Differential equations on homogeneous spaces

1.我们将经典的Capelli恒等式推广到子式的情形，并证明了由该恒等式定义的算子给出了刻画GL（n）的退化级数的微分方程组的生成元.此外，我们还研究了以李代数矩阵幂形式表示的微分算子和刻画经典李群退化级数的微分方程组的生成元。证明了这些方程刻画了经典李群各种边界上函数空间的泊松变换象.在真实的半单李群G上定义了一般超几何函数。利用这些Capelli恒等式，我们研究了GL（n）的退化级数之间的交织算子，证明了当它们对应于真实的Grassmannian上的Radon变换时，它们给出了Gelfand超几何函数的推广，并从表示论的观点给出了它们的自然解释.利用微分方程刻画了Radon变换在真实的Grassmaninans上的像.研究了包含在G的一个闭子群H的表示的一个诱导表示中的真实的半单李群G的表示的重数。我们从两个方面得到了它的好的估计，并得到了（G，H）的重数为有限或一致有界的充要几何条件。这里我们假设H的可约性是在H的表示的维数为无穷大的情况下。例如，若G = U（p + 1，q），H = U（p，q），则G的任一全纯离散级数表示在H的任一不可约容许表示导出的表示中的重数至多为1.构造了完全可积量子系统在经典Weyl群作用下的不变量的所有高阶积分。

1. We generalized the classical Capelli identities to the case of minors and proved that the operators defined by the identities give the generators of the system of differential equations which characterizes the degenerate series of GL(n). Moreover we studied differential operators in the form of poweres of matrices of Lie algebra and generators of the system of differential equations characterizing any degenerate series of classical Lie groups. We proved that the equations characterize the image of Poisson transforms of the space of functions on various boundaries of classical Lie groups.2. We defined general hypergeometric function on a real semisimple Lie group G. Using this Capelli identities, we studied intertwining operators between degenerate series of GL(n) and proved that when they correspond to Radon transforms on real Grassmannians, they give generalization of Gelfand's hypergeometric functions and their natural interpretation form the vie point of representation theory. We also characterized the image of Radon transformation on real Grassmaninans in terms of differential equations.3. We studied the multiplicity of the representation of a real semisimple Lie group G contained in a induced representation of G from a representation of a closed subgroup H of G. We obtained its good estimate from both sides and got a necessary and sufficient geometrical condition for (G, H) so that the multiplicity is finite or uniformly bounded. Here we assume the reductibity of H in the case when the dimension of the representation of H is infinite. For example, if G = U(p + 1, q) and H = U(p, q), the multiplicity of any holomorphic discrete series representation of G in the represenation induced from any irreducible admissible representaion of H is at most one.4. We constructed all the higer integrals of the completely integrable quantum system invariant under the action of a classical Weyl group.