Induced representations of solvable Lie groups and their applications

可解李群的归纳表示及其应用

• 批准号: 10640177
• 负责人: INOUE Junko
• 金额: $ 0.96万
• 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: 1998
• 资助国家: 日本
• 起止时间: 1998 至 1999
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-10640177/
• 关键词: solvable Lie group; induced representation; coadjoint orbit; polarization; holomorphically induced representation; solvable Lie group; induced representation; coadjoint orbit; polarization; holomorphically induced representation

I investigated holomorphically induced representations of solvable Lie groups G from real linear forms f of their Lie algebras and weak polarizations at f. A slightly modified holomorphic induction ρ from f and a positive weak polarization at f is non-zero when G is a connected and simply connected Lie group whose Lie algebra is a normal j-algebra, and f belongs to an open coadjoint orbit. In this case, the decomposition or ρ into irreducible representations can be described in terms of the orbit method, and a distributional Frobenius reciprocity holds. I tried to generalize this results of myself : I studied examples in low dimensional (general) exponential Lie groups. I also reviewed the proof of my previous result mentioned above, and modified some technical parts. I expect that for general exponential groups, holomorphic inductions from positive weak polarizations are described similarly in terms of the orbit method. I was also concerned with holomorphic inductions from complex subalgebras h which are isotropic (not necessarily maximally isotropic) for the bilinear form defined by f when G is nilpotent : I studied low dimensional nilpotent Lie groups. General cases are too complicated to treat, but when h+g(f)c, where g(f) is the Lie algebra of the stabilizer of f, is maximally isotropic, and representations appearing in the decomposition are corresponding to flat orbits, I could describe the decomposition using the orbit method. I will try to generalize it for some class of holomorphic inductions.For problems of "smooth operators" of a Hilbert space where an irreducible representation is realized, I mainly investigated examples in low dimensional exponential groups. For non-unimodular groups, we need to modify the definition of "smooth operators". I plan to find a good definition in order to use it in the theory of Fourier transforms in further research.

我研究了来自lie代数的真实线性形式F和f时极化弱极化的可溶剂lie基团G的全态诱导的表示。当G是一个连接的且简单地连接的谎言组时，F f f f f f f f f f f f f f f f f f f f f f f f f的较大较小触发ρ，而f的弱极化为非零。在这种情况下，我可以试图概括自己的结果：我研究了低维（一般）指数式谎言组的示例。我还回顾了上述结果的证明，并修改了一些技术零件。我预计对于一般指数群，从轨道方法方面类似地描述了来自正弱极化的全体形态诱导。我还关注复杂亚构象H的溶性诱导，这些诱导是在g具有nilpotent时F的双线性形式的各向同性（不一定是各向同性）的诱导：我研究了低维生型lie层。 General cases are too complicated to treat, but when h+g(f)c, where g(f) is the Lie algebra of the stabilizer of f, is maximum isotropic, and representations appearing in the decomposition are corresponding to flat orbits, I I will tr​​y to generalize it for some class of holomorphic inductions.For problems of "smooth operators" of a Hilbert space where an irreducible representation is realized, I mainly investigated低维指数组的示例。对于非潜在组，我们需要修改“平滑运算符”的定义。我计划找到一个良好的定义，以便在进一步的研究中使用傅立叶变换理论。