Quasi-coherent Hecke category and Demazure Descent

Let $G$ be a reductive algebraic group with a Borel subgroup $B$. We define the quasi-coherent Hecke category for the pair $(G,B)$. For any regular Noetherian $G$-scheme $X$ we construct a monoidal action of the Hecke category on the derived category of $B$-equivariant quasi-coherent sheaves on $X$. Using the action we define the Demazure Descent Data on the latter category and prove that the Descent category is equivalent to the derived category of $G$-equivariant sheaves on $X$.