On a question related to a basic convergence theorem of Harish-Chandra

TL;DR

This paper provides elementary proofs of a convergence theorem related to Harish-Chandra's work on spherical functions for certain real rank groups, with applications discussed.

Contribution

It offers new elementary proofs of a key convergence theorem for specific groups, expanding understanding of Harish-Chandra's foundational results.

Findings

Convergence established for real rank one groups.

Convergence established for several real rank two groups.

Applications of the convergence theorem are explored.

Abstract

In his first 1958 paper on zonal spherical functions Harish-Chandra proved an extremely delicate convergence theorem which was basic to his subsequent definition of his Schwartz space and his theory of cusp forms. This paper gives elementary proofs that a related integral converges for for groups of real rank one, several groups of real rank 2 (including $SO(n,2), Sp_4(R)$ and $Sp_4(C)$), $GL(n,R)$ and $GL(n,C)$. In fact, a stronger result has been proved in <cite>raphael</cite>. Applications of the question are also studied.