Timeline for Real forms of complex reductive groups
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19 events
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| Sep 25, 2020 at 14:02 | comment | added | LSpice | (Also, yes to your question; one lets $G$ act transitively on the set of tori, with point stabiliser $N_G(T)$, so, as in general, the parameterisation of rational orbits is $\ker(H^1(\mathbb R, N_G(T)) \to H^1(\mathbb R, G))$. KL are observing that the $N_G(T)$-valued cohomology can be replaced by $W$-valued cohomology, because they are dealing with a field over which connected groups are cohomologically trivial. In fact, I guess that means they don't even need $G$ sc for this purpose ….) | |
| Sep 25, 2020 at 13:57 | comment | added | LSpice | The matching with the Kazhdan–Lusztig statement (Fixed-point varieties in affine flag manifolds) is that their group $G$ is simply connected, so the $G$-valued cohomology is trivial (I think … at least it's true $p$-adically). @MikhailBorovoi's reference: Borovoi and Timashev - Galois cohomology of real semisimple groups via Kac labelings. | |
| Sep 25, 2020 at 0:06 | history | became hot network question | |||
| Sep 24, 2020 at 21:47 | vote | accept | Marc Besson | ||
| Sep 24, 2020 at 20:20 | answer | added | Mikhail Borovoi | timeline score: 7 | |
| Sep 24, 2020 at 18:21 | comment | added | Mikhail Borovoi | In Serre's book "Galois cohomology", read also Section III.1 and Subsection III.4.5. All this (Sections I.5, III.1, and III.4.5) can be read without reading other parts of Serre's book. You cannot read Kazhdan-Lusztig without reading Serre's book... | |
| Sep 24, 2020 at 17:35 | history | edited | Mikhail Borovoi |
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| Sep 24, 2020 at 16:41 | comment | added | Marc Besson | Okay, will do. Thank you very much for your reference and comments. | |
| Sep 24, 2020 at 16:40 | comment | added | Mikhail Borovoi | Read Serre GC I.5, then compute yourself the conjugacy classes in question, and you will see yourself what the correct formula is. | |
| Sep 24, 2020 at 16:35 | comment | added | Marc Besson | How does this match with the statement in Kazhdan-Lusztig "Fixed Point Varieties on Affine Flag Manifolds" , where they say that the set of conjugacy classes of maximal tori in $G(\mathcal{K})$ correspond to $H^1(\Gamma, N(\overline{F}))$? My impression is that they were working with the short exact sequence $1 \rightarrow T \rightarrow N_G(T) \rightarrow W \rightarrow 1$; are you using Galois Cohomology for $1 \rightarrow N_G(T) \rightarrow G \rightarrow G/(N_G(T))$, where we have the scheme parametrizing tori on the right? I'm a little confused | |
| Sep 24, 2020 at 16:32 | comment | added | Mikhail Borovoi | Do read Section 5 of Chapter I in Serre's book "Galois cohomology". | |
| Sep 24, 2020 at 16:30 | comment | added | Mikhail Borovoi | How to use it? You compute both sets, and you compute the map. Thus you get the kernel. In the case $G={\rm SU}_2$ the kernel is trivial, while in the case ${\rm SL}(2,{\Bbb R})$ it is nontrivial. | |
| Sep 24, 2020 at 16:24 | comment | added | Mikhail Borovoi | I would say that they are parametrized (in the case $k={\Bbb R}$) by $${\rm ker}[H^1({\Bbb R},N_G(T))\to H^1({\Bbb R},G)].$$ | |
| Sep 24, 2020 at 16:21 | comment | added | Mikhail Borovoi | You write: "I know that conjugacy classes of tori should be parametrized by $H^1(\operatorname{Gal}(k'/k), N_G(T))$ (at least I think this) but I'm not sure how to use this." This is not quite correct. | |
| Sep 24, 2020 at 16:16 | comment | added | Marc Besson | Dear Mikhail, this looks like the closest reference I could possibly expect. Thank you very much. | |
| Sep 24, 2020 at 16:07 | comment | added | Mikhail Borovoi | Have a look at this preprint. | |
| Sep 24, 2020 at 16:06 | history | edited | YCor | CC BY-SA 4.0 |
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| Sep 24, 2020 at 16:03 | comment | added | Marc Besson | I would also be delighted if anyone felt like there was a good reference (even if it only deals with $\mathbb{C}/\mathbb{R}$) for this material. | |
| Sep 24, 2020 at 16:00 | history | asked | Marc Besson | CC BY-SA 4.0 |