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

李群的全纯诱导表示通常是从李代数的f的实线性和f处的复极化开始构造的。本文研究了从弱极化或一般复子代数n.出发的可解李群的全纯诱导表示。首先，设G为连通和单连通李群，其李代数为正规j代数。当f属于开伴G轨道，且n在f处为正弱极化时，如果适当选择模函数定义的某项0，则G的全纯诱导表示是非零的。它分解成不可约表示的直接和，用轨道法描述。在这个研究过程中，我再次复习和检查了上面的0项和利用正规j代数的代数结构构造缠结算子。我对上述结果的论文进行了修改，并已发表。我研究了低维指数群G和f的弱极化或复杂子代数n的一些情况，它们是各向同性的（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。在计算中使用的半不变向量的描述，本质上依赖于李代数的每一个代数结构。在进一步的研究中，我将尝试找到更好的适合于处理一般设置的描述。对于指数群的不可约表示，我还处理了另一个问题，即寻找与傅里叶变换兼容的“好”算子或表示空间的“好”子空间。我已经尝试使用Ludwig引入的“光滑算子”来描述“好的”子空间，并且我计划在进一步的研究中继续进行。

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.