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Systems of differential equations attached to representations of Lie groups

Research Project

Project/Area Number 12440034
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Research Field Basic analysis
Research InstitutionThe University of Tokyo

Principal Investigator

OSHIMA Toshio  The University of Tokyo, Graduate School of Mathematical Sciences, Professor, 大学院・数理科学研究科, 教授 (50011721)

Co-Investigator(Kenkyū-buntansha) TERADA Itaru  The University of Tokyo, Graduate School of Mathematical Sciences, Associate Professor, 大学院・数理科学研究科, 助教授 (70180081)
MATUMOTO Hisayosi  The University of Tokyo, Graduate School of Mathematical Sciences, Associate Professor, 大学院・数理科学研究科, 助教授 (50272597)
ODA Takayuki  The University of Tokyo, Graduate School of Mathematical Sciences, Professor, 大学院・数理科学研究科, 教授 (10109415)
KOBAYASHI Toshiyuki  Kyoto University, Research Institute of Mathematical Sciences, Professor, 数理解析研究所, 教授 (80201490)
SEKIGUCHI Hideko  The University of Tokyo, Graduate School of Mathematical Sciences, Associate Professor, 大学院・数理科学研究科, 助教授 (50281134)
松木 敏彦  京都大学, 人間環境学研究科, 助教授 (20157283)
Project Period (FY) 2000 – 2003
Project Status Completed (Fiscal Year 2003)
Budget Amount *help
¥8,200,000 (Direct Cost: ¥8,200,000)
Fiscal Year 2003: ¥2,800,000 (Direct Cost: ¥2,800,000)
Fiscal Year 2002: ¥1,500,000 (Direct Cost: ¥1,500,000)
Fiscal Year 2001: ¥1,300,000 (Direct Cost: ¥1,300,000)
Fiscal Year 2000: ¥2,600,000 (Direct Cost: ¥2,600,000)
Keywordssymmetric space / Poisson transform / Verma module / Hardy space / completely integrable system / Radon transform / unversal enveloping algebra / primitive ideal / c-函数 / 最小多項式 / Verma 加群 / 単因子 / 量子化 / 完約リー環
Research Abstract

(1)The annihilator of the generalized Verma module of the scalar type for a reductive Lie algebra is invesitigated. The condition that the annihilator gives the gap between the generalized Verma module and the usual Verma module is clearified and its necessary and sufficient condition is shown when its highest weight is dominant. Moreover a good sufficient condition is obtained in general.
(2)By the dual map of the faithful finite dimensional representation of the reductive Lie algebra and its infinite dimensional representation, the characteristic polynomials and minimal polynomials of matrices are defined and the generator system of the annihilators of generalized Verma modules of scalar type is constracted., A sufficient condition that the ideal gives the gap is applied to some problems in the integral geometry.
(3)Fatou's theorems and Hardy-type spaces on a Riemannian symmetric space of the noncompact type for the general eigenvalue of the invariant differential operators are studied in general. These theorems can be localized only when the rank of the space is equals to one.
(4)On a compactification of a Euclidean space defined by a root lattice of the classical type, Shrodinger operators are classified which allow commuting differential operators of the forth order under the condition that their potentials are meromorphic at an infinite point.
(5)The totality of k-dimensional linear subspaces of n-dimensional spaces over R, C or H is the Grassmannian manifold. The Radon transforms on these Grassmannian manifolds naturally extend the Gelfand-Aomoto's theory of generalized hypergeometric functions. Twisted Radon transforms over totally geodesic submanifolds are studied and interesting examples are given whose images are characterized by differential equations.

Report

(5 results)
  • 2003 Annual Research Report   Final Research Report Summary
  • 2002 Annual Research Report
  • 2001 Annual Research Report
  • 2000 Annual Research Report
  • Research Products

    (75 results)

All Other

All Publications (75 results)

  • [Publications] Toshio Oshima: "Annihilators of generalized Verma modules of the scalar type for classical Lie algebras"UTMS 2001-29, University of Tokyo. 29. 1-29 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Toshio Oshima: "Harmonic analysis on semisimple symmetric spaces"Sugaku Expositions. 15. 151-170 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Salem Ben Said, T.Oshima, N.Shimeno: "Fatou's theorems and Hardy-type spaces for eigenfunctions of the invariant differential operators on symmetric spaces"Intern.Math.Research Notice. 16. 915-931 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Hiroyuki Ochiai, T.Oshima: "Commuting differential operators of type B2"Funkcialaj Ekvacioj. 46. 297-336 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Toshio Oshima: "A calculation of c-functions for semisimple symmetric spaces"Amer.Math.Soc.Transl., Advances in the Mathematical Sciences, Series 2. 210. 307-330 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Hiroyuki Oda, T.Oshima: "Minimal polynomials and annihilators of generalized Verma modules of the scalar type"UTMS 2004-3. 3. 1-56 (2004)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Toshio Oshima: "A quantization of conjugacy classes of matrices"Advances in Math.. (to appear).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] H.Hayata, H.Koseki, T.Oda: "Matrix coefficients of the Pj-principal series and the middle discrete series of SU(2,2)"Adv.Studies in Pure Math.. 26. 49-75 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Takayuki Oda, M.Tsuzuki: "Automorphic Green function associated with the secondary spherical functions"Publ.RIMS, Kyoto Univ.. 19. 451-533 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Hisayosi Matumoto: "On the representations of Sp(p,q) and SO^* (2n) unitamy induced from derived funcor modules"Composito Math. (to appear).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Toshiyuki Kobayashi: "Discontinuous groups for non-Riemannian homogeneous space"Mathematics Unlimited - 2001 and Beyond. 723-748 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Toshiyuki Kobayashi: "Introduction to actions of discrete groups on pseudo-Riemannian homogeneous manifolds"Acta Appl.Math.. 73. 113-131 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Toshiyuki Kobayashi: "Branching problems of unitary representations"Proc.International Congress of Mathematicians. 2. 615-627 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Toshiyuki Kobayashi, S.Nasrin: "Multiplicity one theorem in the orbit method"Amer.Math.Soc.Transl, Advances in the Mathematical Sciences, Series 2. 210. 161-169 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Toshiyuki Kobayashi, B.Orsted: "Analysis on minimal representations of O(p,q), Part I"Advances in Mathematics. 180. 486-512 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Toshiyuki Kobayashi, B.Orsted: "Analysis on minimal representations of O(p,q), Part II"Advances in Mathematics. 180. 513-550 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Toshiyuki Kobayashi, B.Orsted: "Analysis on minimal representations of O(p,q), Part III"Advances in Mathematics. 180. 551-596 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Toshiyuki Kobayashi: "Geometry of multiplicity-free representations of GL(n), visible actions on flag varieties, and triunity"Acta Applicandae Mathematicae. 180. 129-146 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] M.Fujii, H.Ochiai: "An algorithm for solving linear ordinary differential equations of Fuchsian type with regular singular points"Interdisciplinary Information Sciences. 9. 189-200 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Nobukazu Shimeno: "Boundary value problems for various boundaries of Hermitian symmetric spaces"J.of Funct.Anal.. 170. 265-285 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Hideko Sekiguchi: "Combinatorial formula of the dimension of global solutions to a generalized hypergeometric system M32(μ)"Japanese Journal of Math.. 27. 311-326 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Hideko Sekiguchi: "The Penrose transform for Sp(n,R) and singular uninary representations"J.Math.Soc.Japan. 54. 216-253 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] A.Gyoja, K.Taniguchi, K.Nishiyama: "Invariants for Representations of Weyl Groups, Two-sided Cells, and Modular Representations of Iwahori-Hecke Algebras"Adv.Studies in Pure Math.. 28. 105-114 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Kenji Taniguchi: "On the symmetry of commuting differential operators with singularities along hyperplanes"Int.Math.Res.Notices. (to appear).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Toshihiko Matsuki: "Classification of two involutions on compact semisimple Lie groups and loot systems"J.of Lie Theory. 12. 41-68 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Simon Gindikin, T.Matsuki: "Stein extensions of Riemann symmetric spaces and dualities of orbits on flag manifolds"preprint. 1-37

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Hiroyuki Ochiai: "Classification of commuting differential operators with two variables, Yang-Baxter systems, non-linear models and their applications"World Sci.. 183-187 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Kyo Nishiyama, H.Ochiai, K.Taniguchi: "Bernstein degree and associated cycles of Harish-Chandra modules"Asterisque. 273. 15-80 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] S.Kato, H.Ochiai: "The degree of orbits of multiplicity-free actions"Asterisque. 273. 139-158 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] N.Kurokawa, H.Ochiai, M.Wakayama: "Multiple trigonometry and zeta functions"J.Ramanujan Math.Soc.. 17. 101-113 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] 寺田 至: "ヤング図形のはなし"日本評論社. 202 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] T.Oshima: "Annihilators of generalized Verma modules of the scalar type for classical Lie algebras"UTMS 2001-29 (University of Tokyo). 29 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] T.Oshiina: "Harmonic analysis on semisimple symmetric spaces"Sugaku Exposition. 15. 151-170 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] S.Ben Said, T.Oshima, N.Shimeno: "Fatou's theorems and Hardy-type spaces for eigenfunctions of the invariant differential operators on symmetric spaces"Intern.Math.Research Notice. 16. 915-931 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] H.Ochiai, T.Oshima: "Commuting differential operators of type B_2"Funkcialaj Ekvacioj. 46. 297-336 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] T.Oshima: "A calculation of c-functions for semisimple symmetric spaces"Amer.Math.Soc.Transl., Advances in the Mathematical Sciences, Series 2. 210. 307-330 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] H.Oda, T.Oshima: "Minimal polynomials and annihilators of generalized Verma modules of the scalar type"UTMS 2004-3,(University of Tokyo). 56 (2004)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] T.Oshima: "A quantization of conjugacy classes of matrices"Advances in Math.. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] T.Hayata, H.Koseki, T.Oda: "Matrix coefficients of the P_j-principal series and the middle discrete series of SU(2, 2)"Adv.Studies in Pure Math. 26. 49-75 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] T.Oda, M.Tsuzuki: "Automorphic Green function associated with the secondary spherical functions"Publ.RIMS, Kyoto Univ.. 19. 451-533 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] H.Matumoto: "On the representations of Sp(p, q) and SO^*(2n) unitary induced from derived funcor modules"Composito Math.. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] T.Kobayashi: "Introduction to actions of discrete groups on pseudo-Riemannian homogeneous manifolds"Acta Appl.Math. 73. 113-131 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] T.Kobayashi: "Branching problems of unitary representations"Proc.International Congress of Mathematicians. Vol.II. 615-627 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] T.Kobayashi, S.Nasrin: "Multiplicity one theorem in the orbit method"Amer.Math.Soc.Transl., Advances in the Mathematical Sciences, Series 2. 210. 161-169 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] T.Kobayashi, B.Orsted: "Analysis on minimal representations of O(p, q), Part I"Advances in Mathematics. 180. 486-512 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] T.Kobayashi, B.Orsted: "Analysis on minimal representations of O(p, q), Part II"Advances in Mathematics. 180. 513-550 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] T.Kobayashi, B.Orsted: "Analysis on minimal representations of O(p, q), Part III"Advances in Mathematics. 180. 551-596 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] T.Kobayashi: "Geometry of multiplicity-free representations of GL(n), visible actions on fag varieties, and triunity"Acta Applicandae Mathematicae. 129-146 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] T.Matsuki: "Classification of two involutions on compact semisimple Lie groups and root systems"J.of Lie Theory. 12. 41-68 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] S.Gindikin, T.Matsuki: "Stein extensions of Riemann symmetric spaces and dualities oforbits on flag manifolds"(preprint). 37

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] K.Nishiyama, H.Ochiai, K.Taniguchi: "Bernstein degree and associated cycles of Harish-Chandra modules"Asterisque. 273. 13-80 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] S.Kato, H.Ochiai: "The degree of orbits of multiplicity-free actions"Asterisque. 273. 139-158 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] N.Kurokawa, H.Ochiai, M.Wakayama: "Multiple trigonometry and zeta functions"J.Ramanujan Math.Soc.. 17. 101-113 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] M.Fujii, H.Ochiai: "An algorithm for solving linear ordinary differential equations o f Fuchsian type with regular singular points"Interdisciplinary Information Sciences. 9. 189-200 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] N.Shimeno: "Boundary value problems for various boundaries of Hermitian symmetric spaces"J.of Funct.Anal.. 170. 265-285 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] H.Sekiguchi: "Combinatorial formula o f the dimension of global solutions to a generalized hypergeometric system M^^〜_<3,2>(υ)"Japanese Journal of Math.. 27. 311-326 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] H.Sekiguchi: "The Penrose transform for Sp(n, R) and singular uninary representations"J.Math.Soc. Japan. 54. 216-253 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] A.Gyoja, K.Nishiyama, K.Taniguchi: "Invariants for Representations of Weyl Groups, Two-sided Cells, and Modular Representations of Iwahori-Hecke Algebras"Adv.Studies in Pure Math.. 28. 105-114 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] K.Taniguchi: "On the symmetry of commuting differential operators with singularities along hyperplanes"Int.Math.Res.Notices.. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] I.Terada: "Young Diagrams"Nihon-Hyoron-sha. 202 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] T.Kobayashi: "Discontinuous groups for non-Riemannian homogeneous spaces, Mathematics Unlimited -2001, Beyond"Springer-Verlag, Berlin. 723-748 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] H.Ochiai: "Classification of commuting differential operators with two variables, Yang-Baxter systems, non-linear models and their applications"World Sci.. 183-187 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Toshio Oshima: "A calculation of c-functions for semisimple symmetric spaces"A.M.S.Translations. 210. 307-330 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] Hiroyuki Ochiai: "Commuting differential operators of type B_2"Funkcialaj Ekvacioj. 46. 297-336 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] Hiroshi Oda: "Minimal polynomials and annihilators of generalized Verma modules of the scalar type"UTMS. 3. 1-56 (2004)

    • Related Report
      2003 Annual Research Report
  • [Publications] Toshio Oshima: "A quantization of conjugacy classes of matrices"Advances in Math.. (2004)

    • Related Report
      2003 Annual Research Report
  • [Publications] Toshio Oshima: "Harmonic analysis on semisimple symmetric spaces"Sugaku Expositions. 15. 151-170 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] Salem Ben Said: "Fatou's theorems and Hardy-spaces for eigenfunctions of the invariant differential operators on symmetric spaces"Intern, Math. Research Notices. 16. 915-931 (2003)

    • Related Report
      2002 Annual Research Report
  • [Publications] Toshio Oshima: "A quantization of conjugacy classes of matrices"Advances in Math.. (in press). (2003)

    • Related Report
      2002 Annual Research Report
  • [Publications] Hiroyuki Ochiai: "Commuting differential operators of type B_2"Funkcialaj Ekvacioj. (in press). (2003)

    • Related Report
      2002 Annual Research Report
  • [Publications] Toshio Oshima: "Annihilators of generalized Verma modules of the scalar type for classical Lie algebras"UTMS. 29. 1-27 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Toshiyuki Kobayashi: "Discontinuous groups for non-Riemannian homogeneous spaces"Mathematics Unlimited-2001 and Beyond. 723-747 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Toshio Oshima: "A quantization of conjugacy classes of matrices"UTMS. 38. 1-15 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] Hiroyuki Ochiai: "Commuting differential operators of type B2"RIMS Kokyu-roku. 1171. 36-67 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] 小林俊行: "岩波講座 現代数学の基礎Lie群とLie環"岩波書店. 610 (2000)

    • Related Report
      2000 Annual Research Report

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Published: 2000-04-01   Modified: 2016-04-21  

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