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Geometric harmonic analysis on homogeneous Siegel domains

Research Project

Project/Area Number 14340044
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Research Field Basic analysis
Research InstitutionKYUSHU UNIVERSITY (2005)
Kyoto University (2002-2004)

Principal Investigator

NOMURA Takaaki  Kyushu University, Faculty of Mathematics, Professor, 大学院・数理学研究院, 教授 (30135511)

Co-Investigator(Kenkyū-buntansha) UMEDA Toru  Kyoto University, Department of Mathematics, Associate Professor, 大学院・理学研究科, 助教授 (00176728)
KIKUCHI Katsuhiko  Kyoto University, Department of Mathematics, Research Associate, 大学院・理学研究科, 助手 (50283586)
ISHI Hideyuki  Yokohama City University, International College of Arts of Sciences, Research Associate, 国際総合科学部, 助手 (00326068)
Project Period (FY) 2002 – 2005
Project Status Completed (Fiscal Year 2005)
Budget Amount *help
¥9,100,000 (Direct Cost: ¥9,100,000)
Fiscal Year 2005: ¥2,000,000 (Direct Cost: ¥2,000,000)
Fiscal Year 2004: ¥2,200,000 (Direct Cost: ¥2,200,000)
Fiscal Year 2003: ¥2,300,000 (Direct Cost: ¥2,300,000)
Fiscal Year 2002: ¥2,600,000 (Direct Cost: ¥2,600,000)
KeywordsSiegel domain / Cayley transform / pseudoinverse map / normal j-algebra / homogenous cone / symmetric cone / tube domain / Jordan algebra / 等質錘 / 対称錘 / 分裂可解リー群 / ポアッソン核 / ラプラシアン / 有界等質領域 / 非可換調和解析
Research Abstract

The main research results are listed as follows.
1.We have introduced a family of Cayley transforms parametrized by admissible linear forms, and proved that every member maps the Siegel domain onto a bounded domain birationally and biholomorphically. This family contains the Cayley transform introduced by Penney in 1996 as well as the Cayley transform introduced by the head investigator in 2001. This result is already published as a research paper.
2.We have proved that the harmonicity of the Poisson-Hua kernel is equivalent with the symmetry of the domain. Moreover, even if we vary the Koehler metric of the Siegel domain in the standard way, a necessary and a sufficient condition for the Poisson-Hua kernel to be harmonic is that, in addition to the symmetry of the domain, the metric considered is a positive number multiple of the Bergman metric. This result, too, is already published as a research paper.
3.We have established that the analytically continued pseudoinverse map preserves th … More e tube domain if and only if the convex cone defining the tube domain is symmetric. In the actual theorem, the pseudoinverse map is equipped with a parameter, so that the theorem is a little more general. The result is published as a research paper (joint paper with a doctor course student Chifune Kai).
4.We investigated the convexity of the images of homogeneous tube domains under the Cayley transforms mentioned in 1 above. A necessary and sufficient condition for the images of the Cayley and the dual Cayley transforms to be convex is that the tube domains are symmetric. The actual theorem is proved in a little more general way by considering the family of Cayley transforms. The result is published as a research paper (joint with a doctor course student Chifune Kai).
5.We have found a series of non-symmetric open convex cones such that the degrees of the associated basic relative invariants are 1,2,...,r, where r is the rank of the open convex cone. Moreover in the course of joint research with Ishi we have shown that every basic relative invariant appears as an irreducible factor of the detgerminant of the right multiplication operator of the complexified clan. So far basic relative invaraats were obtained as a mere algorithmic resultant. We are currently writing a joint paper about this results. Less

Report

(5 results)
  • 2005 Annual Research Report   Final Research Report Summary
  • 2004 Annual Research Report
  • 2003 Annual Research Report
  • 2002 Annual Research Report
  • Research Products

    (34 results)

All 2005 Other

All Journal Article (21 results) Publications (13 results)

  • [Journal Article] A characterization of symmetric tube domains by convexity of Cayley transform images2005

    • Author(s)
      Chifune Kai, Takaaki Nomura
    • Journal Title

      Differential Geom. Appl. 23・1

      Pages: 38-54

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] 等質Siegel領域の対称性条件をめぐって2005

    • Author(s)
      野村隆昭
    • Journal Title

      数学 57・4

      Pages: 350-368

    • NAID

      10017177392

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Annual Research Report 2005 Final Research Report Summary
  • [Journal Article] The gradient maps associated to non-homogeneous cones2005

    • Author(s)
      Hideyuki Ishi
    • Journal Title

      Proc. Japan Acad. 81・3

      Pages: 44-46

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] On a certain 8-dimensional non-symmetric homogeneous convex cone2005

    • Author(s)
      Takaaki Nomura
    • Journal Title

      表現論シンポジウム報告集

      Pages: 109-114

    • NAID

      130008004802

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Annual Research Report 2005 Final Research Report Summary
  • [Journal Article] A characterization of symmetric tube domain by convexity of Cayley transform images2005

    • Author(s)
      Chifune Kai, Takaaki Nomura
    • Journal Title

      Differential Geom.Appl. 23-1

      Pages: 38-54

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Focusing on symmetry conditions for homogeneous Siegel domains (in Japanese)2005

    • Author(s)
      Takaaki Nomura
    • Journal Title

      Sugaku 57-4

      Pages: 350-368

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] The gradient maps associated to non-homogeneous cones2005

    • Author(s)
      Hideyuki Ishi
    • Journal Title

      Proc.Japan Acad. 81-3

      Pages: 44-46

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] On a certain 8-dimensional non-symmetric homogeneous convex cone2005

    • Author(s)
      Takaaki Nomura
    • Journal Title

      Reports on representation theory

      Pages: 109-114

    • NAID

      130008004802

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] A characterization of symmetric tube domains by convexity of Cayley transform images2005

    • Author(s)
      Chifune Kai, Takaaki Nomura
    • Journal Title

      Differential Geom.Appl. 23・1

      Pages: 38-54

    • Related Report
      2005 Annual Research Report
  • [Journal Article] The gradient maps associated to non-homogeneous cones2005

    • Author(s)
      Hideyuki Ishi
    • Journal Title

      Proc.Japan Acad. 81・3

      Pages: 44-46

    • Related Report
      2005 Annual Research Report
  • [Journal Article] A characterization of symmetric cones through pseudoinverse maps2005

    • Author(s)
      Chifune Kai, Takaaki Nomura
    • Journal Title

      J.Mats.Soc.Japan 57・1

      Pages: 195-215

    • NAID

      10016419774

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Cayley変換像の凸性による対称管状領域の特徴付け2005

    • Author(s)
      甲斐千舟, 野村隆昭
    • Journal Title

      数理解析研究所講究録 1410

      Pages: 113-123

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Gelfand対の分類について2005

    • Author(s)
      菊地克彦
    • Journal Title

      数理解析研究所講究録 1410

      Pages: 9-29

    • Related Report
      2004 Annual Research Report
  • [Journal Article] On symplectic representations of normal j-algebras and their application to Xu's realizations of Siegel domains

    • Author(s)
      Hideyuki Ishi
    • Journal Title

      Differential Geom. Appl. (To appear in)(accepted)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Wavelet transform for semidirect product groups with not necessarily commutative normal subgroups

    • Author(s)
      Hideyuki Ishi
    • Journal Title

      J. Fourier Anal, Appl. (To appear in)(accepted)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] On symplectic representations of normal j-algebras and their application to Xu's realizations of Siegel domains

    • Author(s)
      Hideyuki Ishi
    • Journal Title

      Differential Geom.Appl. (To appear)(accepted)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Wavelet transform for semidirect product groups with not necessarily commutative normal subgroups

    • Author(s)
      Hideyuki Ishi
    • Journal Title

      J.Fourier Anal.Appl. (To appear)(accepted)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] On symplectic representations of normal j-algebras and their application to Xu's realizations of Siegel domains

    • Author(s)
      Hideyuki Ishi
    • Journal Title

      Differential Geom.Appl. (To appear)(accepted)

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Wavelet transforms for semidirect product groups with not necessarily commutative normal subgroups

    • Author(s)
      Hideyuki Ishi
    • Journal Title

      J.Fourier Anal.Appl. (To appear)(accepted)

    • Related Report
      2005 Annual Research Report
  • [Journal Article] A characterization of symmetric tube domains by convexity of Cayley transform images

    • Author(s)
      Chifune Kai, Takaaki Nomura
    • Journal Title

      Diff.Geom.Appl. (発表予定)(To appear in)

    • Related Report
      2004 Annual Research Report
  • [Journal Article] The gradient maps associated to non-homogenous cones

    • Author(s)
      Hideyuki Ishi
    • Journal Title

      Proc.Japan Acad. (発表予定)(To appear in)

    • Related Report
      2004 Annual Research Report
  • [Publications] Takaaki Nomura: "Geometric norm equality related to the harmonicity of the Poisson kernel for homogeneous Siegel domains"J.Funct.Anal.. 198. 229-267 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] Takaaki Nomura: "Family of Cayley transforms of a homogeneous Siegel domain parametrized by admissible"Diff.Geom.Appl.. 18. 55-78 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] 甲斐千舟, 野村隆昭: "擬逆元写像による対称錐の特徴付け"数理解析研究所講究録. 1348. 150-162 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] Toru Umeda: "Application of Koszul complex to Wronski relations for U(gl(n))"Comm.Math.Helv.. 78. 663-680 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] 伊師英之: "Heisenberg群上の不変CR-Laplacian型作用素に関する固有空間について"数理解析研究所講究録. 1348. 23-32 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] 野村隆昭: "Berezin核をめぐって"数理解析研究所講究録. 1352. 24-24 (2004)

    • Related Report
      2003 Annual Research Report
  • [Publications] Takaaki Nomura: "A symmetry characterization for homogeneous Siegel domains related to Berezin trasnforms"Banach Center Publ.. 55. 323-334 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] Takaaki Nomura: "Cayley transforms and symmetry conditions for homogeneous Siegel domains"Adv. Study Pure Math.. 37. 253-265 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] 野村隆昭: "等質Siegel領域のPoisson核とCayley変換の幾何学的関係"数理解析研究所講究録. 1245. 73-91 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] Toru Umeda: "Application of Koszul complex to Wronski relations for U(gl(n))"Comm.Math.Helv.. (To appear)(印刷中).

    • Related Report
      2002 Annual Research Report
  • [Publications] 伊師英之: "等質Siegel領域のShilov境界上の調和解析"数理解析研究所講究録. 1245. 59-72 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] Hideyuki Ishi: "Decomposition of the L^2 function space on the Shilov boundary of a homogeneous Siegel domain"上智大学数学講究録. 45. 121-136 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] 梅田亨, 黒川信重, 若山正人, 砂田利一: "ゼータ研究所だより"日本評論社. 225 (2002)

    • Related Report
      2002 Annual Research Report

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Published: 2002-04-01   Modified: 2016-04-21  

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