  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.
