On The Motive of G-bundles

Let \(G\) be a reductive algebraic group over a perfect field \(k\) and \(\cG\) a \(G\)-bundle over a scheme \(X/k\). The main aim of this article is to study the motive associated with \(\cG\), inside the Veovodsky Motivic categories. We consider the case that \(\charakt k=0\) (resp. \(\charakt k\geq 0\)), the motive associated to \(X\) is geometrically mixed Tate (resp. geometrically cellular) and \(\cG\) is locally trivial for the Zariski (resp. étale) topology on \(X\) and show that the motive of \(\cG\) is geometrically mixed Tate. Moreover for a general \(X\) we construct a nested filtration on the motive associated to \(\cG\) in terms of weight polytopes. Along the way we give some applications and examples.