Genuine Gelfand–Graev functor and the quantum affine Schur–Weyl duality

We explicate relations among the Gelfand–Graev modules for central covers of reductive p-adic groups, the Euler–Poincaré polynomial of the Arnold–Brieskorn manifold, and the quantum affine Schur–Weyl duality. These three objects and their relations are dictated by a permutation representation of the Weyl group. Our main result concerns the connection between the Gelfand–Graev module and quantum affine Schur-Weyl duality, which only holds for certain covers of GL(r). In this case, the Gelfand–Graev functor is essentially the quantum affine Schur–Weyl functor. This has two significant consequences. First, the commuting algebra of the Iwahori-fixed part of the Gelfand–Graev representation is the quotient of a quantum group. Second, intertwining operators on the p-adic group side are matched with R-matrices on the quantum group side, reproving a result of Brubaker–Buciumas–Bump.