D-convolution categories and Hopf algebras

For a smooth affine algebraic group $G$, one can attach various D-module categories to it that admit convolution monoidal structure. We consider the derived category of D-modules on $G$, the stack $G/G_{ad}$ 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 $G/G_{ad}$, which when restricted to the heart, recovers the braiding of Bezrukavnikov-Finkelberg-Ostrik.