Abstract: A quasimap from a curve to a GIT quotient is a map to the stack quotient that is generically stable. An open subset of quasimaps from P^1 to the flag variety, usually called Laumon space or handsaw quiver variety, is known to be closely related to the representation theory of gl_n. In particular, one can construct an action of gl_n on the cohomology of Laumon spaces via geometric correspondences. In this talk, I will explain how to construct an action of gl_n on the cohomology of the full moduli space of quasimaps and explore its relation to tilting modules in Category O.