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We will explain how some new algebraic solutions of the sixth Painlevé equation arise
from complex reflection groups, thereby extending some results of Hitchin and Dubrovin--
Mazzocco for real reflection groups. The problem of finding explicit formulae for these
solutions will be addressed elsewhere.
This article is concerned with moduli spaces of connections on bundles on Riemann surfaces, where the structure group of the bundle may vary in different regions of the surface. Here we will describe such moduli spaces as complex symplectic manifolds, generalising the complex character varieties of Riemann surfaces.
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