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A note on the Sauvageot density principle

Yugo Takanashi

TL;DR

This note identifies and remedies a gap in Sauvageot’s density principle by introducing a module-Stone–Weierstrass framework and a measure-theoretic analogue to distinguish points within fibers of the infinitesimal-character map. It proves a strengthened generic irreducibility result and develops two approximation theorems—one for fiberwise, locally compact settings and one for measure spaces—leading to a robust Sauvageot density principle. The authors then apply these tools to (i) extend Shin’s automorphic Plancherel density theorem and (ii) construct spherical cusp forms with prescribed local properties via globalization and Plancherel asymptotics. The results restore the validity of density arguments in the automorphic setting and enable new density-constrained constructions of cusp forms with targeted local and archimedean behavior.

Abstract

In this short note, we address a gap in the proof of Sauvageot's density principle, which was pointed out in a paper by Nelson-Venkatesh.

A note on the Sauvageot density principle

TL;DR

This note identifies and remedies a gap in Sauvageot’s density principle by introducing a module-Stone–Weierstrass framework and a measure-theoretic analogue to distinguish points within fibers of the infinitesimal-character map. It proves a strengthened generic irreducibility result and develops two approximation theorems—one for fiberwise, locally compact settings and one for measure spaces—leading to a robust Sauvageot density principle. The authors then apply these tools to (i) extend Shin’s automorphic Plancherel density theorem and (ii) construct spherical cusp forms with prescribed local properties via globalization and Plancherel asymptotics. The results restore the validity of density arguments in the automorphic setting and enable new density-constrained constructions of cusp forms with targeted local and archimedean behavior.

Abstract

In this short note, we address a gap in the proof of Sauvageot's density principle, which was pointed out in a paper by Nelson-Venkatesh.

Paper Structure

This paper contains 17 sections, 25 theorems, 73 equations.

Table of Contents

  1. Introduction
  2. Notation and assumptions
  3. Acknowledgment
  4. The space of infinitesimal characters
  5. Definition
  6. A ring of functions and hermitian involutions
  7. The space of induced representations
  8. On the generic irreducibility theorem
  9. The unitary dual and the Plancherel measure
  10. A variant of the Stone-Weierstrass theorem
  11. Approximation on locally compact Hausdorff spaces
  12. Approximation on measure spaces
  13. The Sauvageot density principle
  14. Applications
  15. Some lemmas
  16. ...and 2 more sections

Key Result

Theorem 2.2

We assume that $F$ is a $p$-adic field. Let $\pi$ be an irreducible representation of $G$.

Theorems & Definitions (68)

  • Definition 2.1
  • Theorem 2.2: Renard2010-p-adiques*VI 5.4, Théorème
  • Definition 2.4
  • Definition 2.5
  • Lemma 2.6
  • Definition 3.1
  • Lemma 3.2
  • proof
  • Remark 3.3
  • Lemma 3.5
  • ...and 58 more