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

Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 40)

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.

Keywords

Singular Point Weyl Group Spectral Type Dynkin Diagram Fixed Index 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Arinkin, D.: Rigid irregular connections on ℙ1. Compos. Math. 146, 1323–1338 (2010) MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Crawley-Boevey, W.: On matrices in prescribed conjugacy classes with no common invariant subspaces and sum zero. Duke Math. J. 118, 339–352 (2003) MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Kac, V.C.: Infinite Dimensional Lie Algebras, 3rd edn. Cambridge Univ. Press, Cambridge (1990) MATHCrossRefGoogle Scholar
  4. 4.
    Katz, N.: Rigid Local Systems. Annals of Mathematics Studies, vol. 139. Princeton University Press, Princeton (1996) MATHGoogle Scholar
  5. 5.
    Kawakami, H., Nakamura, A., Sakai, H.: Degeneration scheme of 4-dimensional Painlevé type equations. RIMS Kôkyûroku 1765, 108–123 (2011) (in Japanese) Google Scholar
  6. 6.
    Kostov, V.P.: On some aspects of the Deligne-Simpson problem. J. Dyn. Control Syst. 9, 303–436 (2003) MathSciNetCrossRefGoogle Scholar
  7. 7.
    Hiroe, K.: Linear differential equations on ℙ1 and root systems (2012). arXiv:1010.2580v4, 49pp.
  8. 8.
    Robba, P.: Lemmes de Hensel pour les opérateurs différentiels; applications à la réduction formelle des équations différentielles. Enseign. Math. 26, 279–311 (1980) MathSciNetMATHGoogle Scholar
  9. 9.
    Oshima, T.: Classification of Fuchsian systems and their connection problem (2008). arXiv:0811.2916, 29pp. RIMS Kôkyûroku Bessatsu (to appear)
  10. 10.
    Oshima, T.: Special Functions and Linear Algebraic Ordinary Differential Equations. Lecture Notes in Mathematical Sciences, vol. 11. The University of Tokyo, Tokyo (2011) (in Japanese), typed by K. Hiroe Google Scholar
  11. 11.
    Oshima, T.: Fractional Calculus of Weyl Algebra and Fuchsian Differential Equations. MSJ Memoirs, vol. 28. Mathematical Society of Japan, Tokyo (2012) Google Scholar
  12. 12.
    Takemura, K.: Introduction to middle convolution for differential equations with irregular singularities. In: New Trends in Quantum Integrable Systems: Proceedings of the Infinite Analysis 09, pp. 393–420. World Scientific, Singapore (2010) Google Scholar

Copyright information

© Springer-Verlag London 2013

Authors and Affiliations

  1. 1.Research Institute for Mathematical SciencesKyoto UniversityKyotoJapan
  2. 2.Graduate School of Mathematical SciencesThe University of TokyoMeguro-ku, TokyoJapan

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