2000 Fiscal Year Final Research Report Summary
Representations of Lie groups
| Project/Area Number |
10640218
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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 |
Global analysis
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| Research Institution | Meijo University |
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Principal Investigator |
OKAMOTO Kiyosato Meijo University, Professor, 理工学部, 教授 (60028115)
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| Co-Investigator(Kenkyū-buntansha) |
SAITO Kimiaki Meijo University, Professor, 理工学部, 教授 (90195983)
OZAWA Tetsuya Meijo University, Professor, 理工学部, 教授 (20169288)
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| Project Period (FY) |
1998 – 2000
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| Keywords | unitary representations / homogeneous space / functional analysis / global analysis / Poisson integral on the classical domain / Cauchy integral on the classical domain / Eigenfuctions of Laplacian / Invariant differential operators |
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| Research Abstract |
Unitary representations of Lie groups are realized by the theory of Kirillov-Kostant using the symplectic structure on the adjoint orbits of Lie groups. The head investigator Okamoto worked with the investigator Ozawa about the symplectic structure.
The natural intertwining operator between the irreducible representation realized on the vector space of all smooth sections of homogeneous vector bundles on the boundary of classical domains and the representation realized on the vector space of all smooth sections of homogeneous vector bundles on the classical domains gives us the generalization of the Poisson integral. This generalized Poisson integral in cludes the Cauchy integral as a special case.
The most important fact here is that the invariant differential operator becomes the identity operator on the image of the intertwining operator. It follows that the generalized Poisson integral is an eigenfunction of invariant differential operators. In particular, if we consider the usual functions on the classical domains the results of Hua follows easily from this facts.
For the theory of automorphic functions on the classical domains, it is very important to generalize this to the case of vector bundle. One encounters, however, the crucial difficulty at once owing to the non commutativity of the operators.
In the course of computing the examples we found interesting formulas which contain the example given by Hua.
On the other hand, the head investigator Okamoto coorperated with investigator Saito about the integrability of integrals on the white noise which arises from the Feynman path integral for the infinite dimensional Lie groups.
The head investigator gave a talk at the symposium held at the research institute of Kyoto university.
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