An inequality of Harish-Chandra

TL;DR

This paper provides an elementary proof of a key inequality from Harish-Chandra's work on harmonic analysis, correcting previous misprints and simplifying the original proof involving the Tarski-Seidenberg Theorem.

Contribution

It offers a simplified, self-contained proof of Harish-Chandra's inequality, clarifying and correcting earlier inaccuracies in the original exposition.

Findings

Elementary proof of Harish-Chandra's inequality

Correction of misprints in the original text

Enhanced understanding of integral convergence in harmonic analysis

Abstract

In paper I of his masterpiece Harmonic Analysis on Real Reductive Groups, Harish-Chandra included an important inequality that is useful in proving that certain key integrals depending on a parameter converge for large values of the parameter. His proof involved the Tarski-Seidenberg Theorem. The purpose of this note is an elementary proof of the inequality which is an expansion of the idea in my Real Reductive Groups I. This exposition fixes several critical misprints in the original and can be considered to be an erratum for the book.