Equivariant multiplicities and mirror symmetry for Hilbert schemes

TL;DR

This paper explores the geometry and mirror symmetry of Hilbert schemes on elliptic surfaces, computing multiplicities of core Lagrangians and proposing dualities inspired by Langlands correspondence.

Contribution

It introduces new computations of equivariant multiplicities for core and wobbly components and proposes a novel mirror duality conjecture involving upward flows and Procesi bundles.

Findings

Computed scheme-theoretic multiplicities of core Lagrangians.

Extended equivariant multiplicity to wobbly components and computed for two points.

Proposed a mirror duality between upward flows and modified Procesi bundles.

Abstract

Following Hausel-Hitchin, we investigate core Lagrangians and upward flows in Hilbert schemes of points on elliptic surfaces. We compute the scheme-theoretic multiplicities of core Lagrangians, as well as the equivariant multiplicities of the very stable ones. Furthermore, we extend the notion of equivariant multiplicity to wobbly components and compute it for Hilbert schemes of two points. Inspired by Eisenstein series functor in Dolbeault Langlands correspondence, we propose that upward flows of very stable ideals are mirror dual to modified Procesi bundles, and justify this claim through numerical checks. Finally, we make some conjectures about extending this picture to wobbly upward flows.