Timeline for Conjugation of root subgroups by the Weyl group
Current License: CC BY-SA 4.0
18 events
| when toggle format | what | by | license | comment | |
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| Sep 11, 2021 at 23:19 | vote | accept | Joshua Ruiter | ||
| Sep 8, 2021 at 21:53 | comment | added | LSpice | Ah, yes. Indeed, notice that, in Lemma 1.3, Deodhar restricts to a non-identity element of the root group. That is the restriction I mentioned in my answer. (Deodhar is taking rational points, and the set of rational points of the open subset of $V_a$ I mentioned in which no $u_\alpha$ equals $0$ is precisely $V_a(k) \setminus \{0\}$.) | |
| Sep 8, 2021 at 21:22 | comment | added | Joshua Ruiter | The Deodhar reference was incorrect, but I have fixed it. One of his papers has the exact same title as one of his thesis; I meant to refer to the thesis. | |
| Sep 8, 2021 at 21:20 | history | edited | Joshua Ruiter | CC BY-SA 4.0 |
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| Sep 8, 2021 at 0:43 | comment | added | LSpice | I have written an answer to try to make things clearer. However, I cannot find Lemma 1.3 in the Deodhar reference. Is that really what you meant? | |
| Sep 7, 2021 at 15:31 | answer | added | LSpice | timeline score: 1 | |
| Sep 7, 2021 at 13:24 | comment | added | Joshua Ruiter | I see. This has been helpful, but I still do not see how this answers my question. You seem to be implying that the answer is obvious (at least for question 1), but it is still unclear to me. | |
| Sep 7, 2021 at 3:09 | comment | added | LSpice | Finally, in the generality in which you are working, you should not be thinking about an exponential map $\exp : \mathfrak g \to G$. There can't be such a thing on the level of schemes, since it should restrict to a group map $\exp : \mathfrak s \to S$, which can only be trivial. The $X_\alpha$ are the best we can do in place of the exponential map. | |
| Sep 7, 2021 at 2:50 | comment | added | LSpice | Also, my point is that $V_\alpha \to G$ is not an embedding of group schemes in general, only of schemes; but that issue only arises for multipliable roots, so it's not a problem here. | |
| Sep 7, 2021 at 2:49 | comment | added | LSpice | Unless they mean something different by the notation from the usual, $w_\alpha(t)$ (or $w_\alpha(u)$, whatever) represents the same Weyl-group element. | |
| Sep 7, 2021 at 2:18 | history | edited | Joshua Ruiter | CC BY-SA 4.0 |
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| Sep 7, 2021 at 2:15 | comment | added | Joshua Ruiter | Can you explain more regarding the interaction between the Weyl group action on absolute roots, and the restriction of absolute roots to relative roots? I see how this might be useful for considering conjugation by $w_\alpha(1)$ as it represents a Weyl group element, but it seems less likely to resolve the question of conjugation by general $w_\alpha(u)$. | |
| Sep 7, 2021 at 2:13 | comment | added | Joshua Ruiter | I am aware that $V_\alpha \to G$ is an embedding of group schemes, though I did not mention it. I have not seen other people phrase it as such, but in all the cases I have studied, $V_\alpha$ is essentially the root space $\mathfrak{g}_\alpha$, and $X_\alpha:V_\alpha \to G$ is just the restriction of the exponential map $\operatorname{exp}:\mathfrak{g} \to G$. | |
| Sep 7, 2021 at 0:56 | comment | added | LSpice | Wait, one more comment: I see now you assume the relative root system is reduced, so multipliable roots are not a problem. So once again I'm not sure where the subtlety is …. | |
| Sep 7, 2021 at 0:51 | comment | added | LSpice | Notice, by the way, that (as far as I can tell—the generality is more than that in which I am used to working) the embedding $V_\alpha \to G$ is an embedding just of schemes, not of groups. (You didn't say otherwise, but I misread at first.) | |
| Sep 7, 2021 at 0:47 | comment | added | LSpice | The relative root groups are generated by the absolute root groups with the right restriction, with a predictable modification for multipliable relative roots. The relative Weyl group acts on absolute roots and preserves the restriction. Are you asking for something more subtle than that? EDIT: Oh, I see that your $V_\alpha$ is not the root group. So your worry is precisely about multipliable relative roots, right? | |
| S Sep 6, 2021 at 19:45 | review | First questions | |||
| Sep 6, 2021 at 19:54 | |||||
| S Sep 6, 2021 at 19:45 | history | asked | Joshua Ruiter | CC BY-SA 4.0 |