Quantum imaginary Schur-Weyl duality

Abstract

We study quiver Hecke algebras of untwisted affine type $A$ with an arbitrary choice of parameters and establish a duality with the Iwahori-Hecke algebra of the symmetric group. The parameter $t$ of the Iwahori-Hecke algebra is explicitly determined by the parameters defining the quiver Hecke algebra. This duality provides a deformation of the imaginary Schur-Weyl duality introduced by Kleshchev and Muth. Furthermore, we prove that the characters of simple modules in the imaginary strata are computed in terms of the dual canonical basis and Kazhdan-Lusztig polynomials, and the characters of standard modules coincide with the PBW vectors of the corresponding quantum group under certain assumptions. In addition, we examine other untwisted affine types, where the quiver Hecke algebra is known to be independent of the choice of parameters and the imaginary Schur-Weyl duality with the symmetric group has been established. As in type $A$, we apply this duality to the computation of characters of simple and standard modules.