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Let $G$ be a compact Lie group.

Is each conjugacy class a closed subset of $G$? Define the conjugacy equivalent relation $g\sim h$ if $g$ is conjugate to $h$.Is $G/\sim$ a Haussdoef space with the quotient topology?Is a there a natural manifold structure on it? If the answer is "yes", is there a natural Riemannian meyric on it such that the the quotient map would be a partial isometry?(After we fix a left invariant metric on $G$)

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  • $\begingroup$ The fact that conjugacy classes are closed is clear from definition-chasing and the fact that images of compact spaces in Hausdorff spaces are closed. $\endgroup$
    – user44191
    Commented Apr 20, 2020 at 18:08
  • $\begingroup$ @user44191 yes thanjs.you are right a compactness argumment works. But let $G$ be a connected open lie group which is very far from being abelian for example it does not have a normal subgroup. Does it imopy that the neutral element is an accumulation point for some conjugacy class? The matrix case $\begin{pmatrix} 1&\epsilon\\0&1\end{pmatrix}$ is a motivation.Hoever the matrix group is not simple. $\endgroup$
    – Ali Taghavi
    Commented Apr 20, 2020 at 20:23
  • $\begingroup$ @user44191 I mean that what are some non trivial(non abelian, etc) examples of open connected Lie groups whose all conjugacy classes are closed. $\endgroup$
    – Ali Taghavi
    Commented Apr 20, 2020 at 20:27

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After you choose a maximal torus $T$, the space of conjugacy classes is identified with with the quotient $T / W(T)$ of $T$ by the Weyl group, see https://en.wikipedia.org/wiki/Maximal_torus#Weyl_group. The quotient is not a manifold in general (if $G$ is simply connected, then it can be identified with a Weyl chamber).

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  • $\begingroup$ Thanks for your answer. What can be said about the "metric" part of my question?What about the clisedness if conjugacy classes(in Lie group case or even in compact topiligical group case)? $\endgroup$
    – Ali Taghavi
    Commented Apr 20, 2020 at 18:04
  • $\begingroup$ In the non compact case, what kind of accumulation points can one imagine for conjugacy classes? for example in general linear group? $\endgroup$
    – Ali Taghavi
    Commented Apr 20, 2020 at 23:34

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