Categorical representation theory and the coarse quotient

We show that a localized version of the 2-category of all categories with an action of a reductive group is equivalent to the 2-category of categories with an action of sheaves on a space defined only using the data of the Weyl group action on a maximal torus. As an application of our methods, we upgrade the equivalence of [54] and [78], which identifies the category of bi-Whittaker D-modules on a reductive group with the category of W [superscript aff]-equivariant sheaves on a maximal Cartan subalgebra which satisfy Coxter descent, to a monoidal equivalence (which equips the bi-Whittaker category with a symmetric monoidal structure), and compute a restriction on the essential image of parabolic restriction of very central adjoint equivariant sheaves, providing evidence for a conjecture of [9] on the essential image of enhanced parabolic restriction. Along the way, we develop a 'pointwise' criterion for a W-equivariant sheaf on a maximal Cartan of a semisimple Lie algebra to descend to a sheaf on the coarse quotient, and use this to reprove a result of Lonergan which states that such a sheaf descends to the coarse quotient of a maximal Cartan by the Weyl group if and only if for any simple reflection the sheaf descends to the coarse quotient of a maximal Cartan by the order two subgroup generated by that reflection.