A Classification of Roots of Symmetric Kac-Moody Root Systems and Its Application

Abstract

We study Weyl group orbits in symmetric Kac-Moody root systems and show a finiteness of orbits of roots with a fixed index. We apply this result to the study of the Euler transform of linear ordinary differential equations on the Riemann sphere whose singular points are regular singular or unramified irregular singular points. The Euler transform induces a transformation on spectral types of the differential equations and it keeps their indices of rigidity. Then as a generalization of the result by Oshima (in Fractional calculus of Weyl algebra and Fuchsian differential equations, MSJ Memoirs 28, 2012), we show a finiteness of Euler transform orbits of spectral types with a fixed index of rigidity. © Springer-Verlag London 2013.