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

1.利用轨道方法，将指数可解李群的一个不可约酉性表示实现为连通子群在与R^n可微同胚的齐次空间的L^2-函数空间上的诱导表示，当群是幂零的时，C^∞向量空间与R^n的Schwartz函数空间重合，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。作为与法国梅茨大学的合作，我们研究了指数群G的不可约表示的C^∞向量，得到了如下结果：取一个包含G的李代数的导出理想的幂零理想，并选择一个与此理想相适应的实极化，我们在齐次空间上实现了L^2-函数空间上的表示。我们讨论了一个与幂零理想相关的由具有某种快速递减性质的函数组成的子空间，并证明了它可以嵌入为包含G的群的一个不可约表示的C^∞向量空间。利用这个空间，我们还刻画了有限秩G上的L^1-函数的傅里叶变换像所包含的函数空间。2.研究了广义全纯诱导表示，并对一些低维可解李群的例子给出了相应的半不变向量的刻画。它们的描述取决于特定的群组结构。

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.