Timeline for A subgroup of the Weyl group
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 24, 2013 at 14:31 | vote | accept | Mikhail Borovoi | ||
| Aug 23, 2013 at 21:20 | comment | added | Mikhail Borovoi | @Jim Humphreys: Sure! Carter's Proposition 13.1.2 gives the affirmative answer to my question. | |
| Aug 23, 2013 at 20:37 | comment | added | Jim Humphreys | For the Weyl group it doesn't really make any difference. I think Helgason's book deals more directly with your issue, but the formalism for roots and Weyl groups doesn't change. Satake and Tits also have lecture note treatments covering the Lie groups, as do some textbooks I don't have at hand. Carter's treatment is quite concrete in any case. | |
| Aug 23, 2013 at 18:17 | comment | added | Mikhail Borovoi | @Jim Humphreys: Thank you, it was very helpful. I am interested in twisting compact groups over $\mathbb{R}$, rather than split groups over finite fields. | |
| Aug 23, 2013 at 17:10 | answer | added | Jim Humphreys | timeline score: 4 | |
| Aug 23, 2013 at 13:30 | comment | added | Jim Humphreys | P.S. The early sections of Chapter 13 in Carter's book deal with your situation, though his notation differs from yours. Some of the arguments are classification-free, involving generation of a reflection subgroup of the Weyl group. Then there is case-by-case discussion of the actual possibilities. He is of course treating all possible diagram symmetries. | |
| Aug 23, 2013 at 13:20 | comment | added | Jim Humphreys | This set-up is well-studied in connection with the construction of quasi-split groups over various fields such as finite fields. I think Carter's old book Simple Groups of Lie Type may be a good source for an affirmative answer to your question, but I'd need to check more carefully. There are also accounts by Tits, Satake, etc. | |
| Aug 23, 2013 at 11:28 | history | asked | Mikhail Borovoi | CC BY-SA 3.0 |