Timeline for Centralizers of subtori in reductive groups, derived subgroups
Current License: CC BY-SA 3.0
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 21, 2017 at 11:22 | vote | accept | Tippy Tipper | ||
| May 20, 2017 at 14:36 | comment | added | nfdc23 | OK, thanks for clarifying. I have now given an answer that isn't as comprehensive in terms of general nilpotents in the Lie algebra but focuses on the more limited framework of the question posed to allow all characteristics on equal footing. | |
| May 20, 2017 at 13:32 | comment | added | Paul Levy | The existence of the ${\rm SL}_2$-subgroup fails in small characteristics, even in type $A$. But if the characteristic is good then we still get a cocharacter $\lambda:k^\times\rightarrow G$ (unique up to $G^e$-conjugacy) which is directly analogous to the maximal torus ${\rm diag}(t,t^{-1})$ in $H$. This cocharacter is called an associated cocharacter for $e$, and the kernel is either trivial or $\mu_2$. Then all of the remarks above should hold (in good characteristic) on systematically replacing "$H={\rm PGL}_2$'' by "${\rm ker}\, \lambda=\mu_2$". | |
| May 20, 2017 at 13:12 | comment | added | nfdc23 | Is this method based on Lie algebras applicable in all positive characteristics? | |
| May 20, 2017 at 11:58 | history | answered | Paul Levy | CC BY-SA 3.0 |