Categorical cell decomposition of quantized symplectic algebraic varieties

Abstract

We prove a new symplectic analogue of Kashiwara’s equivalence from $\mathcal{D}$–module theory. As a consequence, we establish a structure theory for module categories over deformation-quantizations that mirrors, at a higher categorical level, the Białynicki-Birula stratification of a variety with an action of the multiplicative group ${\mathbb{G}}_m$. The resulting categorical cell decomposition provides an algebrogeometric parallel to the structure of Fukaya categories of Weinstein manifolds. From it, we derive concrete consequences for invariants such as $K$–theory and Hochschild homology of module categories of interest in geometric representation theory.