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1998 Fiscal Year Final Research Report Summary

THE EXPLICIT EXPRESSION OF C FUNCTION ON SEMISIMPLE LIE GROUPS AND ITS APPLICATION

Research Project

Project/Area Number 09640190
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field 解析学
Research InstitutionHIROSHIMA UNIVERSITY

Principal Investigator

EGUCHI Masaaki  FAC.INT.and SCI.HIROSHIMA UNIVERSITY PROF., 総合科学部, 教授 (30037220)

Co-Investigator(Kenkyū-buntansha) YOSHIDA Kiyoshi  FAC.INT.and SCI.PROF., 総合科学部, 教授 (80033893)
FURUSHIMA Mikio  FAC.INT.and SCI.PROF., 総合科学部, 教授 (00165482)
SHIBATA Tetutaro  FAC.INT.and SCI.ASS.PROF., 総合科学部, 助教授 (90216010)
KONNO Hitoshi  FAC.INT.and SCI.ASS.PROF., 総合科学部, 助教授 (00291477)
KOIZUMI Shin  ONOMICHI JUNIOR COLL.PROF, 教授 (90205310)
Project Period (FY) 1997 – 1998
KeywordsC Function / Intertwining Operator / Representation Theory / Fourier Transform / Lie Group / Homogeneou Space / Hardy Theorem / Heisenberg-Weyl
Research Abstract

1. The Harish-Chandra C-function plays an essential role in harmonic analysis on semisimple Lie groups, because it closely relates to the Plancherel measure, the reducibility of the principal series representations and gives many information for the analysis. After a time, many peoples studied the Harish-Chandra C-function. However, even now, the explicit expressions of the Harish-Chandra C-functions are not known except for a few semisimple Lie groups and special cases.
In this research we succeeded to give the explicit formulae of the Harish-Chandra C-functions for Spin(m, 1) and SU(n, 1). By the product formula for the Harish-Chandra C-function, the problem of computing the Harish-Chandra C-functions of semisimple Lie groups of general rank is reduced to the real rank one case. For this reason, it is crucial to compute the Harish-Chandra C-function for Spin(n, 1) and SU(n, 1). The reason for restricting our attention to the cases Spin(n, 1) and SU(n, 1) is that no multiple irreducible unitary representations of M occur in any irreducible unitary representation of K.
2. In Euclidean space, various forms of the uncertainty principle between a function and its Fourier transform are known. The Hardy theorem asserts that if a measurable function f on R satisfies |f(x)| <less than or equal> C exp{-ax^2} and |f(y)| <less than or equal> C exp{-by^2} then f = O(a.e.) whenever ab> 1/4. This result is generalized to some semisimple Lie groups by A.Sitaram and M.Sundari, and M.Sunclari, M.0. Cowling and J.F.Price We get an analogue of the Hardy theorem for the Cartan motion group and also an L^p version of the Hardy theorem for the motion group.

  • Research Products

    (22 results)

All Other

All Publications (22 results)

  • [Publications] M.Eguchi: "The expressions of the Harish-Chandra C-functions of semisimple Lie groups Spin(n,1),SU(n,1)" J.Math.Soc.Japan. 51・4印刷中. (1999)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] M.Eguchi: "An analogue of the Hardy theorem for the Cartan motion group" Proc.Japan Academy. 74・10. 149-151 (1998)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] S.Koizumi: "The Gangolli estimate for the coefficients of the Harish-Chandra expansions of the Eisenstein integrals and the expressions of the Harish-Chandra C-functions" Mem.Integarated Arts and Sci.,Hiroshima Univ. 24. 141-144 (1998)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] H.Konno: "An Elliptic Algebra U_<q,p>(sl_2) and the Fusion RSOS Model" Communications in Mathematical Physics. 195. 373-403 (1998)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] M.Jimbo: "Quasi-Hopf Twistors for Elliptic Quantum Groups" Transformation Groups. 掲載予定. (1999)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] M.Jimbo: "Elliptic Algebra U_<q,p>(sl_2):Drinfeld Currents and Vertex Operators" Communications in Mathematical Physics. 199. 605-647 (1999)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] T.Shibata: "Two-parameter nonlinear Sturm-Liouville problems" Proc.Edinburgh Math.Soc.41. 225-245 (1998)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] T.Shibata: "Asymptotic profiles of variational eigenvalues of two parameter nonlinear Sturm-Liouville problems" Mathematical Methods in Applied Sciences. 21. 1619-1635 (1998)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] T.Shibata: "Nonlinear multiparameter Sturm-Liouville problems" Asymptotic Analysis. 18. 173-192 (1998)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] M.Furushima: "Non-projective compactifications of C^3(III):A remark on indices" Hiroshima Math.J.掲載予定. (1999)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Y.Mizutani: "Self-similar radial soluitons to a parabolic system modelling chemotaxis via vari-ational method" Hiroshima Mathematical Journal. 29. 145-160 (1999)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] M.Eguchi: "THE expressions of the Harish-Chandra C-functions of semisimple Lie groups Spin (n, 1), SU (n, 1)" J.Math.Soc.Japan. 51・4(to appear). (1999)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] M.Eguchi: "An analogue of the Hardy theorem for the Cartan motion group" Proc.Japan Academy. 74・10. 149-151 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] S.Koizumi: "The Gangolli estimate for the coefficints of the Harish-Chandra expansions of the Eisenstein intrgrals and the expressions of the Harish-Chandra C-functions" Men.Integarated Arts and Sci.Hi-roshima Univ. 24. 141-144 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] H.Konno: "An Elliptic Algebra U_<q, p> (sl_2) and the Fu-sion RSOS Model" Communications in Mathematical Physics. 195. 373-403 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] M.Jimbo, J.Shiraishi: "Quasi-Hopf Twistors for El-liptic Quantum Groups" Transformation Groups. (to appear). (1999)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] M.Jimbo, J.Shiraishi: "Elliptic Algebra U_<q, p> (Sl_2) : Drinfeld Currents and Vertex Operators" Communications in Mathematical Physics. 199. 605-647 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] T.Shibata: "Two-Parameter nonlinear Sturm-Liouville problems" Proc.Edinburgh Math.Soc.bf41. 225-245 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] T.Shibata: "Asymptotic profiles of variational eigen-values of two parmeter nonlinear Sturm-Liouville problems" Mathematical Methods in Applied Sciences. 21. 1619-1635 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] T.Shibata: "Nonlinear multiparameter Sturm-Liouville problems" Asymptotic Analysis. 18. 173-192 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] M.Furushima: "Non-projective compactifications of C^3 (II) (New Examples)" Kyushu J.Math.52. 149-162 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Y.Mizutani: "Self-similar radial soluitons to a parabolic system modelling chemotaxis via variational method" Hiroshima Mathemati-cal Journal. 29. 145-160 (1999)

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

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Published: 1999-12-08  

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