K?theory of regular compactification bundles

Let G be a split connected reductive algebraic group. E ? B $\mathcal {E}\longrightarrow \mathcal {B}$ × $G\times G$ -torsor over smooth base scheme and X regular compactification of G. We describe the Grothendieck ring associated fibre bundle ( ) : = {E}(X):=\mathcal {E}\times _{G\times G} X$ , as an algebra canonical toric flag . These are relative versions corresponding results on in case when is point, generalize classical rings projective bundles, bundles bundles.