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

TL;DR

This paper explores the interplay between representation theory, the Radon transform, and the heat equation on noncompact Riemannian symmetric spaces, highlighting holomorphic extensions and applications to integral transforms.

Contribution

It provides a concise exposition of the representation theory of symmetric spaces and discusses their holomorphic extension, with applications to heat and Radon transforms.

Findings

Holomorphic extension of representation theory to the complex crown.

Applications of the heat transform on symmetric spaces.

Analysis of the Radon transform in this geometric context.

Abstract

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.