Induced representations of solvable Lie groups and their applications

当G是李代数为正规j-代数的连通且单连通的李群，且f属于开余共轭轨道时，f的一个略微修正的全纯诱导ρ和f处的正弱极化是非零的。在这种情况下，分解或ρ成不可约的表示可以用轨道方法来描述，并且分布的弗罗贝尼乌斯互易成立。我试图推广我自己的结果：我研究了低维(一般)指数李群中的例子。我还回顾了前面提到的结果的证明，并对一些技术部分进行了修改。我期望对于一般指数群，正弱极化的全纯诱导可以类似地用轨道方法来描述。当G是幂零的时，我还研究了复子代数h的全纯诱导，其中复子代数h是各向同性的(不一定是最大各向同性的)，对于f定义的双线性形式：我研究了低维幂零李群。一般情况太复杂，难以处理，但当h+g(F)c(其中g(F)是f的稳定器的李代数)是最大各向同性的，并且在分解中出现的表示对应于平坦轨道时，我可以用轨道方法来描述分解。对于实现不可约表示的Hilbert空间的光滑算子问题，我主要研究了低维指数群中的例子。对于非么模群，我们需要修改“光滑算子”的定义。我计划找到一个好的定义，以便在进一步的研究中将其用于傅里叶变换理论。

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.