Complex crowns of Riemannian symmetric spaces and non-compactly causal symmetric spaces

冠域边界处$K$-有限向量轨道映射的全纯延拓的多项式增长性

Polynomial growth of holomorphic extensions of orbit maps of $K$-finite vectors at the boundary of the crown

本研究将Krötz-Stanton延拓定理推广到任意连通半单李群，证明了$K$-有限向量轨道映射全纯延拓在冠域边界的多项式增长性，且其边值存在于分布向量空间中。

Representation theory, Radon transform and the heat equation on a Riemannian symmetric space

Let X=G/K be a Riemannian symmetric space of the noncompact type. We give a short exposition of the representation theory related to X, and discuss its holomorphic extension to the complex crown, a G-invariant subdomain in the complexified symmetric space X_\C=G_\C/K_\C. Applications to the heat transform and the Radon transform for X are given.

The image of the heat kernel transform on Riemannian symmetric spaces of the noncompact type

The heat kernel or Bargmann-Segal transform on a noncompact Riemannian symmetric space X=G/K maps a square integrable function on X to a holomorphic function on the complex crown. In this article we determine the range of this transform.

Holomorphic H-spherical distribution vectors in principal series representations

Let G/H be a semisimple symmetric space. The main tool to embed a principal series representation of G into L^2(G/H) are the H-invariant distribution vectors. If G/H is a non-compactly causal symmetric space, then G/H can be realized as a boundary component of the complex crown $Ξ$. In this article we construct a minimal G-invariant subdomain $Ξ_H$ of $Ξ$ with G/H as Shilov boundary. Let $π$ be a spherical principal series representation of G. We show that the space of H-invariant distribution vectors of $π$, which admit a holomorphic extension to $Ξ_H$, is one dimensional. Furthermore we give…

Paley-Wiener Theorem for Line Bundles over Compact Symmetric Spaces and New Estimates for the Heckman-Opdam Hypergeometric Functions

Paley-Wiener type theorems describe the image of a given space of functions, often compactly supported functions, under an integral transform, usually a Fourier transform on a group or homogeneous space. Several authors have studied Paley-Wiener type theorems for Euclidean spaces, Riemannian symmetric spaces of compact or non-compact type as well as affine Riemannian symmetric spaces. In this article we prove a Paley-Wiener theorem for homogeneous line bundles over a compact symmetric space $U/K$. The Paley-Wiener theorem characterizes f with sufficiently small support in terms of holomorphic …

Surjectivity of Convolution Operators on Noncompact Symmetric Spaces

Let $μ$ be a $K$-invariant compactly supported distribution on a noncompact Riemannian symmetric space $X=G/K$. If the spherical Fourier transform $\widetildeμ(λ)$ is slowly decreasing, it is known that the right convolution operator $c_μ\colon f\mapsto f*μ$ maps $\mathcal E(X)$ onto $\mathcal E(X)$. In this paper, we prove the converse of this result. We also prove that $c_μ$ has a fundamental solution if and only if $\widetildeμ(λ)$ is slowly decreasing.