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Differential equations on homogeneous spaces

Research Project

Project/Area Number 09440048
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Research Field 解析学
Research InstitutionUniversity of Tokyo

Principal Investigator

OSHIMA Toshio  Graduate school of Mathematical Sciences, Univ. of Tokyo, Professor, 大学院・数理科学研究科, 教授 (50011721)

Co-Investigator(Kenkyū-buntansha) TERADA Itaru  Graduate school of Mathematical Sciences, Univ. of Tokyo, Ass. Prof., 大学院・数理科学研究科, 助教授 (70180081)
MATSUMOTO Hisayoshi  Graduate school of Mathematical Sciences, Univ. of Tokyo, Ass. Prof., 大学院・数理科学研究科, 助教授 (50272597)
ODA Takayuki  Graduate school of Mathematical Sciences, Univ. of Tokyo, Professor, 大学院・数理科学研究科, 教授 (10109415)
SHIMENO Nobukazu  Graduate school of Mathematical Sciences, Univ. of Tokyo, Lecturer, 理学部, 講師 (60254140)
KOBAYASHI Toshiyuki  Graduate school of Mathematical Sciences, Univ. of Tokyo, Ass. Prof., 大学院・数理科学研究科, 助教授 (80201490)
落合 啓之  九州大学, 大学院・数理学研究科, 助教授 (90214163)
Project Period (FY) 1997 – 1999
Project Status Completed (Fiscal Year 1999)
Budget Amount *help
¥10,700,000 (Direct Cost: ¥10,700,000)
Fiscal Year 1999: ¥2,800,000 (Direct Cost: ¥2,800,000)
Fiscal Year 1998: ¥2,400,000 (Direct Cost: ¥2,400,000)
Fiscal Year 1997: ¥5,500,000 (Direct Cost: ¥5,500,000)
Keywordshomogenneous space / representation theory / invariant differential equations / spherical function / integrable system / Poisson transform / semisimple Lie group / hypergeometric function / 不変微分作用素 / 超幾何微分方程式 / 完全積分可能系 / 対称空間 / 超幾何関数
Research Abstract

1. We generalized the classical Capelli identities to the case of minors and proved that the operators defined by the identities give the generators of the system of differential equations which characterizes the degenerate series of GL(n). Moreover we studied differential operators in the form of poweres of matrices of Lie algebra and generators of the system of differential equations characterizing any degenerate series of classical Lie groups. We proved that the equations characterize the image of Poisson transforms of the space of functions on various boundaries of classical Lie groups.
2. We defined general hypergeometric function on a real semisimple Lie group G. Using this Capelli identities, we studied intertwining operators between degenerate series of GL(n) and proved that when they correspond to Radon transforms on real Grassmannians, they give generalization of Gelfand's hypergeometric functions and their natural interpretation form the vie point of representation theory. We also characterized the image of Radon transformation on real Grassmaninans in terms of differential equations.
3. We studied the multiplicity of the representation of a real semisimple Lie group G contained in a induced representation of G from a representation of a closed subgroup H of G. We obtained its good estimate from both sides and got a necessary and sufficient geometrical condition for (G, H) so that the multiplicity is finite or uniformly bounded. Here we assume the reductibity of H in the case when the dimension of the representation of H is infinite. For example, if G = U(p + 1, q) and H = U(p, q), the multiplicity of any holomorphic discrete series representation of G in the represenation induced from any irreducible admissible representaion of H is at most one.
4. We constructed all the higer integrals of the completely integrable quantum system invariant under the action of a classical Weyl group.

Report

(4 results)
  • 1999 Annual Research Report   Final Research Report Summary
  • 1998 Annual Research Report
  • 1997 Annual Research Report
  • Research Products

    (54 results)

All Other

All Publications (54 results)

  • [Publications] Toshio Oshima: "Complete integrable system with a symmetry in coordinates"Asian Mathematical Journal. 2. 935-956 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Toshiyuki kabayashi: "A vanishing theosem for modular symbols on locally symmetric spaces"Comm. Math. Helv.. 73. 45-70 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Takahiro Hayata: "Matrix coeffecients of the principal Pj-series and the middle discrete series of S"U(2,2)"Adv. Studies in Pure Math.. 26. 52-76 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Takayuki Oda: "An explicit integral representation of Whittaker functions for the sepresentation of the discrete series-thecase of Sp(2,R)-"J. Math. Kyoto Univ.. 37. 519-530 (1997)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Takuya Miyazaki: "Principal Series Whittaker functious on Sp(2,R), II"Tohoku Math. J.. 50. 243-260 (1997)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Toshiyuki Kobayashi: "Discrete decompasable restrictions of unitary sepresentations of reductive Lie groups"Adv. Studies in Pure Math.. 26. 101-128 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Toshiyuki Kobayashi: "Deformation of compact Clifford-Klein forms of indefirute-Ricmannian homogeneous manifolds"Math. Ann.. 310. 394-408 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Toshiyuki Kobayashi: "Discrete decompasability of the restriction of Aq(Y) with respect to reductive subgroups II"Math. Ann.. 147. 709-729 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Toshiyuki Kobayashi: "L<@D1p@>D1-analysis on homogeneous manifolds of reductive type and sepresentation theory"Proc. Japan Acad.. 73. 62-66 (1997)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Toshihiko Matsuki: "Double coset decompositions of reductive Lie groups arising from two involutions"J. of Algebra. 197. 49-91 (1997)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Nobukazu Shimeno: "Boundary value problems for the Shilov boundary of the Siegel upper half plans"Bull Okayama Univ. Sci.. 33. 53-60 (1997)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Nobukazu Shimeno: "Poisson transforms for some principal series representations of Sp(n,R)"Bull Okayama Univ. Sci.. 34. 29-39 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Nobukazu Shimeno: "Boundary value problems for various boundaries of Hermition symmetric spaces"J. of Funct. Anal. 170. 265-285 (200)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Hiroyuki Ochiai: "Classical character of commuting differential operators with two varialles"Proc. APCTP-NANKAI symp. On Yang-Baxter Systems. 183-187 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Kyo Nishiyama: "Bernstein degree of singular unitary highest weight representations of the mitaplectic group"Proc. Japan Acad.. 75. 9-11 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] 寺田至: "群論、岩波講座、現代数学の基礎"岩波書店. 293 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] 小林俊行: "Lie群とLie環、岩波講座、現代数学の基礎"岩波書店. 610 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] T. Oshima: "Com integrable system with a symmetry in coordinates"Asian Math-ematical Journal, the special 70th birthday issue of Prof. Sato. 2-4. 935-956 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] T. Kobayashi and T. Oda: "A vanishing theorem, for modular symbols on locally symmetric spaces"Comm. Math. Helv.. 73. 45-70 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] T. Hayata. H. Koseki and T. Oda: "Matrix coefficients of the principal PィイD2JィエD2-series and the middle discrete series of SU(2, 2)"Adv. Studies in Pure Math.. 26. 27 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] T. Oda: "An explicit integral representation of Wittaker functions for the representation of the discrete series - the case of Sp(2; R)-"J. Math. Kyoto Univ.. 37. 519-530 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] T. Miyazaki and T. Oda: "Principal series Wittaker functions of Sp(2; R), II"Tohoku Math. J.. 50. 243-260 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] T. Kobayashi: "Discrete decomposable restrictions of unitary representations of reductive Lie groups"Adv. Studies in Pure Math.. 26. 29 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] T. Kobayashi: "Deformation of compact Clifford-Klein forms of indefinite-Riemannian homogeneous manifolds"Math. Ann.. 310. 394-408 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] T. Kobayashi: "Discrete decomposability of the restriction of AィイD2qィエD2(λ) with respect to reductive subgroups II : Micro -local analysis and asymptotic K-support"Ann. Math.. 147. 709-729 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] T. Kobayashi: "LィイD1pィエD1-analysis on homogeneous manifolds of reductive type and representation theory"Proc. Japan Acad.. 73 Ser. A. 62-66 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] T. Matsuki: "Double coset decompositions of reductive Lie groups arising from two involutions"J. of Algebra. 197. 49-91 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] N. Shimeno: "Boundary value problems for the Shilov boundary of the Siegel upper half plane"Bull Okayama Univ. Sci.. 33. 53-60 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] N. Shimeno: "Poisson transforms for some principal series representations of Sp(n, R)"Bull Okayama Univ. Sci.. 34. 29-39 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 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
      1999 Final Research Report Summary
  • [Publications] H. Ochiai: ""Quotients of some prehomogeneous vector spaces", Journal of Algebra"192. 61-73 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] K. Nishiyama and H. Ochiai: "Bernstein degree of singular unitary highest weight representations of the metaplectic group"Proc. Japan Acad.. 75 Ser. A. 9-11 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] H. Ochiai: "Classification of commuting differential operators with two variables"Proceeding of APCTP-NANKAI symposium on Yang-Baxter systems, nonlinear models and their applications, World Scientific. 183-187 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] T. Kobayashi and T. Oshima: "Lie Groups and Lie Algebras"Iwanami Shoten. 293 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] T. Kobayashi and T. Oshima: "Lie Groups and Lie Algebras, Iwanami Kouza"Iwanami Shoten. 610 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Toshio Oshima: "Completely integrable systems with a symmentry in coordenates"Asian Math. Journal. 2-4. 935-956 (1998)

    • Related Report
      1999 Annual Research Report
  • [Publications] Toshiyuki Kobayashi: "Discrete decomposable restrictions of unitary representations of reductive Lie groups"Adv. Studies in Pure Math. 26. 97-125 (2000)

    • Related Report
      1999 Annual Research Report
  • [Publications] Takahiro Hayata: "Matrix coefficients of the Principal P_J-series and teh middle discrete series of SU(2,2)"Adv. Studies in Pure Math. 26. 48-74 (2000)

    • Related Report
      1999 Annual Research Report
  • [Publications] Kyo Nishiyama: "Bernstein degree of singular unitary heightst weight representations of the metaplectic group"Proc. Japan Acad.. SerA 75. 9-11 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] Shinichi Shimeno: "Boundary value problems for various boundaries of Hermitian Symmetric spaces"J. of Funct. Anal.. 170. 265-285 (2000)

    • Related Report
      1999 Annual Research Report
  • [Publications] 小林俊行: "半単純リー群のユニタリ表現の離散的分岐の理論とその展開"数学. 51-4. 337-356 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] 小林俊行: "岩波講座 現代数学の基礎「Lie群とLie環I,II」"岩波書店. 610 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] Toshio Oshima: "Completely integrable systems with a symmetry in coordinates" Asian J.Math. 2. 1-21 (1988)

    • Related Report
      1998 Annual Research Report
  • [Publications] Toshiyuki Kobayashi: "Deformation of compact Clifford-Klein forms of infinite-Riemannian homogeneous manifolds" Math.Ann.310. 394-408 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] Toshiyuki Kobayashi: "Conformal geometry and branchiay laws for unitary representations attached to minimal nilpotent orbits" C.R.Acad.Sci.Paris. 326. 925-930 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] Toshiyuki Kobayashi: "A vanishing theorem for modular symbols on lecally symmetric spaces" Comment.Math.Helv.73. 45-70 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] Hiroyuki Ochiai: "A p-adic property of Taylor series of exp (x+x^P/P)" Hokkaido Math.J.(1999)

    • Related Report
      1998 Annual Research Report
  • [Publications] Hiroyuki Ochiai: "Classification of commuting differential operators with two variables" Proceeding of APCTP-NANKAI symposium on Yang-Baxter systems.

    • Related Report
      1998 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, (1999)

    • Related Report
      1998 Annual Research Report
  • [Publications] Toshio Oshima: "Differentioal equations characterizing degenerate series representations of classical groups" Advanced Studies in Pure Mathmatics. 26. (1998)

    • Related Report
      1997 Annual Research Report
  • [Publications] Toshiyuki Kobayashi: "Deformation of compact Clifford-Klein forms of indetinite-toemannian homogeneous manifolds." Journal of Functional Analysis.(1998)

    • Related Report
      1997 Annual Research Report
  • [Publications] Toshiyuki Kobayashi: "L^p-analysis on hoogeneous maniflods of reductive type and representation theory." Proceedings of Japan Academy. 73. 62-66 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] Toshiyuki Kobayashi: "Discrete decemposalrility of the restrictions of A of (λ)with respect to reductive lie subgrerass II" Annals of Mathematics. (1998)

    • Related Report
      1997 Annual Research Report
  • [Publications] Toshiko Matsuki: "Double coret decompositions of reductive Lie groups arising from two involutions." Journal of Algebra. 197. 49-91 (1997)

    • Related Report
      1997 Annual Research Report

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Published: 1997-04-01   Modified: 2016-04-21  

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