Mirabolic Langlands duality and the quantum Calogero-Moser system

TL;DR

This paper establishes a spectral decomposition equivalence between categories of twisted D-modules and quasi-coherent sheaves on moduli stacks related to mirabolic vector bundles and local systems, linking geometric Langlands duality with the quantum Calogero-Moser system.

Contribution

It constructs a generic spectral decomposition for twisted D-modules on a moduli stack of mirabolic bundles, connecting geometric Langlands duality with quantum integrable systems.

Findings

Spectral decomposition of derived categories of twisted D-modules.

Equivalence with quasi-coherent sheaves on mirabolic local systems.

Solution to the quantum Calogero-Moser system for genus 1 curves.

Abstract

We give a generic spectral decomposition of the derived category of twisted D-modules on a moduli stack of mirabolic vector bundles on a curve X in characteristic p: that is, we construct an equivalence with the derived category of quasi-coherent sheaves on a moduli stack of mirabolic local systems on X. This equivalence may be understood as a tamely ramified form of the geometric Langlands equivalence. When X has genus 1, this equivalence generically solves (in the sense of noncommutative geometry) the quantum Calogero-Moser system.