2007 Fiscal Year Final Research Report Summary
Systems of differential equations with group actions and their applications
| Project/Area Number |
16340034
|
|---|
| Research Category |
Grant-in-Aid for Scientific Research (B)
|
| Allocation Type | Single-year Grants |
|---|
| Section | 一般 |
|---|
| Research Field |
Basic analysis
|
|---|
| Research Institution | The University of Tokyo |
|---|
Principal Investigator |
OSHIMA Toshio The University of Tokyo, Graduate School of Mathematical Sciences, Professor (50011721)
|
|---|
| Co-Investigator(Kenkyū-buntansha) |
ODA Takayuki The University of Tokyo, Graduate School of Mathematical Sciences, Professor (10109415)
KOBAYASHI Toshiyuki The University of Tokyo, Graduate School of Mathematical Sciences, Professor (80201490)
MATUMOTO Hisayosi The University of Tokyo, Graduate School of Mathematical Sciences, Associate Professor (50272597)
SEKIGUCHI Hideko The University of Tokyo, Graduate School of Mathematical Sciences, Associate Professor (50281134)
TERADA Itaru The University of Tokyo, Graduate School of Mathematical Sciences, Associate Professor (70180081)
|
|---|
| Project Period (FY) |
2004 – 2007
|
| Keywords | Differential equation / Representation theory / Completely integrable system / Hypergeometric function / Integral geometry / Regular singularities / Whittaker function / Lie group |
|---|
| Research Abstract |
1. A conjecture for the classification of completely integrable quantum systems related to classical root systems is given and it is proved under a suitable condition. In particular the classification is complete if the systems have a regular singularity at an infinite point, which are most important cases. Higher order operators corresponding to the integrable Schrodinger operators are explicitly given and the complete integrability is proved. The relation between the systems are cleared.
2. The generators of the annihilator of a generalized Verma module of a scalar type for reductive Lie algebra are constructed in two ways by quatization of elementary divisors and by that of minimal polynomials in linear algebra. These correspond to generalization of Capelli identity and Hua operators. These also give the differential equations for degenerate series representations on generalized flag manifolds and some applications to integral geometry including Radon and Poisson transformations.
3. The condition for the existence of Whittaker model for degenerate series is obtained and the multiplicity of the realization is calculated under algebraic sense and also under the moderate growth condition. The differential equations satisfied by K-finite vectors in the realization is also obtained and the condition that the vectors are expressed by classical Whittaker functions is obtained.
4. A general theory of systems of partial differential equations of a little wider class than those with regular singularities is studied and their multi-valued holomorphic solutions are constructed.
5. The subsystems of a root system are classified and the homomorphisms between subsystems are classified.
6. Confluent limits, restrictions to singular sets and different real forms of Heckman-Opdam hypergeometric systems are studied. It is proved that the Whittaker vector with the moderate growth is obtained by this limit of Heckman-Opdam hypergeometric function.
|
Research Products
(96 results)
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
[Presentation] ルート系の部分系の分類2006
Author(s)
大島利雄
Organizer
玉原表現論ワークショップ
Place of Presentation
玉原国際セミナーハウス
Year and Date
2006-10-07
Description
「研究成果報告書概要(和文)」より
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
[Book] 岩波書店2005
Author(s)
小林俊行, 大島利雄
Total Pages
610
Publisher
Lie群と表現論
Description
「研究成果報告書概要(和文)」より
-
-
