Noncommutative Counterparts of the Springer Resolution

TL;DR

This paper explores a noncommutative structure related to the Springer resolution of nilpotent elements in semisimple Lie algebras, linking it to various areas like geometric Langlands duality and modular representation theory.

Contribution

It introduces a new noncommutative structure or t-structure on the Springer resolution, revealing connections across multiple representation theoretic and geometric frameworks.

Findings

Identifies a common t-structure in diverse mathematical contexts

Links noncommutative geometry to modular representation theory

Provides insights into derived equivalences and stability conditions

Abstract

Springer resolution of the set of nilpotent elements in a semisimple Lie algebra plays a central role in geometric representation theory. A new structure on this variety has arisen in several representation theoretic constructions, such as the (local) geometric Langlands duality and modular representation theory. It is also related to some algebro-geometric problems, such as the derived equivalence conjecture and description of T. Bridgeland's space of stability conditions. The structure can be described as a noncommutative counterpart of the resolution, or as a $t$-structure on the derived category of the resolution. The intriguing fact that the same $t$-structure appears in these seemingly disparate subjects has strong technical consequences for modular representation theory.