Timeline for Split rank of inner forms
Current License: CC BY-SA 3.0
15 events
| when toggle format | what | by | license | comment | |
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| Jan 30, 2018 at 12:30 | answer | added | thierry stulemeijer | timeline score: 3 | |
| Jan 27, 2018 at 22:13 | comment | added | Cheng-Chiang Tsai | By the way, I typed the split rank of the quasi-split unitary group wrong - my apology for any confusion. | |
| Jan 27, 2018 at 22:12 | history | edited | Cheng-Chiang Tsai | CC BY-SA 3.0 |
A mistake corrected.
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| Jan 26, 2018 at 22:44 | comment | added | LSpice | @JimHumphreys, I guess it's clear now, but $G$ and $G^*$ are groups over $F$, with $G$ arbitrary and $G^*$ quasisplit (over $F$), such that their base changes $G_{\overline F}$ and $G^*_{\overline F}$ to the separable closure $\overline F$ of $F$ are isomorphic (over $\overline F$) (plus a further condition to make the forms inner). Cheng-Chiang's $F$ and $G^*$ are $k$ and $\overline G$, in the notation of Section 6.5 of Borel–Tits (and my $\overline F$ is their $K$). | |
| Jan 26, 2018 at 22:03 | comment | added | Jim Humphreys | @LSpice: I got a little confused by the formulation, being unsure about how to understand notation like $G$ and $G^*$, or a statement about "isomorphism" of these groups without reference to a field. | |
| Jan 26, 2018 at 4:13 | vote | accept | Cheng-Chiang Tsai | ||
| Jan 26, 2018 at 3:12 | comment | added | LSpice | @JimHumphreys, since the question is about split ranks, surely considering an already $F$-split group $G$, where $G = G^*$, does not shed any additional light? | |
| Jan 26, 2018 at 3:10 | answer | added | LSpice | timeline score: 14 | |
| Jan 26, 2018 at 1:56 | comment | added | thierry stulemeijer | At least one can check that this is true for absolutely simple groups, by just going through and comparing the table of indices in Tits article "Classification of algebraic semisimple groups" and in Springer book "Linear Algebraic groups" (by noting that when Springer says the index appears for outer type, than it means for you it's an inner form of a non-split quasi-split algebraic group). This leaves out two things to do: give a conceptual explanation of this fact, and then understand how tori interfere in this situation. | |
| Jan 26, 2018 at 1:37 | comment | added | paul garrett | To go through the classical groups might give an interesting data-point set, since, obviously, no general fact that fails (for relatively elementary reasons, as opposed to general algebraic-group reasons) there can be true generally. It would engender some intuition, I'd think, though, who knows, the exceptional groups can always do weird things. Even D4 and "triality". Still, classical groups are a good test sample... | |
| Jan 26, 2018 at 1:34 | history | edited | Cheng-Chiang Tsai | CC BY-SA 3.0 |
added 5 characters in body
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| Jan 26, 2018 at 1:31 | comment | added | Cheng-Chiang Tsai | I am worried that the variety of languages I know is limited (i.e. this is the only language I am familiar with). But thanks and let me try to add some elaborations and example to the question. | |
| Jan 26, 2018 at 1:26 | history | edited | Cheng-Chiang Tsai | CC BY-SA 3.0 |
an example added
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| Jan 26, 2018 at 0:25 | comment | added | Jim Humphreys | To make this question a little more concrete, maybe a famliar example such as an $F$-split simple group $G$ of type $A_n$ would be helpful? (Anyway I guess you are following the language of Borel-Tits here, as in the Tits classification.) | |
| Jan 25, 2018 at 21:11 | history | asked | Cheng-Chiang Tsai | CC BY-SA 3.0 |