1999 Fiscal Year Final Research Report Summary
Differential operators of gradient type on symmetric spaces and representations of Lie algebras
| Project/Area Number |
09440002
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Algebra
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| Research Institution | Hokkaido University |
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Principal Investigator |
YAMASHITA Hiroshi Hokkaido Univ. Grad. School of Sci., Asso. Prof., 大学院・理学研究科, 助教授 (30192793)
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| Co-Investigator(Kenkyū-buntansha) |
SHIBUKAWA Youichi Hokkaido Univ. Grad. School of Sci., Asso. Prof., 大学院・理学研究科, 助手 (90241299)
SAITO Mutsumi Hokkaido Univ. Grad. School of Sci., Asso. Prof., 大学院・理学研究科, 助教授 (70215565)
YAMADA Hiro-fumi Hokkaido Univ. Grad. School of Sci., Asso. Prof., 大学院・理学研究科, 助教授 (40192794)
NISHIYAMA Kyo Kyoto Univ. Fac. Of Int. Hum. St., Asso. Prof., 総合人間学部, 助教授 (70183085)
HIRAI Takeshi Kyoto Univ. Grad. School of Sci., Prof., 大学院・理学研究科, 教授 (70025310)
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| Project Period (FY) |
1997 – 1999
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| Keywords | semisimple Lie group / Harish-Chandra module / nilpotent orbit / differential operator of gradient type / multiplicity / generalized Whittaker model / discrete series / highest weight representation |
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| Research Abstract |
The purpose of this project is to study the embeddings of irreducible Harish-Chandra modules into various induced representations of a semisimple Lie group, by using the invariant differential operators of gradient type on certain homogeneous vector bundles over the Riemannian symmetric space. The kernel of such a differential operator realizes the maximal globalization of the dual Harish-Chandra module, and the determination of the embeddings in question is reduced to specifying the equivariant functions in this kernel space.
First, the generalized Gelfand-Graev representations form a family of induced modules parametrized by the nilpotent orbits. Concerning the Harish-Chandra modules with highest weights for a simple Lie group of Hermitian type, the generalized Whittaker models associated with the holomorphic nilpotent orbits are specified. Namely, it is shown that each highest weight module embeds, with nonzero and finite multiplicity, into the generalized Gelfand-Graev representation attached to the unique open orbit in its associated variety. As for the unitary highest weight module, the space of the embeddings can be completely described in terms of the principal symbol of the differential operator of gradient type.
Second, we consider a simple Lie group of quaternionic type. The 0th n-homology spaces, or equivalently, the embeddings into the principal series, of the Borelde Siebenthal discrete series are described, by using the Schmid differential operator of gradient type. We find in particular that the n-homology space has exactly two exponents if the real rank of the group is not one.
Third, the relationship between the multiplicities in the associated cycles and the differential operators of gradient type are clarified for certain Harish-Chandra modules with irreducible associated varieties. The multiplicity can be written down by means of the principal symbol of a gradient type differential operator.
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