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.

谎言基团的全态诱导表示形式通常是从lie代数的F的真实线性开始构建的，在f处构建复杂的极化。在这项研究中，我调查了来自弱极化或一般复杂的子代数n.fir的可溶剂谎言组的霍明型诱导的表示，让G为一个连接的，简单地连接的谎言组，其谎言代数为正常的J-代数。当F属于开放式连接G-orbit，而N是F处的正弱极化，如果适当选择模块化函数定义的某些项O定义的某些项O，则holomor形诱导的g表示为零。它分解为直接的不可约表示总和，这是通过轨道方法描述的。在这项研究的过程中，我使用正常J-Algebras的代数结构进行了审查并再次检查上述术语O和交织运算符的构建。我修改了上述结果的论文，并已发表。我研究了一些低维指数组G和弱极化或复杂的亚地骨N的病例，这些案例是f的各向同性（不一定是最大的）。在某些情况下，我实际上获得了它们的非零表示和分解。在计算中使用的半不变矢量的描述基本上取决于lie代数的每个代数结构。我将尝试找到适合在进一步研究中处理一般环境的更好的描述。对于指数式群体的不可约表示，我还将另一个问题处理，以找到与傅立叶变换兼容的代表空间的“良好”操作员或“良好”子空间。我试图通过使用路德维希（Ludwig）引入的“平滑运算符”来表征“良好”子空间，并计划在进一步的研究中继续进行。