Demazure Descent Data and Braid group actions on categories

We recall the classical notion of Demazure operators acting on the $K$-theory of a $G$-variety $X$, $G$ being a reductive algebraic group.
Then we propose a categorification of the algebra generated by Demazure operators and introduce the notion of Demazure Descent Data (DDD) on a category. We define the descent category for a DDD on a triangulated category $C$.
We explain how DDD arises naturally from a monoidal action of the tensor category of quasicoherent sheaves on $B\setminus G/B$ on a category. A natural example of such picture is provided by the derived category of quasicoherent sheaves on $X/B$ for a scheme $X$ with an action of the reductive group $G$. The descent category in this case is the derived category of quasicoherent sheaves on $X/G$.
Next we replace the category of quasicoherent sheaves by DG-modules over the algebra of differential forms on $X$. We explain how an analog of the construction above gives rise to a braid group action of a category.