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 G k ‐torsor for a subfield k ⊂ F and a split semisimple algebraic group G k over k with( G k ) F = G . For any generically split generic flag variety X , we show that the Chow ringCH X 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.