0
$\begingroup$

Let $K$ be a semisimple compact Lie group.

In here H.C. Wang defines a C-subgroup as a closed subgroup $U$ of $K$ such that the semisimple part of $U$ equals the semisimple part of the centralizer of a torus of $K$.

At the end of the paper he introduces M-manifolds to be homogeneous spaces $K/U$ where $K$ is a simple compact Lie group and $U$ is a semisimple C-subgroup.

At some point he says that in this case $U$ is the centralizer $Q$ of a torus of $K$. Up to this point I only know that $U$ (semisimple C-subgroup) is the semisimple part of some centralizer $Q$. Why does it have to be all $Q$?

thanks

David

Improve this question
$\endgroup$
9
  • $\begingroup$ The centralizer of a torus contains a maximal torus. But any torus in $G$ (assuming $G$ is semisimple) satisfies your stated criterion, even if it isn't maximal. So I suspect you've misread something. $\endgroup$
    – Allen Knutson
    Commented Feb 7, 2015 at 3:55
  • $\begingroup$ you mean every torus is a C-subgroup? $\endgroup$
    – David P
    Commented Feb 7, 2015 at 16:56
  • $\begingroup$ According to your second paragraph, $U$ is a C-subgroup if $U' = Z(S)'$ for $S$ a torus. If $S$ is a maximal torus, then $Z(S) = S$ so $Z(S)' = S' = 1$. Then if $U$ is any torus, $U' = 1$ too, making it a C-subgroup. $\endgroup$
    – Allen Knutson
    Commented Feb 7, 2015 at 17:40
  • $\begingroup$ Ok, all tori are C-subgroups but there might be others and I'm asking about the semisimple ones. $\endgroup$
    – David P
    Commented Feb 8, 2015 at 8:37
  • 2
    $\begingroup$ $K = Spin(5), S = Spin(2), C_K(S) = Spin(2) \times Spin(3), C_K(S)' = 1 \times Spin(3) = U.$ This may be the point at which you should write the author. $\endgroup$
    – Allen Knutson
    Commented Feb 11, 2015 at 12:50

0

Reset to default

You must log in to answer this question.