# OSHIMA Toshio

Department of Mathematical Sciences, University of Tokyo
3-8-1, Komaba, Tokyo 153-8914, Japan

Last update is July. 2012.

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## Papers and Preprints.

1. An elementary approach to the Gauss hypergeometric function, Josai Mathematical Monographs 6(2013), 3-23.
- abstract: We give an introduction to the Gauss hypergeometric function, the hypergeometric equation and their properties in an elementary way. Moreover we explicitly and uniformly describe the connection coefficients, the reducibility of the equation and the monodromy group of the solutions.
2. Quantization of linear algebra and its application to integral geometry, (with H. Oda), Geometric Analysis and Integral Geometry, Contemporary Mathematics 598 (2013), 189-208.
- abstract: In order to construct good generating systems of two-sided ideals in the universal enveloping algebra of a complex reductive Lie algebra, we quantize some notions of linear algebra, such as minors, elementary divisors, and minimal polynomials. The resulting systems are applied to the integral geometry on various homogeneous spaces of related real Lie groups.
3. A classification of roots of symmetric Kac-Moody root systems and its application, (with K. Hiroe), Symmetries, Integral Systems and Representations, Springer Proceedings of Mathematics and Statics 40 (2012), 195-241.
- 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 Euler transform of linear ordinary differential equations on the Riemann sphere whose singular points are regular singular or unramified irregular singular points.
4. Finite multiplicity theorems (with T. Kobayashi), Adv. Math. 248 (2013), 912-944.
- abstract: We find upper and lower bounds of the multiplicities of irreducible admissible representations of a semisimple Lie group occurring in the induced pepresentations from irreducible represenations of a closed subgroup. We give geometric criteria for the finiteness of the multiplicities.
5. Boundary value problems on Riemannian Symmetric Spaces of the noncompact Type (with N. Shimeno), Lie Groups: Structure, Actions, and Representations, Progress in Mathematics, 307 (2013), 273-308, Birkhäuser
- abstract: We characterize the image of the Poisson transform on each distinguished boundary of a Riemannian symmetric space of noncompact type by a system of differential equations.
6. Fractional calculus of Weyl algebra and Fuchsian differential equations, MSJ Memoirs 28, ixi+203 pages, 2012, Correction (older version: arXiv:1102.2792v1).
Corrections for UTMS 2011-5, Feb. 24, 2011.
- abstract: We give a unified interpretation of confluence, reducibility, contiguity relation and Katz's middle convolutions for linear ordinary differential equations with polynomial coefficients and their generalization to partial differential equations. The integral representation and series expansion of their solutions are also within our interpretation. As applications to Fuchsian differential equations on the Riemann sphere, we construct a universal model of Fuchsian differential equations with a given spectral type, in particular, we construct single ordinary differential equations without apparent singularities corresponding to the rigid local systems, whose existence was an open problem presented by Katz. Furthermore we obtain an explicit solution of the connection problem for the rigid Fuchsian differential equations. We give many examples calculated by our fractional calculus.
7. Heckman-Opdam hypergeometric functions and their specializations, RIMS Kokyuroku Bessatsu B20 (2010), 129-162 (with N. Shimeno),
- abstract: We discuss three topics, confluence, restrictions and real forms for the Heckman-Opdam hypergeometric functions.
8. Katz's middle convolution and Yokoyama's extending operation, arXiv:0812.1135, 18 pages, 2008.
- abstract: We give a concrete relation between Katz's middle convolution and Yokoyama's extension and show the equivalence of both algorithms using these operations for the reduction of Fuchsian systems.
9. Classification of Fuchsian systems and their connection problem, arXiv:0811.2916, 29 pages, 2008, to appear in RIMS Kokyuroku Bessatsu.
- abstract: We review the Deligne-Simpson problem, a combinatorial structure of middle convolutions and their relation to a Kac-Moody root system discovered by Crawley-Boevey. We show with examples that middle convolutions transform the Fuchsian systems with a fixed number of accessory parameters into fundamental systems whose spectral type is in a finite set and we give an explicit connection formula for solutions of Fuchsian differential equations without moduli.
10. Heckman-Opdam hypergeometric functions and their specializations, Harmonische Analysis und Darstellungstheorie Topologischer Gruppen, Mathematisches Forschungsinstitut Oberwolfach, Report No. 49/2007, 38-40.
11. A classification of subsystems of a root system, preprint, 47pp, 2006, math.RT/0611904, submitted.
- abstract: We classify isomorphic classes of the homomorphisms of a root system $\Xi$ to a root system $\Sigma$ which keep Cartan integers invariant. We examine different types of isomorphic classes defined by the Weyl group of $\Sigma$, that of $\Xi$ and the automorphisms of $\Sigma$ or $\Xi$ etc. We also distinguish the subsystem generated by a subset of a fundamental system. We introduce the concept of the dual pair for root systems which helps the study of the action of the outer automorphism of $\Xi$ on the homomorphisms.
12. Commuting differential operators with regular singurarities, (preprint, 30pp, 2006, math.AP/0611899), Algebraic Analysis of Differential Equations --- from Microlocal Analysis to Exponential Asymptotics --- Festschrift in Honor of Takahiro Kawai Springer-Verlag, Tokyo, 2007. 195-224.
- abstract: We study a system of partial differential equations defined by commuting family of differential operators with regular singularities. We construct ideally analytic solutions depending on a holomorphic parameter. We give some explicit examples of differential operators related to $SL(n,\mathbb R)$ and completely integrable quantum systems.
13. Completely integrable systems associated with classical root systems, (2005, preprint), math-ph/0502028, SIGMA 3(2007), 061, 50pages,
- abstract: We explicitly construct sufficient integrals of completely integrable quantum and classical systems associated with classical root systems, which include Calogero-Moser-Sutherland models, Inozemtsev models and Toda finite lattices with boundary conditions. We also discuss the classification of the completely integrable systems.
14. A class of completely integrable quantum systems associated with classical root systems, (19pp, preprint, 2004), Indag. Mathem. 16(2005), 655-677.
- abstract: We classify the completely integrable systems associated with classical root systems whose potential functions are meromorphic at an infinite point.
15. Minimal polynomials and annihilators of generalized Verma modules of the scalar type, (56pp, UTMS 2004-3, DVI file, PDF file), Journal of Lie Theory 16(2006), 155-219 (with H. Oda).
- abstract: We calculate the minimal polynomial associated to the pair of any finite dimensional representation of any semisimple Lie algebra and a generarized Verma module of the scalar type of the Lie algebra. Using this minimal polynomial, we give a generator system of a generalized Verma module of the scalar type. In the classical limit, it gives the generator system of the defining ideal of a coadjoint orbit.
16. A calculation of c-functions for semisimple symmetric spaces, Lie Groups and Symmetric Spaces, in the memory of F. I. Karpelevich, edited by Gindikin, AMS Translation Series, 210(2003), 315-339
17. Fatou's theorems and Hardy-type spaces for eigenfunctions of the invariant differential operators on symmetric spaces, International Mathematics Research Notices 16(2003), 915-931 (with S. B. Said and N. Shimeno).
18. Harmonic analysis on semisimple symmetric spaces, Sugaku Expositions, 15(2002), 151-170, AMS, translated from Sugaku 37(1985), 97-112.
19. Annihilators of generalized Verma modules of the scalar type for classical Lie algebras, (29pp, preprint, UTMS, 2001-29, DVI file), "Harmonic Analysis, Group Representations, Automorphic forms and Invariant Theory", in honor of Roger Howe, Vol. 12 (2007), Lecture Notes Series, Institute of Mathematical Sciences, National University of Singapore, 277-319.
-abstract: We construct a generator system of the annihilator of a Verma mudule of a classical reductive Lie algebra induced from a character of a parabolic subalgebra as an analogue of the minimal polynomial of a matrix. In a classical limit it gives a generator system of the defining ideal of any semisimple co-adjoint orbit of the Lie algerba.
20. A quantization of conjugacy classes of matrices, 16pp, preprint, UTMS, 2000-38,   DVI file, PDF file. Adv. in Math. 196(2005), 124-146.
- abstract: We construct a generator system of the annihilator of a generalized Verma module of $\mathfrak {gl}(n,\mathbb C)$ induced from any character of any parabolic subalgebra as an analogue of minors and elementary divisors. The generator system has a quantization parameter $\epsilon$ and it generates the defining ideal of the conjugacy class of square matrices at the classical limit $\epsilon=0$.
21. Dimensions of spaces of generalized spherical functions, Amer. J. Math., 118(1996), 637-652, (with J. Huang and N. Wallach).
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22. Generalized Capelli identities and boundary value problems for GL(n) , Structure of Solutions of Differential Equations, Katata/Kyoto 1995, 307-335, World Scientific, 1996.
- abstract: The Capelli identity is extended to the case of minors. The operators appearing in the generalized identities give the annihilator of the degenerate principal series for $GL(n)$ and characterizes the image of the Poisson transform of the hyperfunctions on several boundaries of $GL(n)$. Hypergeometric functions are defined through realizations of some special sections of the degenerate principal series and the realizations on boundaries of $GL(n)$ generalize Gelfand's hypergeometric functions. Related Radon transforms for Grassmannians are discussed.
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23. Commuting families of differential operators invariant under the action of a Weyl group, J. Math. Sci. Univ. Tokyo, 2(1995), 1-75 (with H. Sekiguchi).
- abstract: We study the family of commuting differential operators invariant under the natural action of an irreducible classical Weyl group. When their highest order terms generate the invariant differential operators with constant coefficients, we determine the potential function of the Schrodinger operator in the family.
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24. Commuting differential operators of type B2, UTMS 94-65, Dept. of Mathematical Sciences, Univ. of Tokyo, 1994, 31pp (with H. Ochiai), Funkcialaj Ekvacioj 46(2003), 297-336.
- abstract: We determine the completely integrable quantum systems with $B_2$ symmetry. We also study the reducibility of the systems of differential equations defined by these quantum systems.
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25. Completely integrable systems with a symmetry in coordinates, UTMS 94-6, Dept. of Mathematical Sciences, Univ. of Tokyo, 1994, 22pp. Asian J. Math., 2(1998), 935-955.
- abstract: We explicitly construct the integrals of completely integrable quantum or classical systems whose potential functions are invariant under the action of a classical Weyl group. Our potential functions and integrals are expressed by the Weierstrass elliptic function.
26. Commuting families of symmetric differential operators, Proc. Japan. Acad. 70(1994), 62-66 (with H. Ochiai and H. Sekiguchi).
- abstract: This paper is a resume of main results in the following three papers.
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27. Multiplicities of representations on homogeneous spaces, Abstracts from the conference `Harmonic Analysis on Lie Groups held at Sandbjerg Gods', August 26-30, 1991, 2pp, edited by N. V. Pedersen, 1991, Mathematical Institute, Copenhagen University.
28. Paley Wiener theorems on a symmetric space and its applications, Differential Geometry and Its Applications, 1(1991), 247-278 (with Y. Saburi and M. Wakayama).
29. Embeddings of discrete series into principal series, The Orbit Method in Representation Theory Proceeding of a Conference Held in Copenhagen, August to September 1988, 1990, Birkhauser, 147-175 (with T. Matsuki).
30. Asymptotic behavior of Flensted-Jensen's spherical trace functions with respect to spectral parameters, Algebraic Analysis, Geometry and Number Theory, Proceedings of JAMI Inaugural Conference, Supplement to Amer. J. Math., The Johns Hopkins University Press, 1989, 313-323.
31. A note on Ehrenpreis' fundamental principle on a symmetric space, Algebraic Analysis, edited by T. Kawai and M. Kashiwara, Academic Press, 1988, 681-697 (with Y. Saburi and M. Wakayama).
32. A method of harmonic analysis on semisimple symmetric spaces, Algebraic Analysis, edited by T. Kawai and M. Kashiwara, Academic Press 1988, 667-680.
33. Boundedness of certain unitarizable Harish-Chandra modules, Advanced Studies in Pure Math., 14(1988), 651-660 (with M. Flensted-Jensen and H. Schlichtkrull).
34. A realization of semisimple symmetric spaces and construction of boundary value maps, Advanced Studies in Pure Math., 14(1988), 603-650.
35. Asymptotic behavior of spherical functions on semisimple symmetric spaces, Advanced Studies in Pure Math., 14(1988), 561-601.
36. Open problems suggested by T. Oshima, Open Problems in Representation Theory of Lie Groups, Proceeding of Eighteenth International Symposium, Division of Mathematics, The Taniguchi Foundation, edited by T. Oshima, 1987, 21-23.
37. Discrete series for semisimple symmetric spaces, Proceedings of the International Congress of Mathematicians, 1984, 901-904
38. The restricted root systems of semisimple symmetric pairs, Advanced Studies in Pure Math., 4(1984), 433-497 (with J. Sekiguchi).
39. Boundary value problems for systems of linear partial differential equations with regular singularities, Advanced Studies in Pure Math., 4(1984), 391-432.
40. A description of discrete series for semisimple symmetric spaces, Advanced Studies in Pure Math., 4(1984), 331-390 (with T. Matsuki).
41. A definition of boundary values of solutions of partial differential equations with regular singularities, Publ. RIMS Kyoto Univ., 22(1983), 1203-1230.
42. Fourier analysis on semisimple symmetric spaces ,Non-Commutative Harmonic Analysis, Proceedings, 1980, Lect. Notes in Math., 880(1981), 357-369, DVI file, PDF file.
43. Micro-local analysis of prehomogeneous vector spaces, Invent. Math., 62(1980), 117-179 (with M. Sato, M. Kashiwara and T. Kimura).
44. Eigenspaces of invariant differential operators on an affine symmetric space , Invent. Math., 57(1980), 1-81 (with J. Sekiguchi).
45. Orbits on affine symmetric spaces under the action of the isotropy subgroups , J. Math. Soc. Japan, 32(1980), 399-414 (with T. Matsuki).
46. Poisson transformations on affine symmetric spaces, Proc. Japan Acad. 55A(1979), 323-327.
47. A study of Feynman integrals by micro-differential equations, Comm. Math. Phys., 60(1978), 97-130 (with M. Kashiwara and T. Kawai).
48. On analytic equivalence of glancing hypersurfaces, Sci. Papers College Gen. Ed. Univ. Tokyo, 28(1978), 51-57.
49. A realization of Riemannian symmetric spaces, J. Math. Soc. Japan, 30(1978), 117-132.
50. Eigenfunctions of invariant differential operators on a symmetric space, Ann. of Math., 107(1978), 1-39 (with M. Kashiwara, A. Kowata, K. Minemura, K. Okamoto and M. Tanaka).
51. Systems of differential equations with regular singularities and their boundary value problems, 106(1977), 145-200 (with M. Kashiwara).
52. Boundary value problem on symmetric homogeneous spaces, Proc. Japan Acad., 53A(1977), 81-83 (with J. Sekiguchi).
53. Holonomy structure of Landau singularities and Feynman integrals, Publ. RIMS Kyoto Univ., 12 Suppl(1977), 387-438 (with M. Sato, T. Miwa and M. Jimbo)
54. Introduction to microlocal analysis, Publ. RIMS Kyoto Univ., 12 Suppl(1977), 267-300 (with T. Miwa and M. Jimbo).
55. Boundary value problem with regular singularity and Helgason-Okamoto conjecture, Publ. RIMS Kyoto Univ., 12 Suppl(1977), 257-265 (with K. Minemura).
56. Structure of a single pseudo differential equation in a real domain, J. Math. Soc. Japan, 28(1976), 80-85 (with T. Kawai and M. Kashiwara).
57. Structure of cohomology groups whose coefficients are micro-function solution sheaves of systems of pseudo-differential equations with multiple characteristics II, Proc. Japan Acad., 50(1974), 549-550 (with T. Kawai and M. Kashiwara).
58. Structure of cohomology groups whose coefficients are micro-function solution sheaves of systems of pseudo-differential equations with multiple characteristics I, Proc. Japan Acad., 50(1974), 420-425 (with T. Kawai and M. Kashiwara).
59. On the global existence of solutions of systems of linear differential equations with constant coefficients, J. Math. Soc. Japan, 26(1974), 575-586.
60. A proof of Ehrenpreis' fundamental principle in hyperfunctions, Proc. Japan Acad., 50(1974), 14-18.
61. Singularities in contact geometry and degenerate pseudo-differential equations , J. Fac. Sci. Univ. Tokyo Sec. IA 21(1974), 43-87.
62. On the theorem of Cauchy-Kowalevsky for first order linear differential equations with degenerate principal symbols, Proc. Japan Acad., 49(1973), 83-87.

## Others

1. Fractional calculus of Weyl algebra and its application to differential equations on the Riemann sphere, Mathematics - String theory theminar, IPMU, Univ. of Tokyo, September 8, 2009
2. Fractional calculus of Weyl algebra and its applications, Representation Theory of Real reductive Lie groups, held at University of Utah, July 30, 2009.
3. Classification of Fuchsian systems and their connection problem, Differential equations and symmetric spaces, held at the Univ. of Tokyo, January 15, 2009
4. Completely integrable systems with coordinate symmetries, "Non Commutative Harmonic Analysis" held at RIMS, Kyoto University, August 1994, 20pp(in Japanese)
5. Capelli identities, degenerate series and hypergeometric functions, Okinawa Symposium on Representation Theory of Lie Groups, December, 1995, 19pp.
6. Twisted Radon transformation on Grassmanians, "Integral Geometry in Representation Theory" held at MSRI, Berkeley, October 8, 2001.  DVI files, PS files.(copy of slides: 12in x 9 in, 14pp)

oshima@ms.u-tokyo.ac.jp