Tensor product varieties, perverse sheaves and stability conditions

TL;DR

This paper establishes an isomorphism between classes of simple perverse sheaves and tensor products of modules in quantum algebra, linking geometric and algebraic structures and clarifying stability conditions.

Contribution

It introduces a new isomorphism connecting perverse sheaves with tensor products of quantum group modules and compares stability conditions in different localization frameworks.

Findings

Isomorphism between simple perverse sheaves and tensor product modules

Identification of simple perverse sheaves with canonical basis elements

Equivalence of stability conditions via supports and singular supports

Abstract

We show that the space spanned by the class of simple perverse sheaves in [Zh08] without localization is isomorphic to the tensor product of a Verma module with a tensor product of irreducible integrable modules of the quantum enveloping algebra associated with a graph. Under the isomorphism, the simple perverse sheaves get identified with the canonical basis elements of the tensor product module. We also show that the two stability conditions coincide in the localization process in [Zh08], by using supports and singular supports of complexes of sheaves, respectively.