Relative Quasimaps and Tilting Module of $U(\mathfrak{gl}_n)$

We study the equivariant cohomology of the moduli space of quasimaps from $\mathbb{P}^1$ with one marked point to the flag variety. This moduli space has an open subset isomorphic to the Laumon space. The equivariant cohomology of the Laumon space carries a natural action of $U(\mathfrak{gl}_n)$ constructed via geometric correspondences. We extend this construction to the entire quasimap moduli space and relate it to tilting modules of $U(\mathfrak{gl}_n)$.