Timeline for Is any F-stable maximal torus contained in some F-stable Borel subgroup? [closed]
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 19, 2015 at 17:27 | history | closed |
Dima Pasechnik Will Sawin Daniel Loughran Alex Degtyarev YCor |
Not suitable for this site | |
| May 19, 2015 at 14:45 | answer | added | grghxy | timeline score: 1 | |
| May 19, 2015 at 13:56 | comment | added | Hebe | @JayTaylor Thank you for your advice. But in your example, why can $F$ be assumed to act trivially on the Weyl group? Suppose so, and if $T$ is contained in an $F$-stable Borel subgroup $B$, and there exists an element $m\in N_G(T_0)$ such that $g^{-1}B_0g=m^{-1}Bm$. Hence, by $F(B)=B$ and $F(B_0)=B_0$, we obtain that $gm^{-1}F(m)F(g^{-1})\in N_G(B_0)=B_0$. But if we want to get $gF(g^{-1})\in B_0$ so that $gF(g^{-1})\in B_0\cap N_G(T_0)=T_0$, how can $m^{-1}F(m)$ be deleted? Note that $F(m)\in mT_0$. | |
| May 19, 2015 at 12:29 | answer | added | John Binder | timeline score: 1 | |
| May 19, 2015 at 11:25 | comment | added | Dima Pasechnik | I'm voting to close this question as it has been answered in the comments | |
| May 19, 2015 at 9:18 | review | Close votes | |||
| May 19, 2015 at 17:30 | |||||
| May 19, 2015 at 9:09 | comment | added | Jay Taylor | It's very hard to understand what's going on just by reading Digne--Michel, I wouldn't advise it. You might want to look at my answer to this question mathoverflow.net/questions/203602/…. Good luck. | |
| May 19, 2015 at 9:03 | comment | added | Hebe | @JayTaylor I see. Thank you. Actually I am reading the book "Representations of finite groups of Lie type" by Francois DIGNE and Jean MICHEL. Now I think that I had better turn to read the book recommended by you. | |
| May 19, 2015 at 8:58 | comment | added | Jay Taylor | Look at Example 4.3.3 in Geck's "An Introduction to Algebraic Geometry and Algebraic Groups" and the results around there. | |
| May 19, 2015 at 8:57 | comment | added | Jay Taylor | I think you should really read Chapter 4 of Geck's book. It has all the answers to your questions. The answer here is no. There always exists an $F$-stable maximal torus and Borel subgroup $T_0\leqslant B_0$ in $G$. If you take $T = gT_0g^{-1}$ with $g^{-1}F(g) = n \in N_G(T_0)$ then this is always $F$-stable. For convenience, assume $F$ acts trivially on the Weyl group $N_G(T_0)/T_0$. Then the maximal torus $T$ is contained in an $F$-stable Borel subgroup if and only if $n \in T_0$. Hence there are lots of $F$-stable maximal tori that are not contained in $F$-stable Borels. | |
| May 19, 2015 at 8:07 | history | asked | Hebe | CC BY-SA 3.0 |