Quiver Varieties with Multiplicities, Weyl Groups of Non-Symmetric Kac–Moody Algebras, and Painlevé Equations

To a finite quiver equipped with a positive integer on each of its vertices, we associate a holomorphic symplectic manifold having some parameters. This coincides with Nakajima's quiver variety with no stability parameter/framing if the integers attached on the vertices are all equal to one. The construction of reflection functors for quiver varieties are generalized to our case, in which these relate to simple reflections in the Weyl group of some symmetrizable, possibly non-symmetric Kac–Moody algebra. The moduli spaces of meromorphic connections on the rank 2 trivial bundle over the Riemann sphere are described as our manifolds. In our picture, the list of Dynkin diagrams for Painlevé equations is slightly different from (but equivalent to) Okamoto's.