Differential equations on homogeneous spaces

1. 我们将经典Capelli恒等式推广到小调情形，并证明了由这些恒等式定义的算子给出了表征GL(n)退化级数的微分方程组的生成子。此外，我们还研究了李代数矩阵幂形式的微分算子，以及刻画经典李群的任意简并级数的微分方程组的生成子。我们证明了这些方程刻画了函数空间在经典李群的各种边界上泊松变换的象。在实数半单李群g上定义了一般超几何函数。利用Capelli恒等式，研究了GL(n)的退化级数之间的缠结算子，证明了它们对应于实数Grassmannians上的Radon变换时，给出了Gelfand超几何函数的推广，并从表征论的角度给出了它们的自然解释。我们还用微分方程刻画了Radon变换在实格拉斯曼人身上的像。从G的闭子群H的表示出发，研究了G的诱导表示中包含的实数半单李群G的表示的多重性，得到了它在两边的良好估计，并得到了（G， H）多重性是有限或一致有界的一个充分必要几何条件。这里我们假设当H表示的维数为无穷大时H的可约性。例如，若G = U（p + 1, q）且H = U(p, q)，则在由H的任何不可约容许表示导出的表示中，G的任何全纯离散级数表示的多重性最多为1。在经典Weyl群作用下，构造了完全可积量子系统不变量的所有高积分。

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.