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I will be happy for any help.

I am interested in any reference or answer about the following question. We know that any automorphism of simple Lie type group is direct of inner, diagonal, field or graph automorphism. In general I like to know what is the structure of $C_{Aut(S)}(\sigma)$ where $S$ is simple lie type and $\sigma$ is an automorphism. "This is my major question" Maybe your answer is the following: Case by case is different. There if I be clear, is it possible to discuss about the centralizer of a diagonal or graph automprphism?

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  • $\begingroup$ You say "Lie group" in the title and "Lie type group" in the question, the latter usually refers to finite groups. Could you clarify? $\endgroup$
    – YCor
    Commented Jan 10, 2014 at 14:56
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    $\begingroup$ The 'inner', 'diagonal', 'field' etc suggest this is a question about groups of Lie type. I would suggest you go to vol 3 of Gorenstein, Lyons and Solomon's rewriting of the classification, who classify centralizers of all involutions. I don't know of any source that classifies every centralizer explicitly, but Carter's books might help. $\endgroup$
    – Nick Gill
    Commented Jan 10, 2014 at 14:59
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    $\begingroup$ As Nick points out, there is detailed case-by-case information in the literature (such as GLS) on outer automorphisms. For a simple group of Lie type, the group itself forms a large normal subgroup of the full automorphism group, where your question about centralizers is posed. I suspect there is no unified way to answer this. $\endgroup$
    – Jim Humphreys
    Commented Jan 10, 2014 at 21:37
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    $\begingroup$ P.S. Centralizers of inner automorphisms are also studied to some extent in case-by-case fashion, while using some general theory. See for instance the recent Liebeck-Seitz book and references there. $\endgroup$
    – Jim Humphreys
    Commented Jan 11, 2014 at 0:43

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