Constructions and decompositions of induced representations of solvable Lie groups and their applications

可解李群的诱导表示的构造与分解及其应用

基本信息

批准号：
12640178
负责人：
INOUE Junko
金额：
$ 1.41万
依托单位：
Tottori University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2000
资助国家：
日本
起止时间：
2000 至 2002
项目状态：
已结题

项目摘要

Holomorphically induced representations of a Lie group are usually constructed starting from a real linear from f of the Lie algebra and a complex polarization at f. In this research, I investigated holomorphically-induced representations of solvable Lie groups from weak polarizations or general complex subalgebra n.First of all, let G be a connected and simply connected Lie group whose Lie algebra is a normal j-algebra. When f belongs to an open coadjoint G-orbit and n is a positive weak polarization at f, the holomorphically-induced representation of G is non-zero if some term o defined by the modular function is suitably chosen. It decomposes into a direct sum of irreducible representations, which is described by the orbit method. In the course of this research, I reviewed and checked again the term o above and the construction of intertwining operators using algebraic structures of normal j-algebras. I revised the paper of the results above, and it has been published.I investigated some cases for low-dimensional exponential groups G and weak polarizations or complex subalgebras n which are isotropic (not necessarily maximally isotropic) for f. In some cases, I actually obtained non-zero representations and decompositions of them. The descriptions of semi-invariant vectors, which are used in computations, essentially depend on each algebraic structure of Lie algebras. I will try to find better descriptions suitable for treating a general setting in further study.For irreducible representations of exponential groups, I also treated another problem to find "good" operators or "good" subspaces of representation spaces which are compatible with the Fourier transforms. I have tried to characterize "good" subspaces by using "smooth operators" introduced by Ludwig, and I plan to proceed with it in further research.

李群的全纯诱导表示通常是从李代数 f 的实线性和 f 处的复极化开始构建的。在这项研究中，我研究了弱极化或一般复子代数 n 中可解李群的全纯诱导表示。首先，令 G 为连通且单连通的李群，其李代数为正规 j 代数。当 f 属于开共交 G 轨道且 n 是 f 处的正弱极化时，如果适当选择由模函数定义的某些项 o，则 G 的全纯诱导表示不为零。它分解为不可约表示的直和，由轨道方法描述。在这项研究的过程中，我再次回顾并检查了上面的o项以及使用正规j代数的代数结构构建交织算子。我修改了上述结果的论文，并已发表。我研究了低维指数群 G 和弱极化或复子代数 n 的一些情况，它们对于 f 是各向同性（不一定是最大各向同性）。在某些情况下，我实际上获得了它们的非零表示和分解。计算中使用的半不变向量的描述本质上取决于李代数的每个代数结构。在进一步的研究中，我将尝试找到适合处理一般设置的更好的描述。对于指数群的不可约表示，我还处理了另一个问题，以找到与傅立叶变换兼容的表示空间的“好”运算符或“好”子空间。我尝试使用路德维希引入的“平滑算子”来表征“好的”子空间，并且我计划在进一步的研究中继续进行。