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1995 Fiscal Year Final Research Report Summary

Analytic and geometric studies of group representations and their applications

Research Project

Project/Area Number 06452010
Research Category

Grant-in-Aid for General Scientific Research (B)

Allocation TypeSingle-year Grants
Research Field 解析学
Research InstitutionKYOTO UNIVERSITY

Principal Investigator

NOMURA Takaaki  Kyoto Univ., Graduate School of Science, Associate Professor, 大学院・理学研究科, 助教授 (30135511)

Co-Investigator(Kenkyū-buntansha) TANIGUCHI Masahiko  Kyoto Univ., Graduate School of Science, Associate Professor, 大学院・理学研究科, 助教授 (50108974)
SHIGEKAWA Ichiro  Kyoto Univ., Graduate School of Science, Associate Professor, 大学院・理学研究科, 助教授 (00127234)
WATANABE Shinzo  Kyoto Univ., Graduate School of Science, Professor, 大学院・理学研究科, 教授 (90025297)
UMEDA Toru  Kyoto Univ., Graduate School of Science, Associate Professor, 大学院・理学研究科, 助教授 (00176728)
HIRAI Takeshi  Kyoto Univ., Graduate School of Science, Professor, 大学院・理学研究科, 教授 (70025310)
Project Period (FY) 1994 – 1995
KeywordsLie Group / Quantum Group / Jordan Algebra / Berezin Fransformation / Diffeomorphism Group / Multiplicity-free Action / Capelli Identity / Dual Pair
Research Abstract

The Head Investigator Nomura, studying algebraic systems by functional-analytic method, has investigated the Riemann-Hilbert manifold of idempotents (Grassmann manifolds) of Hilbert Jordan algebras and described geodesics, the Riemannian metric and the sectional curvature formula by using the Jordan-algebra structure, clarifying the contribution of the algebra. Spectral decomposition of Berezin transformation associated to the multiplicity-free compact Lie group actions has also been studied. Some of the results are already reported at the work-shop in Tottori and in E.Fujita's master thesis (Kyoto university).
Investigator Umeda has studied the invariant theory and the representation theory. A particular focus has been placed upon the research on the the Capelli identity associated to dual pairs under the theme "invariant theory based on quantum symmetry". A formula for the Capelli identity associated to the dual pair $O_-n, {/goth sp} _- {Zm} $ has been obtained. This explains some of the Capelli identities associated to multiplicity-free actions and also the central elements of the universal enveloping algebra of $ {/goth o} _-n} $. Though not an ultimately complete form, the results are reported in G.Ochial's master thesis (Kyoto University).
Investigator Hiral has treated the infinite symmetric groups and the diffeomorphism group of manifolds. Considering the infinite tensor product of natural representations, one finds that a certain infinite symmetric group acts as intertwining operators. It is shown that these two kinds of groups form a dual pair in a certain case and investigation has been done on the irreducible decomposition of this infinite tensor product through the dual pair.

  • Research Products

    (12 results)

All Other

All Publications (12 results)

  • [Publications] 野村 隆昭: "Grassmann manifold of a JH-algebra," Ann. Global Anal. Geom.12. 237-260 (1994)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] 梅田 亨: "A quantum dual pair \{$goth sl}_2,{$goth o}_n)\ and associated Capelli Identity" Lett. Math. Phys.34. 1-8 (1995)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] 平井 武: "Representations of diffeomorphism groups and the infinte symmetric group" Noncompact Lie Gr. and Some of Their Appl.225-237 (1994)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] 渡辺 信三: "Generalized arc-sine laws for one-dimensional diffusion processes and random walks" Proc. Symp. Pure Math.57. 157-172 (1995)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] 重川 一郎: "An example of regular \(r,p)\-capacity and essential self-adjointness of a diffusion operator in infinite dimensions" J. Math. Kyoto Univ.35. 639-651 (1995)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] 谷口 雅彦: "On the contraction of the Teichmuller metrics" J. Math. Kyoto Univ.35. 133-142 (1995)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] T.NOMURA: "Grassmann manifold of a JH-algebra" Ann.Global Anal.Geom.12. 237-260 (1994)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] T.UMEDA: "A quantum dual pair $ ({/goth sl} _-2, {/goth o} _-n) $ and the associated Capelli Identity" Lett.Math.Phys.34. 1-8 (1995)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] T.HIRAI: "Representations of diffeomorphism groups and the infinte symmetric group" Noncompact Lie Gr.and Some of Their Appl.225-237 (1994)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] S.WATANABE: "Generalized arc-sine laws for one-dimensional diffusion processes and random walks" Proc.Symp.Pure Math.57. 157-172 (1995)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] I.SHIGEKAWA: "An example of regular ^< (r, p) >-capacity and essential self-adjointness of a diffusion operator in infinite dimensions" J.Math.Kyoto Univ.35. 639-651 (1995)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] M.TANIGUCHI: "On the contraction of the Teichmuller metrics" J.Math.Kyoto Univ.35. 133-142 (1995)

    • Description
      「研究成果報告書概要(欧文)」より

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Published: 1997-03-04  

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