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

We explicate relations among the Gelfand–Graev modules for central covers, 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. Specifically, our main result shows that for certain covers of $\mathrm{G}\mathrm{L}(r)$ 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.