Quantizations of conical symplectic resolutions I: local and global structure

TL;DR

This paper explores the quantization of conical symplectic resolutions, extending classical representation theory concepts to a broader geometric context, and applying these ideas to various algebraic structures.

Contribution

It generalizes key representation theory results to quantizations of conical symplectic resolutions, including new applications to quiver and hypertoric varieties.

Findings

Extension of Beilinson-Bernstein localization

Development of Harish-Chandra bimodule theory

Group actions on derived categories

Abstract

We re-examine some topics in representation theory of Lie algebras and Springer theory in a more general context, viewing the universal enveloping algebra as an example of the section ring of a quantization of a conical symplectic resolution. While some modification from this classical context is necessary, many familiar features survive. These include a version of the Beilinson-Bernstein localization theorem, a theory of Harish-Chandra bimodules and their relationship to convolution operators on cohomology, and a discrete group action on the derived category of representations, generalizing the braid group action on category O via twisting functors. Our primary goal is to apply these results to other quantized symplectic resolutions, including quiver varieties and hypertoric varieties. This provides a new context for known results about Lie algebras, Cherednik algebras, finite W-algebras, and hypertoric enveloping algebras, while also pointing to the study of new algebras arising from more general resolutions.