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.

Zonal球形构造是重要功能，概括了表示形式的特征。它们的径向成分的特征在于在Weyl群的作用下，微分方程的全体系统系统不变。 Heckman-Opdam将系统中的离散参数概括为连续。另一方面，已知完全可以整合的量子系统在Weyl组或Coxter组或此类不变的群体下是不变的。 Heckman-Opdam的微分方程系统是具有三角电位的完全可集成的系统。在这项研究项目中，我们攻击了该问题，以使经典的Weyl组下的所有完全集成的系统不变，并最终成功地完成了此类系统的完整分类。也就是说，我们证明了潜在功能是通过椭圆函数或其退化，三角函数或有理功能表达的，并明确确定了它们。此外，我们通过显式构建了高阶的积分，证明了系统的完整性。在具有椭圆电位的情况下，它们的完整整合性是一种猜想。 Cherednik还证明了当电势与根系相对应的结果相对应时的整合性。这些系统被认为是Huen对部分微分方程的普通微分方程的概括。现在，当参数采用与表示理论或其他领域重要的综合有关的一些特殊值时，我们计划详细研究系统及其解决方案。