Global Weyl groups and a new theory of multiplicative quiver varieties

Abstract

In previous work a relation between a large class of Kac-Moody algebras and meromorphic connections on global curves was established---notably the Weyl group gives isomorphisms between different moduli spaces of connections, and the root system is also seen to play a role. This involved a modular interpretation of many Nakajima quiver varieties, as moduli spaces of connections, whenever the underlying graph was a complete k-partite graph (or more generally a supernova graph). However in the isomonodromy story, or wild nonabelian Hodge theory, slightly larger moduli spaces of connections are considered. This raises the question of whether the full moduli spaces admit Weyl group isomorphisms, rather than just the open parts isomorphic to quiver varieties. This question will be solved here, by developing a "multiplicative version" of the previous approach. This amounts to constructing many algebraic symplectic isomorphisms between wild character varieties. This approach also enables us to state a conjecture for certain irregular Deligne--Simpson problems and introduce some noncommutative algebras, generalising the "generalised double affine Hecke algebras".

62 pages, 10 figures. Comments welcome. (v3: minor improvements/clarifications)

References in corpus (5)

• Symplectic Manifolds and Isomonodromic Deformations
• Geometry of Multiplicative Preprojective Algebra
• Quivers and difference Painleve equations
• Indecomposable parabolic bundles and the existence of matrices in prescribed conjugacy class closures with product equal to the identity
• Geometry of moduli spaces of meromorphic connections on curves, Stokes data, wild nonabelian Hodge theory, hyperkahler manifolds, isomonodromic deformations, Painleve equations, and relations to Lie theory

Cited by in corpus (16)

• Twisted wild character varieties
• Calabi-Yau structures on topological Fukaya categories
• Diagrams for nonabelian Hodge spaces on the affine line
• Quantizations of multiplicative hypertoric varieties at a root of unity
• Multiplicative preprojective algebras are 2-Calabi-Yau
• Moduli spaces of generalized hyperpolygons
• Calabi-Yau structures for multiplicative preprojective algebras
• D-modules of pure Gaussian type and enhanced ind-sheaves
• Stokes matrices for Airy equations
• On the noncommutative Poisson geometry of certain wild character varieties
• Counting the fission trees and nonabelian Hodge graphs (untwisted case)
• Integrable systems on multiplicative quiver varieties from cyclic quivers
• $\mathrm{Sp}(1)$-symmetric hyperkähler quantisation
• Lecture Notes on Quiver Representations and Moduli Problems in Algebraic Geometry
• Euler continuants in noncommutative quasi-Poisson geometry
• Diagrams and irregular connections on the Riemann sphere