Timeline for Reference for nonquasi-split groups of type $E_6$ and $E_7$ over local fields
Current License: CC BY-SA 4.0
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 21, 2019 at 18:10 | answer | added | Victor Petrov | timeline score: 4 | |
| Jun 21, 2019 at 0:18 | comment | added | Mikhail Borovoi | Let $c\in Z^1(k,G_0^{\rm ad})$ be a representative of $\xi$. Then it is not hard to twist $G_0$ using $c$, and you obtain the desired group $_c G_0$ of type $^3\mathrm{E}_6$ or $^2\mathrm{E}_7$, respectively, and its faithful $k$-irreducible representation $V$ over $k$ with centralizer $D$ in ${\rm End}_k(V)$. | |
| Jun 21, 2019 at 0:09 | comment | added | Mikhail Borovoi | Let $G_0$ be the corresponding simply connected split group, and $G_0^{\rm ad}$ be the adjoint group. Let $\eta(D)\in H^2(k,\mu_d)$ denote the class of $D$ (where $\mu_d=Z(G_0)$ and $d$ is 3 or 2, respectively). Then one should somehow use a construction of Martin Kneser (see also the book by Platonov and Rapinchuk) in order to lift $\eta$ to a cohomology class $\xi\in H^1(k,G_0^{\rm ad})$. Here one should use the assumption that $k$ is a local field. | |
| Jun 20, 2019 at 23:56 | comment | added | Mikhail Borovoi | I suspect that Tits does not give constructions of $^3\mathrm{E}_6$ and $^2\mathrm{E}_7$ because there is no nice constructions! | |
| Jun 19, 2019 at 15:23 | history | asked | Arkandias | CC BY-SA 4.0 |