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It may be useful to expand Steven Sam's partial answer. This isn't really an answer to the original question, but is too long for a comment. The discussion in Chriss-Ginzburg 2.4, which relies on ma …

Borel's lengthy 1953 Annals paper is essentially his 1952 Paris thesis. It was followed by work of Bott, Samelson, Kostant, and others, which eventually answers your side question affirmatively. …

I'm not sure whether the questions here concern just compact connected Lie groups, but in general the connectedness of centralizers involves some subtleties and may not be dealt with definitively in b …

I can't answer this completely, but I can point out some important follow-ups in the literature by Borel and others which need to be be taken into account. Borel's Tohoku paper is reprinted in the s …

Besides Samelson's short 1946 research note linked by Mrc Plm, it's also useful to mention his longer 1952 survey on topology of Lie groups here (see Section 10 and references for various proofs of …

Jesper Grodal's reference to Bourbaki is a reasonable one for these questions, including 1). There are also two volumes in the Springer GTM series which treat many aspects of compact Lie groups, inc …

I think the standard early reference is the paper by Hotta and Springer here, in which they work with $\ell$-adic cohomology (but the results seem to carry over to other settings). What they show is …

The question is stated a bit loosely, but the basic literature goes back about four decades to work of Brieskorn and Deligne. Since I'm not an expert on these matters I can only refer to the basic …

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 …