Perverse sheaves on instanton moduli spaces and AGT conjecture

Abstract: We consider Uhlenbeck partial compactifications of framed moduli spaces of instantons on $\mathbb R^4$ with a type $ADE$ gauge group. We show that a natural class of perverse sheaves behaves nicely under the hyperbolic restriction functor with respect to a Levi subgroup. As an application, we obtain a representation of a W-algebra on the direct sum of intersection cohomology groups, combined with earlier works by Maulik-Okounkov, Schiffmann-Vasserot for type $A$. This proves the AGT conjecture for type $ADE$. Work in progress with Braverman, Finkelberg.