Simply-laced isomonodromy systems

Boalch, Philip

Publications Mathématiques de l'IHÉS, Tome 116 (2012), pp. 1-68.

A new class of isomonodromy equations will be introduced and shown to admit Kac–Moody Weyl group symmetries. This puts into a general context some results of Okamoto on the 4th, 5th and 6th Painlevé equations, and shows where such Kac–Moody Weyl groups and root systems occur “in nature”. A key point is that one may go beyond the class of affine Kac–Moody root systems. As examples, by considering certain hyperbolic Kac–Moody Dynkin diagrams, we find there is a sequence of higher order Painlevé systems lying over each of the classical Painlevé equations. This leads to a conjecture about the Hilbert scheme of points on some Hitchin systems.

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