Timeline for When are all centralizers in a Lie group connected?
Current License: CC BY-SA 3.0
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| Apr 5, 2012 at 8:47 | comment | added | Jesper Grodal | @Mark: Oops!! Sorry, what I said originally was of course not true. I've modified the statement. If you want for all closed $H$, I believe $G = U(n)$ is essentially the only example. I will try to write a more detailed post later, with a precise statement. One easy way to see the $U(n)$ case is to note that the centralizer, essentially by Schur's lemma, has to again be a product of $U(k)$'s where $k$'s are the multiplicities of the reps of $H$ on ${\mathbb C}^n$. However for $Sp(n)$ reps can be either of real, complex or quaterionic type, which can produce non-conn. centralisers... | |
| Apr 5, 2012 at 8:24 | history | edited | Jesper Grodal | CC BY-SA 3.0 |
Changed to more restricted statement.
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| Mar 30, 2012 at 12:57 | comment | added | Mark Grant | Thanks for your answer. I am wondering about $SU(n)$ though. I thought the centre (ie the centralizer of the whole group) was $\mathbb{Z}/n$ in this case? | |
| Mar 26, 2012 at 12:52 | history | answered | Jesper Grodal | CC BY-SA 3.0 |