1994 Fiscal Year Final Research Report Summary
Systems of differential equations invariant under an action of a group
| Project/Area Number |
05452010
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| Research Category |
Grant-in-Aid for General Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Research Field |
解析学
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| Research Institution | Graduate School of Mathematical Sciences, University of Tokyo |
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Principal Investigator |
OSHIMA Toshio 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) |
OSADA Hirofumi University of Tokyo, Graduate School of Mathematical Sciences, associate profess, 大学院・数理科学研究科, 助教授 (20177207)
KATAOKA Kiyooki University of Tokyo, Graduate School of Mathematical Sciences, professor, 大学院・数理科学研究科, 教授 (60107688)
KUSUOKA Shigeo University of Tokyo, Graduate School of Mathematical Sciences, professor, 大学院・数理科学研究科, 教授 (00114463)
KAWAMATA Yujiro University of Tokyo, Graduate School of Mathematical Sciences, professor, 大学院・数理科学研究科, 教授 (90126037)
KOMATSU Hikosaburo University of Tokyo, Graduate School of Mathematical Sciences, professor, 大学院・数理科学研究科, 教授 (40011473)
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| Project Period (FY) |
1993 – 1994
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| Keywords | completely integrable systems / invariant differential operators / hypergeometric functions / spherical functions / unitary representations / seimsimple Lie groups / symmetric spaces |
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| Research Abstract |
The zonal spherical funtions are important functions generalized the characters of representations. Their radial components are characterized by the holonomic systems of differential equations invariant under the action of the Weyl group. Heckman-Opdam generalized the discrete parameters in the system to continuous ones.
On the other hand, known completely integrable quantum systems are invariant under a Weyl group or a Coxter groups or specializations of such invariant ones. Heckman-Opdam's system of differential equations are completely integrable systems with trigonometric potentials.
In this research project we attacked the problem to get all the completely integrable systems invariant under the classical Weyl group and we finally succeeded in the complete classification of such systems. Namely, we proved that the potential functions are expressed by elliptic functions or its degeneration, trigonometric functions or rational functions and determined them explicitely.
Moreover we proved the complete integrability of the systems by the explicit construction of integrals of the higher order. Their complete integrability had been a conjecture in the case of elliptic potential. Cherednik also proved the integrability when the potentials are corresponding to a root system after our results were obtained.
These system are considered to be a generalization of Huen's ordinary differential equation to partial differential equations. Now we are planning to study the systems and their solutions in detail when the parameters take some special values related to important ploblems in representation theory or other fields.
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