Global Weyl groups and a new theory of multiplicative quiver varieties

TL;DR

This paper develops a multiplicative framework to establish Weyl group symmetries for full wild character varieties, extending previous results on moduli spaces of connections and quiver varieties, with implications for irregular Deligne--Simpson problems.

Contribution

It introduces a multiplicative approach to Weyl group isomorphisms for wild character varieties, broadening the scope beyond open parts to full moduli spaces.

Findings

Constructed algebraic symplectic isomorphisms between wild character varieties.

Extended Weyl group symmetries to larger moduli spaces of connections.

Proposed a conjecture for irregular Deligne--Simpson problems and introduced related noncommutative algebras.

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".