In this talk, I will explore cohomological integrality isomorphisms, originally introduced by Kontsevich and Soibelman in the context of quiver representations. These isomorphisms have since been extended to a broader range of settings, including categories of sheaves on K3 surfaces and, more generally, 2- and 3-Calabi-Yau categories. My focus will be on extending these results to the setting of quotient stacks arising from representations of reductive groups and, more broadly, smooth affine algebraic varieties under the action of a reductive group.
The objective is to define and understand the topology, geometry, and enumerative invariants associated with the corresponding Geometric Invariant Theory (GIT) quotients. I will present the core representation-theoretic results that underpin these constructions and discuss their applications to geometry, including potential developments in the study of moduli spaces, derived categories, and enumerative invariants.