A categorical perspective on non-abelian localization

Abstract

In equivariant geometry, a localization (a.k.a., concentration) theorem is typically interpreted as a relationship between the equivariant geometry of a space with a group action and the geometry of its fixed locus. We take a different perspective, that of non-abelian localization: a localization theorem relates the geometry of an algebraic stack that is equipped with a $\Theta$-stratification to the geometry of the centers of this stratification. We establish a ``virtual'' $K$-theoretic non-abelian localization formula, meaning it applies to algebraic derived stacks with perfect cotangent complexes. We also establish a categorical upgrade of this theorem, by introducing a category of ``highest weight $K$-homology cycles'' with respect to the stratification, and relating the category of highest weight cycles on the stack to those on the centers of its $\Theta$-stratification. We apply these results to prove a universal wall-crossing formula, and establish a new finiteness theorem for the cohomology of tautological complexes on the stack of one-dimensional sheaves on an algebraic surface.