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Systems of differential equations invariant under an action of a group

群作用下不变的微分方程组

基本信息

批准号：
05452010
负责人：
OSHIMA Toshio
金额：
$ 3.2万
依托单位：
Graduate School of Mathematical Sciences, University of Tokyo
依托单位国家：
日本
项目类别：
Grant-in-Aid for General Scientific Research (B)
财政年份：
1993
资助国家：
日本
起止时间：
1993 至 1994
项目状态：
已结题

项目摘要

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.

带状球函数是推广表示性质的重要函数。它们的径向分量由在Weyl群作用下不变的完整微分方程组来表征。Heckman-Opdam将系统中的离散参数推广为连续参数，另一方面，已知的完全可积量子系统在Weyl群或Coxter群下是不变的，或者在这类不变群的特殊化下是不变的。Heckman-Opdam微分方程组是一类具有三角势的完全可积系统，本研究课题对这类系统进行了研究，得到了所有在经典Weyl群下不变的完全可积系统，并成功地对这类系统进行了完全分类。即证明了系统的势函数可以用椭圆函数或其退化函数、三角函数或有理函数表示，并明确地确定了它们，同时通过高阶积分的显式构造，证明了系统的完全可积性。它们的完全可积性在椭圆势情形下一直是一个猜想。Cherednik在我们的结果之后，证明了当位势对应于一个根系时的可积性，这些根系被认为是Huen常微分方程到偏微分方程的推广。现在，我们正计划详细研究当参数取某些特殊值时的系统及其解，这些参数与表示论或其他领域中的重要问题有关。