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
–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).
Keywords
quiver variety, complex symplectic quotient, GIT,
quasi-Hamiltonian, wild character variety, Weyl group, wild
Riemann surface, irregular curve, Stokes local system
