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I am looking for the description of the centralizer $Z_G(X)$ , where $G$ is a real simple Lie Group and $X\in \ Lie (G) $ such that $X^d=0,\ X^{d-1}\neq 0 $. It is is helpful to me any references or any suggestion .

There is a unpublished manuscript by R. Proud, on the title “On centralizers of unipotent elements in algebraic groups”. If some one have a copy of this please post it.

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  • $\begingroup$ It would probably be simplest to contact Ross Lawther and/or Donna Testerman, though Richard Proud did not remain active in mathematical research. (By the way, a UCLA thesis by John Kurtzke in the late 1970s led to a couple of published papers on this theme, but only involving good prime characteristics. My recollection is that there were flaws in his work.) $\endgroup$
    – Jim Humphreys
    Commented Dec 16, 2013 at 21:04
  • $\begingroup$ The first version of your question was very narrow, but this one is much too broad. First, you need to clarify what a "nilpotent" element is in the Lie algebra of a real Lie group. Beyond this, the description of centralzers (in the adjoint group) requires much case-by-case study using linear algebra, etc. This depends on classification of (non-compact) real forms of complex Lie algebras and their classes of nilpotents. $\endgroup$
    – Jim Humphreys
    Commented Dec 17, 2013 at 18:07

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