# Differential equations and the algebra of confluent spherical functions on semisimple Lie groups

## TL;DR

This paper explores the algebraic structure of confluent spherical functions on semisimple Lie groups of real rank one and their connection to the Schwartz algebra of spherical functions, enhancing understanding of harmonic analysis on these groups.

## Contribution

It introduces the concept of confluent spherical functions on semisimple Lie groups and examines their algebraic properties and relationship with the Schwartz algebra.

## Key findings

- Confluent spherical functions form an algebra related to the Schwartz algebra.
- The properties of these functions are characterized within the context of semisimple Lie groups.
- The paper establishes a connection between confluent spherical functions and harmonic analysis on Lie groups.

## Abstract

We consider the notion of a confluent spherical function on a connected semisimple Lie group, $G,$ with finite center and of real rank $1,$ and discuss the properties and relationship of its algebra with the well-known Schwartz algebra of spherical functions on $G.$