Homological stability for classical groups

On homology of linear groups over $k[\mathrm{t}]$

“…Better stabilization results for orthogonal groups are available in the work of Essert, cf. [Ess09]. In Essert's work, dealing with the homological contribution from the unipotent radical as above is not necessary -he uses opposition complexes, where stabilizers are Levi subgroups instead of the full parabolic subgroups.…”