Systems of differential equations with group actions and their applications

1.给出了与经典根系有关的完全可积量子系统分类的一个猜想，并在适当的条件下得到了证明。特别是分类是完整的，如果系统有一个规则的奇点在无穷远点，这是最重要的情况。给出了相应于可积Schrodinger算子的高阶算子，并证明了其完全可积性。明确了系统之间的关系.利用线性代数中的初等因子的量化和极小多项式的量化两种方法构造了约化李代数的标量型广义Verma模的零化子的生成元.这些对应于Capelli恒等式和华算子的推广。这些也给出了广义旗流形上退化级数表示的微分方程以及在积分几何中的一些应用，包括Radon变换和Poisson变换.给出了退化级数的Whittaker模型存在的条件，并在代数意义下和适度增长条件下计算了实现的重数。得到了实现中K-有限向量所满足的微分方程组，以及向量用经典Whittaker函数表示的条件.研究了一类比正则奇点更广的偏微分方程组的一般理论，并构造了它们的多值全纯解.对根系的子系统进行了分类，并对子系统之间的同态进行了分类.研究了Heckman-Opdam超几何系统的合流极限、对奇异集的限制和不同的真实的形式。证明了利用Heckman-Opdam超几何函数的这一极限可以得到中等增长的Whittaker向量。

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.