K‐theory of regular compactification bundles

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AbstractLet G be a split connected reductive algebraic group. Let be a ‐torsor over a smooth base scheme and X be a regular compactification of G. We describe the Grothendieck ring of the associated fibre bundle , as an algebra over the Grothendieck ring of a canonical toric bundle over a flag bundle over . These are relative versions of the corresponding results on the Grothendieck ring of X in the case when is a point, and generalize the classical results on the Grothendieck rings of projective bundles, toric bundles and flag bundles.