On Generically Split Generic Flag Varieties

Let G be a split semisimple algebraic group over an arbitrary field F , let E be a G-torsor over F , and let P be a parabolic subgroup of G. The quotient variety X := E/P , known as a flag variety, is generically split, if the parabolic subgroup P is special. It is generic, provided that the G-torsor E over F is a standard generic Gk-torsor for a subfield k ⊂ F and a split semisimple algebraic group Gk over k with (Gk)F = G. For any generically split generic flag variety X, we show that the Chow ring CHX is generated by Chern classes (of vector bundles over X). This implies that the topological filtration on the Grothendieck ring of X coincides with the computable gamma filtration. The results were already known in some cases including the case where P is a Borel subgroup. We also provide a complete classification of generically split generic flag varieties and, equivalently, of special parabolic subgroups for split simple groups.