Cohomological integrality for symmetric quotient stacks

In this talk, I will explore the notion cohomological integrality, originally introduced by Kontsevich and Soibelman in the context of quiver representations. It has then 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 main result is a decomposition of the cohomology of the corresponding quotient stack (which is usually infinite-dimensional) into finitely many pieces involving finite-dimensional subspaces. The objective is to define and understand enumerative invariants associated with the corresponding Geometric Invariant Theory (GIT) quotients. I will present the core representation-theoretic results that underlie these constructions and discuss their applications to geometry, including potential developments in the study of moduli spaces, derived categories, and enumerative invariants.