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Global Weyl groups and a new theory of multiplicative quiver varieties

Philip Boalch

Geometry & Topology 19 (2015) 3467–3536

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 (fission algebras) generalising the deformed multiplicative preprojective algebras (some special cases of which contain the generalised double affine Hecke algebras).

quiver variety, complex symplectic quotient, GIT, quasi-Hamiltonian, wild character variety, Weyl group, wild Riemann surface, irregular curve, Stokes local system
Mathematical Subject Classification 2010
Primary: 14L24, 34M40, 53D05, 53D20, 53D30
Received: 19 August 2014
Accepted: 10 February 2015
Published: 6 January 2016
Proposed: Simon Donaldson
Seconded: Richard Thomas, Jim Bryan
Philip Boalch
Département de Mathématiques
Bâtiment 425
Faculté des Sciences d’Orsay
Université Paris-Sud
91405 Orsay Cedex