Affine Weyl group symmetry of the Garnier system

Abstract

We show that the Garnier system in n-variables has affine Weyl group symmetry of type $B_{n+3}^{(1)}$. We also formulate the $\tau$ functions for the Garnier system (or the Schlesinger system of rank 2) on the root lattice $Q(C_{n+3})$ and show that they satisfy Toda equations, Hirota-Miwa equations and bilinear differential equations.

33 pages