Affine Weyl Group Symmetry of the Garnier System

Abstract

In this paper, we show that the Garnier system in n-variables has affine Weyl group symmetry of type B⁽¹⁾_n+3. We also formulate the τ-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.