Symmetries of categorical representations and the quantum Ng\^o action

We observe that all classical Hamiltonian systems coming from the invariant polynomials on a reductive Lie algebra g can be integrated in a universal way. This is a consequence of Ng\^o's action of the group scheme J of regular centralizers in G on all centralizers: the Hamiltonian flows associated to invariant polynomials integrate to an action of J as commutative symplectic groupoid. We quantize the Ng\^o action, providing a universal integration for all quantum Hamiltonian systems coming from the center Z=Z(Ug) of the enveloping algebra. Namely we extend Kostant's Whittaker description of Z to the action of a commutative quantum groupoid Wh, the bi-Whittaker Hamiltonian reduction of $D_G$, which also integrates all quantum Hamiltonian systems coming from the action of Z. These actions come from a braided tensor functor, the quantum Ng\^o map, from the W-category (modules for Wh) to adjoint-equivariant D-modules on G, giving a categorical family of G-invariant commuting operators on any strong G-category. This action also leads to a notion of Langlands parameters for categorical representations of G, refined central character for character sheaves, and a new symmetry of homology of character varieties. We derive our construction as the Langlands dual form of a simple symmetry principle for groupoids. Namely the symmetric monoidal category of equivariant sheaves (modules for the groupoid algebra) acts centrally on the corresponding convolution category. In particular modules for the nil-Hecke algebra for any Kac-Moody group act centrally on the corresponding Iwahori-Hecke category. We deduce both the Ng\^o action and its quantization from the case of the Langlands dual equivariant affine Grassmannian using the renormalized Geometric Satake theorem of Bezrukavnikov-Finkelberg.