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Differential equations on homogeneous spaces

齐次空间上的微分方程

基本信息

批准号：
09440048
负责人：
OSHIMA Toshio
金额：
$ 6.85万
依托单位：
University of Tokyo
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (B)
财政年份：
1997
资助国家：
日本
起止时间：
1997 至 1999
项目状态：
已结题

项目摘要

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.

1. 我们将经典的 Capelli 恒等式推广到未成年人的情况，并证明由恒等式定义的算子给出了表征 GL(n) 简并级数的微分方程组的生成元。此外，我们还研究了李代数矩阵幂形式的微分算子和表征任意经典李群简并级数的微分方程组生成元。证明了该方程组刻画了经典李群各边界上函数空间泊松变换的图像。 2．我们在实半单李群 G 上定义了一般超几何函数。利用这个 Capelli 恒等式，我们研究了 GL(n) 简并级数之间的交织算子，并证明当它们对应于实格拉斯曼函数上的 Radon 变换时，它们给出了 Gelfand 超几何函数的推广及其自然解释，形成了表示论的观点。我们还用微分方程描述了真实Grassmaninans上Radon变换的图像。 3．我们研究了由 G 的闭子群 H 的表示导出的 G 的导出表示中包含的实半单李群 G 的表示的重数。我们从两边得到了它的良好估计，并得到了 (G, H) 的充分必要几何条件，使得重数是有限的或一致有界的。这里我们假设 H 的表示维数为无穷大时 H 的可还原性。例如，如果G = U(p + 1, q) 且H = U(p, q)，则G 的任何全纯离散级数表示在由H 的任何不可约容许表示导出的表示中的重数最多为1。4．我们在经典Weyl群的作用下构造了完全可积量子系统不变量的所有更高积分。