The global nilpotent cone for universal curves

TL;DR

The paper constructs a conic Lagrangian in the cotangent bundle of the moduli stack of G-bundles over the universal curve, providing a support condition for the Betti geometric Langlands correspondence and proving a family version of local constancy of Hecke operators.

Contribution

It introduces a new conic Lagrangian in the cotangent bundle of the moduli stack of G-bundles over the universal curve, advancing the geometric Langlands program.

Findings

Constructed a conic Lagrangian restricting to the global nilpotent cone.

Established a support condition for the Betti geometric Langlands correspondence.

Proved a family version of local constancy of Hecke operators.

Abstract

We construct a conic Lagrangian in the cotangent bundle of the moduli stack of $G$-bundles over the universal curve, restricting to the global nilpotent cone for each curve. It gives rise to a singular support condition suitable for the Betti geometric Langlands correspondence for families of curves and the automorphic gluing functor studied in arXiv: 2105.12318. We also prove a family version of ``local constancy of Hecke operators," generalizing our earlier result.