of the ideas here generalize nicely to semisimple algebraic groups over an algebraically closed field of arbitrary "good" characteristic (sometimes avoiding 2, 3, 5). Here the basic results on centralizers and component groups, though not so explicitly topological, are much the same. Detailed accounts and tables are found in the 1985 book on finite simple groups by Roger Carter, while a recent AMS monograph by Liebeck and Seitz goes into considerable detail about the structure of centralizers in all characteristics.

The older ideas involving nilpotent orbits or unipotent classes have continued to be explored in research papers, often in connection with representation theory and algebraic geometry. See for example the rethinking of component groups for unipotent elements (in good characteristic) by McNinch and Sommers here.

However you approach the nilpotent elements in the Lie algebra, case-by-case work is inevitable: the centralizers aren't as a rule reductive, but on the other hand the number of orbits is always finite. Subregular orbits have been especially well studied, in connection with Dynkin curves and the like in algebraic geometry.

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This is an excellent answer. Thank you! –  Peter Crooks Feb 21 '13 at 15:06

The component groups $\pi_0(C_G(X))$ are known by work of Alekseevskiĭ, MR557505 = MR2140712, which contains additional references and can be read on Google books. Another good place to look is the book by Collingwood & McGovern, Nilpotent orbits in semisimple Lie algebras.

(Out of curiosity, what other topological invariants are you after?)

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The topological invariants of $C_G(X)$ are a means to obtain topological invariants of the nilpotent adjoint orbit of $X$, such as the integral cohomology algebra. I really appreciate your answer! –  Peter Crooks Feb 21 '13 at 15:06

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