Braid group action on the category of twisted $D$-modules on the flag variety

We recall the geometry of the Springer variety and of the Grothendieck variety for a simple algebraic group $G$. We show that universal twisted differential operators on the flag variety of $G$ provide a quantization for the ring of functions on the Grothendieck variety. We recall the braid group action on the category of coherent sheaves on the Grothendieck variety due to Bezrukavnikov and Riche. Then we quantize the construction and define a braid group action on the category of universal twisted $D$-modules on the flag variety. We explain the notion of invariants for a categorical Braid group action and show that the category of $U(g)$-modules can be realized as the braid group invariants in the category of universal twisted $D$-modules on the flag variety. Finally we propose a conjecture for a similar construction in the case of the quantum group at a root of unity.