Timeline for Topological properties of $K$ orbits in $G/B$
Current License: CC BY-SA 3.0
20 events
| when toggle format | what | by | license | comment | |
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| Jan 7, 2015 at 2:22 | answer | added | Jeffrey Adams | timeline score: 0 | |
| Oct 20, 2013 at 19:27 | vote | accept | Reladenine Vakalwe | ||
| Oct 10, 2013 at 21:06 | answer | added | Faisal | timeline score: 3 | |
| Oct 7, 2013 at 0:14 | comment | added | Dave Anderson | It seems the standard reference for much of what you're asking is R. Steinberg, Endomorphisms of linear algebraic groups, in Mem AMS (1968). I'd also suggest looking in papers of Brion from the 1990's for more facts and references. | |
| Oct 6, 2013 at 16:42 | comment | added | YCor | Still, I'd like a "typical example" of $(G,\sigma)$. | |
| Oct 6, 2013 at 15:39 | comment | added | Reladenine Vakalwe | @YvesCornulier: apart from the reason I have added above to use the notation $K$, there is another one. This $\theta$ is associated to an antiholomorphic involution of the Langlands dual. Here by antiholomorphic, I mean in the sense that it differs from a morphism over $Spec(\mathbb{C})$ by complex conjugation. In this vein a holomorphic involution would be just what I called an involution. I do not know if this is standard terminology, so I avoided it. | |
| Oct 6, 2013 at 15:32 | history | edited | Reladenine Vakalwe | CC BY-SA 3.0 |
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| Oct 6, 2013 at 15:24 | history | edited | Reladenine Vakalwe | CC BY-SA 3.0 |
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| Oct 6, 2013 at 15:00 | comment | added | YCor | @Reladenine: OK. Actually that's surprising because $K$ is a typical notation for a maximal compact subgroup, which can indeed can be obtained as set of fixed points of a antiholomorphic involution (but not a holomorphic one). Could you provide a "typical" example of $(G,\sigma)$ you have in mind? | |
| Oct 6, 2013 at 14:55 | answer | added | Max Horn | timeline score: 1 | |
| Oct 6, 2013 at 13:33 | comment | added | Reladenine Vakalwe | @YvesCornulier: I have clarified in the question. Sorry for causing any confusion. I had assumed that since I was working with algebraic groups over $\mathbb{C}$, involution would automatically be understood to be in this category. | |
| Oct 6, 2013 at 13:27 | history | edited | Reladenine Vakalwe | CC BY-SA 3.0 |
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| Oct 6, 2013 at 12:56 | history | edited | Reladenine Vakalwe | CC BY-SA 3.0 |
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| Oct 6, 2013 at 12:49 | history | edited | Reladenine Vakalwe | CC BY-SA 3.0 |
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| Oct 6, 2013 at 7:31 | comment | added | YCor | But well, what is assumed on $\theta$? continuous/holomorphic/antiholomorphic involutive automorphism? | |
| Oct 6, 2013 at 2:36 | comment | added | Zhaoting Wei | I'm not sure what do you mean by "$K$-orbits" here. If $G$ is complex semisimple and $K$ is the maximal compact subgroup, it is well-known that the $K$-action on $G/B$ is also transitive, hence the $K$-orbit is $G/B$ itself. Maybe what you really mean is the isotropy group of the $K$-action. | |
| Oct 6, 2013 at 2:33 | answer | added | Zhaoting Wei | timeline score: 0 | |
| Oct 6, 2013 at 0:36 | comment | added | Reladenine Vakalwe | @YvesCornulier: No additional hypothesis on $\theta$. But I am happy even with partial answers with additional hypothesis. My understanding is rather minimal at the moment, so anything would help! | |
| Oct 6, 2013 at 0:33 | comment | added | YCor | Do you assume some special hypothesis on the involution $\theta$ implying that $K$ is a maximal compact subgroup? | |
| Oct 6, 2013 at 0:15 | history | asked | Reladenine Vakalwe | CC BY-SA 3.0 |