Parabolic subgroups of twisted Chevalley groups over a semilocal ring

N. A. Vavilov

Zapiski Nauchnykh Seminarov POMI, Rings and modules. Part 2, Tome 94 (1979), pp. 21-36

Résumé

It is shown that, under minor additional assumptions, the standard parabolic subgroups of a Chevalley group $G_\rho(\Phi,R)$ of twisted type $\Phi=A_\ell$, $\ell$ – odd, $D_\ell$ ,$E_6$ over a commutative semilocal ring $R$ with involution $\rho$ are in one-to-one correspondence with the $\rho$-invariant parabolic nets of ideals of $R$ of type $\Phi$, i.e., with the sets, of ideals $\sigma_\alpha$ of $R$ such that: (1) whenever; (2) $\rho\sigma_\alpha=\sigma_{\rho\alpha}$ for all $\alpha$; (3 $\sigma_\alpha=R$ for $\alpha>0$. For Chevalley groups of normal types, analogous results were obtained in Ref. Zh. Mat. 1976, 10A151; 1977, 10A 301; 1978, 6A476.