Quantizations of conical symplectic resolutions I: local and global structure
Tom Braden, Nicholas Proudfoot, Ben Webster
TL;DR
This paper develops a unifying framework for quantizations of conical symplectic resolutions, treating universal enveloping algebras as section rings of quantum deformations and establishing Beilinson–Bernstein–type localization, Harish-Chandra bimodule theory, and braid-group–style actions in a broad geometric setting. It introduces the period map for quantizations, S-structures, and the section ring A, linking quantum Hamiltonian reduction to Kirwan-type period shifts and demonstrating derived/abelian localization in generic regimes via Z-algebras. The authors develop a robust theory of line-bundle quantizations, quantum homogeneous coordinate rings, and abelian/derived localization, including analytic-algebraic comparisons and Kirwan functors, enabling localization to reduced spaces and a categorical Kirwan surjectivity. They then construct a rich convolution/twisting calculus through Harish-Chandra bimodules, characteristic cycles, and twisting bimodules, culminating in a comprehensive action of twisting functors by the fundamental group of a hyperplane-complement modulo a Weyl group. The framework applies to key classes such as quiver varieties and hypertoric varieties, clarifying connections to Lie algebras, Cherednik algebras, finite W-algebras, and hypertoric enveloping algebras, and yielding new avenues for algebras arising from more general resolutions.
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.