大島 利雄（OSHIMA Toshio）

東京大学大学院数理科学研究科
〒153-8914 東京都目黒区駒場３−８−１

Last update is Feb. 2011.

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論文とプレプリント

  1. A classification of roots of symmetric Kac-Moody root systems and its application, (with K. Hiroe) preprint, 43 pages, 2012,
    - 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.
  2. Boundary value problems on Riemannian Symmetric Spaces of the noncompact Type (with N. Shimeno), 26 pages, 2010, to appear in a book celebrating J. Wolf's 75th birthday.
    - abstract: We characterize the image of the Poisson transform on each distinguished boundary of a Riemannian symmetric space of the noncompact type by a system of differential equations.
  3. Fractional calculus of Weyl algebra and Fuchsian differential equations, 195 pages, 2011 (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.
  4. 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.
  5. 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.
  6. Classification of Fuchsian systems and their connection problem, arXiv:0811.2916, 29 pages, 2008.
    - 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.
  7. Heckman-Opdam hypergeometric functions and their specializations, Harmonische Analysis und Darstellungstheorie Topologischer Gruppen, Mathematisches Forschungsinstitut Oberwolfach, Report 49(2007), 38-40.
  8. 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 Ξ to a root system Σ which keep Cartan integers invariant. We examine different types of isomorphic classes defined by the Weyl group of Σ, that of Ξ and the automorphisms of Σ or Ξ 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 Ξ on the homomorphisms.
  9. 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.
  10. Completely integrable systems associated with classical root systems, (preprint, 2005), 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.
  11. 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.
  12. Minimal polynomials and annihilators of generalized Verma modules of the scalar type, (56pp, UTMS, 2004-3, DVI 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.
  13. 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
  14. 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).
  15. Harmonic analysis on semisimple symmetric spaces, Sugaku Expositions, 15(2002), 151-170, AMS, translated from Sugaku 37(1985), 97-112.
  16. 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.
  17. A quantization of conjugacy classes of matrices, (16pp, preprint, UTMS, 2000-38, DVI 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 ε  and it generates the defining ideal of the conjugacy class of square matrices at the classical limit ε = 0.
  18. Dimensions of spaces of generalized spherical functions, Amer. J. Math., 118(1996), 637-652 (with J. Huang and N. Wallach).
    - View Top
  19. 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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  20. 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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  21. 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.
    - View Top
  22. 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.
  23. 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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  24. 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.
  25. 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).
  26. 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).
  27. 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.
  28. 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).
  29. A method of harmonic analysis on semisimple symmetric spaces, Algebraic Analysis, edited by T. Kawai and M. Kashiwara, Academic Press 1988, 667-680.
  30. Boundedness of certain unitarizable Harish-Chandra modules, Advanced Studies in Pure Math., 14(1988), 651-660 (with M. Flensted-Jensen and H. Schlichtkrull).
  31. A realization of semisimple symmetric spaces and construction of boundary value maps, Advanced Studies in Pure Math., 14(1988), 603-650.
  32. Asymptotic behavior of spherical functions on semisimple symmetric spaces, Advanced Studies in Pure Math., 14(1988), 561-601.
  33. 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.
  34. Discrete series for semisimple symmetric spaces, Proceedings of the International Congress of Mathematicians, 1984, 901-904
  35. The restricted root systems of semisimple symmetric pairs, Advanced Studies in Pure Math., 4(1984), 433-497 (with J. Sekiguchi).
  36. Boundary value problems for systems of linear partial differential equations with regular singularities, Advanced Studies in Pure Math., 4(1984), 391-432.
  37. A description of discrete series for semisimple symmetric spaces, Advanced Studies in Pure Math., 4(1984), 331-390 (with T. Matsuki).
  38. A definition of boundary values of solutions of partial differential equations with regular singularities, Publ. RIMS Kyoto Univ., 22(1983), 1203-1230.
  39. Fourier analysis on semisimple symmetric spaces ,Non-Commutative Harmonic Analysis, Proceedings, 1980, Lect. Notes in Math., 880(1981), 357-369, DVI file, PDF file.
  40. Micro-local analysis of prehomogeneous vector spaces, Invent. Math., 62(1980), 117-179 (with M. Sato, M. Kashiwara and T. Kimura).
  41. Eigenspaces of invariant differential operators on an affine symmetric space , Invent. Math., 57(1980), 1-81 (with J. Sekiguchi).
  42. Orbits on affine symmetric spaces under the action of the isotropy subgroups , J. Math. Soc. Japan, 32(1980), 399-414 (with T. Matsuki).
  43. Poisson transformations on affine symmetric spaces, Proc. Japan Acad. 55A(1979), 323-327.
  44. A study of Feynman integrals by micro-differential equations, Comm. Math. Phys., 60(1978), 97-130 (with M. Kashiwara and T. Kawai).
  45. On analytic equivalence of glancing hypersurfaces, Sci. Papers College Gen. Ed. Univ. Tokyo, 28(1978), 51-57.
  46. A realization of Riemannian symmetric spaces, J. Math. Soc. Japan, 30(1978), 117-132.
  47. 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).
  48. Systems of differential equations with regular singularities and their boundary value problems, 106(1977), 145-200 (with M. Kashiwara).
  49. Boundary value problem on symmetric homogeneous spaces, Proc. Japan Acad., 53A(1977), 81-83 (with J. Sekiguchi).
  50. 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)
  51. Introduction to microlocal analysis, Publ. RIMS Kyoto Univ., 12 Suppl(1977), 267-300 (with T. Miwa and M. Jimbo).
  52. Boundary value problem with regular singularity and Helgason-Okamoto conjecture, Publ. RIMS Kyoto Univ., 12 Suppl(1977), 257-265 (with K. Minemura).
  53. 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).
  54. 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).
  55. 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).
  56. On the global existence of solutions of systems of linear differential equations with constant coefficients, J. Math. Soc. Japan, 26(1974), 575-586.
  57. A proof of Ehrenpreis' fundamental principle in hyperfunctions, Proc. Japan Acad., 50(1974), 14-18.
  58. Singularities in contact geometry and degenerate pseudo-differential equations , J. Fac. Sci. Univ. Tokyo Sec. IA 21(1974), 43-87.
  59. On the theorem of Cauchy-Kowalevsky for first order linear differential equations with degenerate principal symbols, Proc. Japan Acad., 49(1973), 83-87.

その他

  1. 特殊関数と代数的線形常微分方程式, 2010年度「代数解析」講義録, 於東大数理科学研究科, 廣惠一希記, Oct. 2011
  2. Fuchs型常微分方程式とKac-Moody root系, 表現論シンポジウム2010，於 伊豆長岡, Nov. 11, 2011
  3. 数式処理(Risa/Asir)とTeXとdviout, TeXユーザの集い2010, ，ポスターセッション, 於 東大生産技術研究所, Oct. 24, 2010
  4. Fractional calculus of Weyl algebra and its application to differential equations on the Riemann sphere, Mathematics - String theory theminar, IPMU, Univ. of Tokyo, Sep. 8, 2009
  5. Fractional calculus of Weyl algebra and its applications, Representation Theory of Real reductive Lie groups, held at University of Utah, July 30, 2009.
  6. Classification of Fuchsian systems and their connection problem, Differential equations and symmetric spaces, held at the Univ. of Tokyo, January 15, 2009
  7. 旗多様体を規定する方程式, slides 15pp, ラドン変換 -積分が拓く新しい世界- ENCOUNTER with MATHEMATICS, 於 中央大学理工, May 30, 2009.
  8. 分かるFuchs型常微分方程式, abstract 21pp, 第47回実函数論・函数解析学合同シンポジウム　特別講演， 於 慶応大学，Aug. 7, 2008.
  9. Heckman-Opdam hypergeometric functions and their specializations, abstract 3pp, talk at "Harmonische Analysis und Darstellungstheorie Topologischer Gruppen", Oberwolfach Workshop, Oct. 19, 2007.
  10. Subsystem of a root system (E8), Slides 16pp, Aug., 2007，於 Tambara Institute of mathematical Siceinces.
  11. Boundary value ploblems for Riemannian symmetric spaces, Slides 13pp, Aug., 2007，於 University of Iceland.
  12. Differential equations attached to generalized flag manifolds and their applications to integral geometry, Slides 15pp, June., 2007，於 Max-Planck Institute for Mathematics.
  13. Table of subroot systems, 2006, 8pp.
  14. 退化系列表現のWhittaker模型，数理解析研究所講究録 1467(2006), 71-78.
  15. Whittaker model of degerete principal series, 9pp, for Beamer, 2006年1月, Singapore.
  16. 線形代数の量子化と積分幾何, 数理解析研究所講究録 1421(2005), 12-25.
  17. Lie群と表現論, 岩波書店，2005, pp.610（小林俊行氏と共著）.
  18. 線形代数の量子化と積分幾何, 日本数学会, 企画特別講演，2004年3月，於 筑波大学，13pp, PDF file.
  19. 確定特異点型の可換微分作用素系と完全積分可能量子系, 数理解析研究 所講究録 1294(2002), 100-109.
  20. Eigenfunctions on a Riemannian symmetric space of the noncompact type with Lp-boundary value, 数理解析研究所講究録 1294(2002), 93-99 (Ben Said氏および示野信一氏と共著）.
  21. Twisted Radon transforms on Grassmannians, "Integral Gemetry in Representation Theory", MSRI at Berkeley, 2001年10月. Slides:12 in x 9 in, 14pp, DVI file, PS file. 
  22. Commuting differential operators of type B2, 数理解析研究所講究録 1171(2000), 36-67 (with H. Ochiai).
  23. Lie 群と Lie 環 I, 岩波講座, 現代数学の基礎 17, 1999, pp.294 （小林俊行氏と共著）.
  24. Capelli identities, degenerate series and hypergeometric functions, 表現論沖縄シンポジウム, 1995年12月, 19pp.
  25. 座標対称性をもつ完全積分可能な量子系, 研究集会「等質空間上の非可換調和解析」, 数理解析研究所, 1994年8月, 895(1995), 24-43
  26. 完全積分可能な量子系, 1994年度 実関数論・関数解析学シンポジウム講演集録, 1-19.
  27. Completely integrable quantum systems with coordinate symmetries and hypergeometric equations, 数理解析研究所講究録, 856(1994), 123-131.
  28. Harmonic Analysis on semisimple symmetric spaces, ユニタリ表現論セミナー報告集, IX, 1989, 90-95.
  29. Asymptotic behavior of Flensted-Jensen's spherical trace functions with respect to spectoral parameters, ユニタリ表現論セミナー報告集, VIII, 1988, 1-16.
  30. 対称空間上のEhrenpreisの基本原理, 数理解析研究所講究録, 642(1988), 229-244(佐分利豊氏、若山正人氏と共著).
  31. 半単純対称空間の離散系列表現の存在条件, 数理解析研究所講究録, 642(1988), 119-133(松木敏彦氏と共著)
  32. ユニタリ化可能なHarish-Chandra加群の有界性, 数理解析研究所講究録, 632(1987), 174-185.
  33. 半単純対称空間の調和解析, 37, 数学、岩波, 1985, 97-112.
  34. 球函数の漸近挙動について, ユニタリ表現論セミナー報告集, IV, 1984, 21-29.
  35. 数表 単純リー環の既約表現の次元, ユニタリ表現論セミナー報告集, I, 1981, 122--140
  36. ある多項式のべき積分(c-函数)の計算, 数理解析研究所講究録, 431(1981), 90-108.
  37. Simple holonomic system の構造について, 数理解析研究所講究録, 416(1981), 21-19.
  38. 確定特異点型の境界値問題と表現論, 8, 上智大学数学講究録, 1979.　
  39. 対称空間上の不変微分作用素のスペクトル, 341(1978), 173-179.
  40. 確定特異点型境界値問題と表現論, 2, 数学、岩波 (1977),
  41. Affine symmetric spaces における境界値問題, 数理解析研究所講究録, 300(1977), 40-53.
  42. 対称空間上の種々の特殊固有函数について, 数理解析研究所講究録, 295(1977), 12--19.
  43. Harmonic analysis on affine symmetric spaces, 数理解析研究所講究録, 287(1977), 77--87 (関口次郎氏と共著)
  44. 一階偏微分方程式、岩波講座、基礎数学, 1976 (小松彦三郎氏と共著）,
  45. 対称空間の種々の境界に対する境界値問題, 数理解析研究所講究録, 281(1976), 211-226.
  46. Local equivalence of differential forms and their deformations, 数理解析研究所講究録, 266(1976), 108-129.
  47. 対称空間における境界値問題について, 数理解析研究所講究録, 249(1975), 10-21.
  48. 確定特異点型境界値問題について, 数理解析研究所講究録, 248(1975), 319-329.
  49. 微分形式の特異点について, 数理解析研究所講究録, 227(1975), 97-108.
  50. Maximally degenerate な台を持つ擬微分方程式について, 数理解析研究所講究録, 226(1975), 29-38.
  51. R^n/Z^n x R での基本原理, 数理解析研究所講究録, 209(1974), 78-86.
  52. 接触幾何学における第二種の特異点の構造と退化した擬微分方程式, 数理解析研究所講究録, 192(1973), 327-381.
  53. 定数係数線型微分方程式系の解の存在について, 数理解析研究所講究録, 168(1972), 76-86.

-*-*-*-*- ティータイムの数学 -*-*-*-*-

予想

  1. Rigid分割3.1.1予想

問題

  1. 三角関数の恒等式

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