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.
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References
Arinkin, D.: Rigid irregular connections on ℙ1. Compos. Math. 146, 1323–1338 (2010)
Crawley-Boevey, W.: On matrices in prescribed conjugacy classes with no common invariant subspaces and sum zero. Duke Math. J. 118, 339–352 (2003)
Kac, V.C.: Infinite Dimensional Lie Algebras, 3rd edn. Cambridge Univ. Press, Cambridge (1990)
Katz, N.: Rigid Local Systems. Annals of Mathematics Studies, vol. 139. Princeton University Press, Princeton (1996)
Kawakami, H., Nakamura, A., Sakai, H.: Degeneration scheme of 4-dimensional Painlevé type equations. RIMS Kôkyûroku 1765, 108–123 (2011) (in Japanese)
Kostov, V.P.: On some aspects of the Deligne-Simpson problem. J. Dyn. Control Syst. 9, 303–436 (2003)
Hiroe, K.: Linear differential equations on ℙ1 and root systems (2012). arXiv:1010.2580v4, 49pp.
Robba, P.: Lemmes de Hensel pour les opérateurs différentiels; applications à la réduction formelle des équations différentielles. Enseign. Math. 26, 279–311 (1980)
Oshima, T.: Classification of Fuchsian systems and their connection problem (2008). arXiv:0811.2916, 29pp. RIMS Kôkyûroku Bessatsu (to appear)
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
Oshima, T.: Fractional Calculus of Weyl Algebra and Fuchsian Differential Equations. MSJ Memoirs, vol. 28. Mathematical Society of Japan, Tokyo (2012)
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)
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Dedicated to Professor Michio Jimbo on the occasion of his 60th birthday.
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Hiroe, K., Oshima, T. (2013). A Classification of Roots of Symmetric Kac-Moody Root Systems and Its Application. In: Iohara, K., Morier-Genoud, S., Rémy, B. (eds) Symmetries, Integrable Systems and Representations. Springer Proceedings in Mathematics & Statistics, vol 40. Springer, London. https://doi.org/10.1007/978-1-4471-4863-0_9
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DOI: https://doi.org/10.1007/978-1-4471-4863-0_9
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