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

Radon transformation on reductive symmetric spaces: support theorems

We introduce a class of Radon transforms for reductive symmetric spaces, including the horospherical transforms, and study some of their properties. In particular we obtain a support theorem that generalizes Helgason's support theorem for the horospherical transform on a Riemannian symmetric space.

Horospherical Cauchy-Radon transform on compact symmetric spaces

Harmonic analysis on noncompact Riemannian symmetric spaces is in a sense equivalent to the theory of the horospherical transform. There are no horospheres on compact symmetric spaces, but we define a complex version of horospherical transform which plays a similar role for the harmonic analysis on them.

The Radon transform and its dual for limits of symmetric spaces

The Radon transform and its dual are central objects in geometric analysis on Riemannian symmetric spaces of the noncompact type. In this article we study algebraic versions of those transforms on inductive limits of symmetric spaces. In particular, we show that normalized versions exists on some spaces of regular functions on the limit. We give a formula for the normalized transform using integral kernels and relate them to limits of double fibration transforms on spheres.

Hurwitz-Radon 数与半单李群的proper作用

Hurwitz-Radon numbers and proper actions of semisimple Lie groups

研究半单李群在伪黎曼对称空间上的proper作用，证明此类作用必须同构于$Spin(n,1)$，并确定Hurwitz-Radon数为最大允许$n$值。

Regularity of the Radon-Nikodym Derivative of a Convolution of Orbital Measures on Noncompact Symmetric Spaces

Let $G/K$ be a Riemannian symmetric space of noncompact type, and let $ν_{a_j}$, $j=1,...,r$ be some orbital measures on $G$ (see the definition below). The aim of this paper is to study the $L^{2}$-regularity (resp. $C^k$-smoothness) of the Radon-Nikodym derivative of the convolution $ν_{a_{1}}\ast...\astν_{a_{r}}$ with respect to a fixed left Haar measure $μ_G$ on $G$. As a consequence of a result of Ragozin, \cite{ragozin}, we prove that if $r \geq \, \max_{1\leq i \leq s}\dim {G_i}/K_i$, then $ν_{a_{1}}\ast...\astν_{a_{r}}$ is absolutely continuous with respect to $μ_G$, i.e., $d\big(ν_{a_…

Beurling's Theorem and characterization of heat kernel for Riemannian Symmetric spaces of noncompact type

We prove Beurling's theorem for rank 1 Riemmanian symmetric spaces and relate it to the characterization of the heat kernel of the symmetric space.