Timeline for Question about adjoint orbits
Current License: CC BY-SA 4.0
11 events
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| Mar 16, 2022 at 15:28 | comment | added | მამუკა ჯიბლაძე | @LSpice In this generality - Vinberg, "Weyl group of a graded Lie algebra" (paywall, sorry). He has detailed analysis of closed orbits, notably the notion of Cartan subspace in an eigenspace of a finite order automorphism, and many other goodies. I don't see all the details though, that's why I am commenting rather than answering. | |
| Mar 16, 2022 at 15:01 | comment | added | LSpice | I agree with @მამუკაჯიბლაძე that this must be a question about $\theta$-groups (I was just coming here to mention Kostant and Rallis - Orbits and representations associated to symmetric spaces), but the papers that I know seem mainly to talk about $K$-orbits on $\mathfrak p$. @მამუკაჯიბლაძე, do you have a reference for your result? (Probably it's in the Kostant–Rallis paper, or in one of Vinberg's; I don't know them well.) | |
| Mar 16, 2022 at 14:49 | comment | added | მამუკა ჯიბლაძე | I believe the following implies what you need: for any finite order automorphism $\theta$ of a semisimple Lie algebra one can find a Cartan subalgebra that meets generically each of the eigenspaces of $\theta$ (this follows from Vinberg's theory of theta-groups) | |
| Mar 16, 2022 at 13:51 | history | edited | Mira | CC BY-SA 4.0 |
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| Mar 16, 2022 at 13:50 | comment | added | Mira | Yes, you are right, I'll fix that. | |
| Mar 16, 2022 at 13:30 | comment | added | LSpice | Yes. Still 'closed' is redundant for an orbit of a compact group, but of course there is no harm in stating it anyway. | |
| Mar 16, 2022 at 1:44 | comment | added | Mira | @LSpice, thanks for your comment! I think your comment is addressed to my question before the edit isn't it ?, I've edited it 1 hour ago because I realize that I forgot to mention some details. | |
| Mar 16, 2022 at 1:24 | comment | added | LSpice | What is the property you want for a compact Lie group $G$? In that context, $\mathfrak k = \mathfrak g$ and $\mathfrak p = \{0\}$, so the decomposition is easy. (Also, in that context, every orbit is the continuous image of $G$, hence compact, hence closed in $\mathfrak g$.) | |
| Mar 15, 2022 at 23:47 | history | edited | Mira | CC BY-SA 4.0 |
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| Mar 15, 2022 at 23:19 | history | edited | Mira | CC BY-SA 4.0 |
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| Mar 15, 2022 at 21:19 | history | asked | Mira | CC BY-SA 4.0 |