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

TL;DR

This paper explores the connections between Gelfand--Graev modules, the Euler--Poincaré polynomial of the Arnold--Brieskorn manifold, and quantum affine Schur--Weyl duality, revealing their relations through Weyl group symmetries.

Contribution

It demonstrates that for certain covers of GL(r), the Gelfand--Graev functor is directly related to quantum affine Schur--Weyl duality, linking representation theory and quantum groups.

Findings

Gelfand--Graev functor relates to quantum affine Schur--Weyl duality for specific covers of GL(r)

The commuting algebra of the Iwahori-fixed Gelfand--Graev representation is a quotient of a quantum group

Relations among modules, polynomials, and duality are governed by Weyl group permutations

Abstract

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.