Gelfand--Graev functor and quantum affine Schur--Weyl duality

We explicate relations among the Gelfand--Graev modules for central covers, the Euler--Poincar\'e 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. Specifically, our main result shows that for certain covers of $\mathrm{GL}(r)$ the Gelfand--Graev functor is related to quantum affine Schur--Weyl duality. Consequently, the commuting algebra of the Iwahori-fixed part of the Gelfand--Graev representation is the quotient of a quantum group.