A Double Hall Algebra Approach to Affine Quantum Schur–Weyl Theory

摘要

We investigate the structure of the double Ringel-Hall algebras associatedwith cyclic quivers and its connections with quantum loop algebras of$mathfrak{gl}_n$, affine quantum Schur algebras and affine Hecke algebras.This includes their Drinfeld-Jimbo type presentation, affine quantum Schur-Weylreciprocity, representations of affine quantum Schur algebras, and connectionswith various existing works by Lusztig, Varagnolo-Vasserot, Schiffmann, Hubery,Chari-Pressley, Frenkel-Mukhin, etc. We will also discuss conjectures on arealization of Beilinson-Lusztig-MacPherson type and Lusztig type integralforms for double Ringel-Hall algebras.