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An inequality of Harish-Chandra

Nolan R Wallach

TL;DR

This note provides an elementary proof of the Harish-Chandra inequality for real reductive groups, clarifying its historical tie to the $Tarski$-$Seidenberg$ framework and correcting misprints in Harmonic Analysis on Real Reductive Groups. By establishing the parabolic-structure setup and a graded-Lie-algebra analysis, it reduces the inequality to a concrete growth bound for a structured map via Lemma 3.1, yielding $a_{ar{P}}(n)^{\rho_P}\ge C(1+\|\log n\|^2)^m$ with constants $C,m>0$. The main technical engine is a general growth lemma for maps $\varphi:W\to V$ with graded components, proven by induction and injectivity of the diagonal blocks. An appendix compiles parabolic-subgroup basics, keeping the results self-contained for applications to the convergence of parameter-dependent integrals in the WPT setting and broader harmonic analysis on real reductive groups.

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.

An inequality of Harish-Chandra

TL;DR

This note provides an elementary proof of the Harish-Chandra inequality for real reductive groups, clarifying its historical tie to the - framework and correcting misprints in Harmonic Analysis on Real Reductive Groups. By establishing the parabolic-structure setup and a graded-Lie-algebra analysis, it reduces the inequality to a concrete growth bound for a structured map via Lemma 3.1, yielding with constants . The main technical engine is a general growth lemma for maps with graded components, proven by induction and injectivity of the diagonal blocks. An appendix compiles parabolic-subgroup basics, keeping the results self-contained for applications to the convergence of parameter-dependent integrals in the WPT setting and broader harmonic analysis on real reductive groups.

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.

Paper Structure

This paper contains 4 sections, 4 theorems, 53 equations.

Table of Contents

  1. Introduction
  2. The set-up
  3. A key lemma
  4. Appendix: Basics on parabolic subgroups

Key Result

Proposition 1

There exist $C>0$ and $m>0$ such that if $X\in \mathfrak{n}_{P}$ then Notice that $\exp:\mathfrak{n}_{P}\rightarrow N_{P}$ is a diffeomrophism. Thus if $\log:N_{P}\rightarrow\mathfrak{n}_{P}$ is the invers map then the inequality reads

Theorems & Definitions (4)

  • Proposition 1
  • Lemma 2
  • Lemma 3
  • Lemma 4