Symplectic Duality of $T^*Gr(k,n)$

Abstract: In this paper, we explore a consequence of symplectic duality (also known as 3d mirror symmetry) in the setting of enumerative geometry. The theory of quasimaps allows one to associate hypergeometric functions called vertex functions to quiver varieties. In this paper, we prove a formula which relates the vertex functions of $T^*Gr(k,n)$ and its symplectic dual. In the course of the proof, we study a family of $q$-difference operators which act diagonally on Macdonald polynomials. Our results may be interpreted from a combinatorial perspective as providing an evaluation formula for a $q$-Selberg type integral.