1998 Fiscal Year Final Research Report Summary
Study of applied representation theory in global analysis
| Project/Area Number |
09640105
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Geometry
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| Research Institution | Kyoto Institute of Technology |
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Principal Investigator |
TSUKAMOTO Chiaki Kyoto Inst.of Tech., Fac.of Textile Science, Associate Professor, 繊維学部, 助教授 (80155340)
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| Co-Investigator(Kenkyū-buntansha) |
YAGASAKI Tatsuhiko Kyoto Inst.of Tech., Fac.of Eng. and Design, Associate Professor, 工芸学部, 助教授 (40191077)
OKURA Hiroyuki Kyoto Inst.of Tech., Fac.of Eng. and Design, Associate Professor, 工芸学部, 助教授 (80135649)
MAITANI Fumio Kyoto Inst.of Tech., Fac.of Eng. and Design, Professor, 工芸学部, 教授 (10029340)
NAKAOKA Akira Kyoto Inst.of Tech., Fac.of Eng. and Design, Professor, 工芸学部, 教授 (90027920)
UCHIYAMA Jun Kyoto Inst.of Tech., Fac.of Textile Science, Professor, 繊維学部, 教授 (70025401)
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| Project Period (FY) |
1997 – 1998
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| Keywords | representaion theory / analytic torsion / spectral zeta funcion / spectral geometry |
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| Research Abstract |
We have investigated the analytic torsion of line bundles over compact Hermitian symmetric spaces. The analytic torsion is defined as the first derivative at zero of the sum of spectral zeta functions derived from the spectral data of Hodge Laplacians.
We first showed that the lodge Laplacians of compact Hermitian symmetric spaces C/H is written by the Casimir operator of the group G of automorphisms, the Casimir operator of the isotropy group K, and the action of K on the fiber. Since these operators act by scalar multiplication on each irreducible G-submodules of the space of sections, we can determine the spectrum of Hodge Lapacian by studying the branching law of C-modules as K-modules, in view of Frobenius' reciprocity law. The muliplicity of each eigen value is computed by Weyl's formula.
We computed the spectral zeta functions associated with the product of the hyperplane section line bundle over complex projective space. These type of spectral zeta functions had been treated by Carletti and Monti-Bragadin. We refined their results on the Dirichlet series and succeeded in computing the analytic torsions for low dimensional projective spaces.
The similar method used in the computation of spectral data arc also applicable to the study of integrability of conformal infinitesimal Zoll deformations.
The other results by investigators are shown in the references.
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