Timeline for Centraliser of regular semisimple element in $G^F$, for a connected reductive algebraic group $G$
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
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| Aug 7, 2023 at 23:10 | comment | added | LSpice | @Riju, re, $C_G(s)$ is not always generated by $T$ and certain root groups; those are both connected, so we can only access $C_G(s)^\circ$ that way. As @MikhailBorovoi's comments 1 2 indicate, $C_G(s)$ will generally fail to be connected (although it is so for $G$ simply connected!), and the failure is measured, algebraically (for any field) and rationally (for any finite field), by a quotient of $C_W(s)$. | |
| Jun 3, 2019 at 0:56 | comment | added | Mikhail Borovoi | Now if $w=nT$ and $wsw^{-1}=s$, and moreover $w\in W^F$, then by Lang's theorem there exists $n_0\in N^F$ such that $w=n_0 T$. Clearly, $n_0\in C_{G^F}(s)$. | |
| Jun 3, 2019 at 0:51 | comment | added | Mikhail Borovoi | Namely, write $N=N_G(T)$, $W=N/T$. Then $C_G(s)\subset N$. For any $w=nT\in W$ we can define $wsw^{-1}$, and for a given regular semisimple element $s$ you have to compute the centralizer $C_W(s)$. | |
| Jun 3, 2019 at 0:34 | comment | added | Mikhail Borovoi | @Riju: A standard approach is first to compute $C_G(s)$, and after that to compute the set of $L$-points $C_{G^F}(s)$ using Galois cohomology (which is not difficult over a finite field $L$). | |
| Jun 2, 2019 at 9:18 | vote | accept | Riju | ||
| Jun 1, 2019 at 9:30 | comment | added | Riju | Fine example! Btw is there any way to find a description of this quantity $C_{G^F}(s)$, in terms of $T^F$ and something more, just like we have a description of $C_{G}(s)$ , which says that it is generated by $T$ and “certain” root subgroups! | |
| Jun 1, 2019 at 2:18 | history | edited | Mikhail Borovoi | CC BY-SA 4.0 |
added 149 characters in body
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| Jun 1, 2019 at 2:01 | history | answered | Mikhail Borovoi | CC BY-SA 4.0 |