Timeline for About the intrinsic definition of the Weyl group of complex semisimple Lie algebras
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 24, 2021 at 12:54 | comment | added | LSpice | Title of @Asaf's reference: Bader and Furman - Boundaries, Weyl groups, and superrigidity. | |
| Aug 12, 2012 at 19:18 | comment | added | Jim Humphreys | @Zhaoting: I should add a follow-up reference, which makes more explicit the abstract definition (but isn't usable without the background theory). See section 1.2 of Lusztig's paper Coxeter orbits and eigenvalues of Frobenius, Invent. Math. 38 (1976), available online at gdz.sub.uni-goettingen.de | |
| Aug 4, 2012 at 5:26 | vote | accept | Zhaoting Wei | ||
| Aug 4, 2012 at 5:26 | vote | accept | Zhaoting Wei | ||
| Aug 4, 2012 at 5:26 | |||||
| Aug 3, 2012 at 14:14 | comment | added | Jim Humphreys | @Zhaoting: This is all worked out carefully by Deligne-Lusztig in their section 1.2. But I'd emphasize that it uses most of the deep structure theory (including conjugation theorems and Bruhat decomposition in the group version) to reach the intrinsic formulation. | |
| Aug 2, 2012 at 23:10 | comment | added | Zhaoting Wei | @Jim Maybe we can look at the set of $G$ -orbits of $X \times X$ and say that "this is the Weyl group". But can we define a multiplication just on this set of $G$-orbits? If we can, then this is what I am seeking for: an intrinsic definition of Weyl group. | |
| Aug 2, 2012 at 18:14 | comment | added | Asaf | @Jim, a related approach defining an abstract "Weyl group" for probably the most general category of groups that should have Weyl groups, appeared in the recent work by Bader and Furman related to Margulis' superrigidity, see for example homepages.math.uic.edu/~furman/preprints/sr-note-published.pdf, and after your introduction, I see it inspired from the Deligne-Lusztig definition. | |
| Aug 2, 2012 at 17:36 | comment | added | Jim Humphreys | @quasi (if I may call you that): See my added paragraph, where I emphasize that the definition itself uses no more than basic definitions. The price for that is having to figure out what it actually means in concrete terms; then you do need more theory. | |
| Aug 2, 2012 at 17:32 | history | edited | Jim Humphreys | CC BY-SA 3.0 |
added 875 characters in body
|
| Aug 2, 2012 at 12:34 | comment | added | user22479 | Of course, this elegant "intrinsic definition" rests on all of the usual conjugacy results, though it isn't clear what the OP is really seeking by asking for a definition which avoids the need to check "the unambiguity". | |
| Aug 2, 2012 at 10:44 | history | answered | Jim Humphreys | CC BY-SA 3.0 |