$T$-equivariant motives of flag varieties

We use the construction of the stable homotopy category by Khan-Ravi to calculate the integral $T$-equivariant $K$-theory spectrum of a flag variety over an affine scheme, where $T$ is a split torus associated to the flag variety. More precisely, we show that the $T$-equivariant $K$-theory ring spectrum of a flag variety is decomposed into a direct sum of $K$-theory spectra of the classifying stack $\text{B}T$ indexed by the associated Weyl group. We also explain how to relate these results to the motivic world and deduce classical results for $T$-equivariant intersection theory and $K$-theory of flag varieties.\par For this purpose, we analyze the motive of schemes stratified by affine spaces with group action, that preserves these stratifications. We work with cohomology theories, that satisfy certain vanishing conditions, which are satisfied for example by motivic cohomology and $K$-Theory.