Harmonic analysis on solvable Lie groups associated with constructions of induced representations

1.根据轨道法，将指数可解李群的不可约酉表示化为连通子群在L^2-函数空间上的酉特征的诱导表示。当群为幂零时，C^∞向量的空间与R^n的Schwartz函数的空间重合，并且它符合对群的傅里叶变换的描述。然而，当群是一般指数时，C^∞向量空间就没有这样简单的描述了。与法国Metz大学的j.l uwig合作，研究了指数群G的C^∞向量的不可约表示，得到了以下结果：取一个包含G的李代数的派生理想的幂零理想，选择一个与该理想相适应的实极化，在齐次空间上实现了L^2函数在空间上的不可约表示。我们处理了一个由与幂零理想相关的速降性质的函数组成的子空间，并证明了它可以嵌入到含有G的群的不可约表示的C^∞向量空间中。利用该空间，我们还描述了有限秩G上L^1函数的傅里叶变换像中包含的函数的一定空间。在此基础上，给出了C^∞向量空间对某些特殊表示类型的具体描述。研究了低维可解李群的广义全纯诱导表示，并给出了相应的半不变向量。它们的描述依赖于特定的群结构。

1.According to the orbit method, an irreducible unitary representation of an exponential solvable Lie group is realized as an induced representation from a unitary character of a connected subgroup on a space of L^2-functions of the homogeneous space diffeomorphic with R^n. When the group is nilpotent, the space of C^∞ vectors coincides with the space of Schwartz functions of R^n, and it fits in with a description of the Fourier transform for the group. However, when the group is general exponential, the space of C^∞ vectors does not have such simple descriptions. As a collaboration with J.Ludwig (University of Metz, France), we investigated C^∞ vectors for irreducible representations of exponential groups G, and obtained the following results : Taking a nilpotent ideal which includes the derived ideal of the Lie algebra of G, and choosing a real polarization adapted to the ideal, we realize the representation on a space of L^2-functions on the homogeneous space. We treat a subspace consisting of functions with some rapidly decreasing property associated with the nilpotent ideal, and show that it can be embedded as the space of C^∞ vectors for an irreducible representation of a group containing G. Using the space, we also describe a certain space of functions included in the image of Fourier transforms of L^1-functions on G of finite ranks. Furthermore, specific descriptions of the space of C^∞ vectors for some special classes of representations are obtained.2.I investigated generalized holomorphically induced representations, and described associated semiinvariant vectors for some examples of low dimensional solvable Lie groups. Their descriptions depend on particular group structures.