Systems of differential equations invariant under an action of a group

带状球面函数是广义表示性质的重要函数。它们的径向分量在Weyl群作用下具有不变的完整微分方程组的特征。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群或这些不变量群的专门化下是不变的。赫克曼-奥普达姆微分方程组是具有三角位势的完全可积系统。在本课题中，我们研究了在经典Weyl群下所有完全可积系统不变的问题，并最终成功地对这类系统进行了完全分类。即证明了势函数可以用椭圆函数或其退化、三角函数或有理函数表示，并明确地确定了它们。通过显式构造高阶积分，证明了系统的完全可积性。在椭圆势的情况下，它们的完全可积性是一个猜想。Cherednik还证明了势对应于根系时的可积性。这些系统被认为是Huen常微分方程到偏微分方程的推广。现在，我们计划在参数取一些特殊值的情况下，对系统及其解进行详细的研究，这些问题与表示理论或其他领域的重要问题有关。

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.