For a smooth affine algebraic group Formula: see text, one can attach various D-module categories to it that admit convolution monoidal structure. We consider the derived category of D-modules on Formula: see text, the stack Formula: see text and the category of Harish-Chandra bimodules. Combining the work of Beilinson–Drinfeld on D-modules and Hecke patterns with the recent work of the author with Dimofte and Py, we show that each of the above categories (more precisely the equivariant version) is monoidal equivalent to a localization of the DG category of modules of a graded Hopf algebra. As a consequence, we give an explicit braided monoidal structure to the derived category of D-modules on Formula: see text, which when restricted to the heart, recovers the braiding of Bezrukavnikov–Finkelberg–Ostrik.
Wenjun Niu (Mon,) studied this question.