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Let G be a finite group of Lie type. How does one describe the semisimple
conjugacy classes of G and how does one describe the centralizer of a
semisimple element ?

The answer to this is given in Carter's book Finite Groups of Lie Type
in Chapter III. However, in practice the computation of the centralizers
seems unnecessarily complicated since it depends on an examination of the
entire root system for G.

If G is GL(n), one can first diagonalize a semisimple element over the
algebraic closure, then group the identical eigenvalues together on
the diagonal and then easily compute the centralizer. In effect, one
is choosing the element first and the ordering of the roots next. This
seems easier than obtaining a collection of roots whose root subgroups
a priori generate the centralizer along with the torus and certain
Weyl group elements and trying to recognize the group they generate.

For example, with GL(n), if one removes those vertices of the
Dynkin diagram that are not trivial on the semisimple element
(assuing the roots were ordered as above), what's left is the
Dynkin diagram of the centralizer.

To what extent can this be done in general ?

Allan Adler
ara@altdorf.ai.mit.edu