| Project/Area Number |
12440034
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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 |
Basic analysis
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| Research Institution | The University of Tokyo |
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Principal Investigator |
OSHIMA Toshio The University of Tokyo, Graduate School of Mathematical Sciences, Professor, 大学院・数理科学研究科, 教授 (50011721)
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| Co-Investigator(Kenkyū-buntansha) |
TERADA Itaru The University of Tokyo, Graduate School of Mathematical Sciences, Associate Professor, 大学院・数理科学研究科, 助教授 (70180081)
MATUMOTO Hisayosi The University of Tokyo, Graduate School of Mathematical Sciences, Associate Professor, 大学院・数理科学研究科, 助教授 (50272597)
ODA Takayuki The University of Tokyo, Graduate School of Mathematical Sciences, Professor, 大学院・数理科学研究科, 教授 (10109415)
KOBAYASHI Toshiyuki Kyoto University, Research Institute of Mathematical Sciences, Professor, 数理解析研究所, 教授 (80201490)
SEKIGUCHI Hideko The University of Tokyo, Graduate School of Mathematical Sciences, Associate Professor, 大学院・数理科学研究科, 助教授 (50281134)
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| Project Period (FY) |
2000 – 2003
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| Keywords | symmetric space / Poisson transform / Verma module / Hardy space / completely integrable system / Radon transform / unversal enveloping algebra / primitive ideal |
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| Research Abstract |
(1)The annihilator of the generalized Verma module of the scalar type for a reductive Lie algebra is invesitigated. The condition that the annihilator gives the gap between the generalized Verma module and the usual Verma module is clearified and its necessary and sufficient condition is shown when its highest weight is dominant. Moreover a good sufficient condition is obtained in general.
(2)By the dual map of the faithful finite dimensional representation of the reductive Lie algebra and its infinite dimensional representation, the characteristic polynomials and minimal polynomials of matrices are defined and the generator system of the annihilators of generalized Verma modules of scalar type is constracted., A sufficient condition that the ideal gives the gap is applied to some problems in the integral geometry.
(3)Fatou's theorems and Hardy-type spaces on a Riemannian symmetric space of the noncompact type for the general eigenvalue of the invariant differential operators are studied in general. These theorems can be localized only when the rank of the space is equals to one.
(4)On a compactification of a Euclidean space defined by a root lattice of the classical type, Shrodinger operators are classified which allow commuting differential operators of the forth order under the condition that their potentials are meromorphic at an infinite point.
(5)The totality of k-dimensional linear subspaces of n-dimensional spaces over R, C or H is the Grassmannian manifold. The Radon transforms on these Grassmannian manifolds naturally extend the Gelfand-Aomoto's theory of generalized hypergeometric functions. Twisted Radon transforms over totally geodesic submanifolds are studied and interesting examples are given whose images are characterized by differential equations.
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