Induced representations of solvable Lie groups and their applications

可解李群的归纳表示及其应用

基本信息

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

项目摘要

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.

我根据李代数的实线性形式 f 和 f 处的弱极化研究了可解李群 G 的全纯诱导表示，当 G 是连通且 f 处的正弱极化时，稍微修改的全纯诱导 ρ 和 f 处的正弱极化非零。简单连接的李群，其李代数是正规的 j-代数，并且 f 属于开共轭轨道在这种情况下，将 ρ 分解为不可约。表示可以用轨道方法来描述，并且分布弗罗贝尼乌斯互易成立。我尝试推广自己的结果：我研究了低维（一般）指数李群的例子，我还回顾了我之前提到的结果的证明。上面，并修改了一些技术部分，我希望对于一般指数群，正弱极化的全纯归纳可以用轨道方法来类似地描述，我也关心复子代数 h 的全纯归纳。当 G 是幂零时，对于由 f 定义的双线性形式，它们是各向同性的（不一定是最大各向同性）：我研究了低维幂零李群，一般情况太复杂而无法处理，但是当 h+g(f)c 时，其中 g( f) 是 f 的稳定子的李代数，是最大各向同性的，分解中出现的表示对应于平坦轨道，我可以使用轨道方法来描述分解，我将尝试。将其推广到一类全纯归纳。对于实现不可约表示的希尔伯特空间的“平滑算子”问题，我主要研究了低维指数群中的例子。对于非幺模群，我们需要修改定义。我计划找到一个很好的定义，以便在进一步的研究中将其用于傅里叶变换理论。