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

Abstract 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 $$\textrm{GL}(r)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtext>GL</mml:mtext> <mml:mo>(</mml:mo> <mml:mi>r</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> . 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.