Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Stable Harmonic Analysis and Stable Transfer

Matthew Sunohara

TL;DR

The paper advances stable harmonic analysis in the local Langlands/Beyond Endoscopy framework by constructing stable transfer operators that realize local functorial transfers between spaces of stable orbital integrals. It develops and leverages stable Paley–Wiener theorems for Harish-Chandra Schwartz functions, test functions, and K-finite test functions, establishing that the stable Fourier transform is an isomorphism and that pullback along tempered L-embeddings preserves stability. A key novelty is the stable Paley–Wiener theorem for Schwartz functions and the weak-* density of stable tempered characters in stable tempered distributions for p-adic groups, conditional on a local Langlands hypothesis. The results yield a concrete, functorial transfer mechanism compatible with Beyond Endoscopy and provide explicit examples (tori and complex groups) illustrating the construction and spectral properties of stable transfer operators, with potential global applications in spectral stability and trace formula comparisons.

Abstract

Langlands posed the question of whether a local functorial transfer map of stable tempered characters can be interpolated by the transpose of a linear operator between spaces of stable orbital integrals of test functions. These so-called stable transfer operators are intended to serve as the main local ingredient in Beyond Endoscopy, his proposed strategy for proving the Principle of Functoriality. Working over a local field of characteristic zero and assuming a hypothesis on the local Langlands correspondence for p-adic groups, we prove the existence of continuous stable transfer operators between spaces of stable orbital integrals of Harish-Chandra Schwartz functions, test functions, and K-finite test functions. This is achieved via stable Paley--Wiener theorems for each of the three types of function spaces. The stable Paley--Wiener theorem for Harish-Chandra Schwartz functions is new and includes the result that stable tempered characters span a weak-* dense subspace of the space of stable tempered distributions, a result previously unknown for p-adic groups.

Stable Harmonic Analysis and Stable Transfer

TL;DR

The paper advances stable harmonic analysis in the local Langlands/Beyond Endoscopy framework by constructing stable transfer operators that realize local functorial transfers between spaces of stable orbital integrals. It develops and leverages stable Paley–Wiener theorems for Harish-Chandra Schwartz functions, test functions, and K-finite test functions, establishing that the stable Fourier transform is an isomorphism and that pullback along tempered L-embeddings preserves stability. A key novelty is the stable Paley–Wiener theorem for Schwartz functions and the weak-* density of stable tempered characters in stable tempered distributions for p-adic groups, conditional on a local Langlands hypothesis. The results yield a concrete, functorial transfer mechanism compatible with Beyond Endoscopy and provide explicit examples (tori and complex groups) illustrating the construction and spectral properties of stable transfer operators, with potential global applications in spectral stability and trace formula comparisons.

Abstract

Langlands posed the question of whether a local functorial transfer map of stable tempered characters can be interpolated by the transpose of a linear operator between spaces of stable orbital integrals of test functions. These so-called stable transfer operators are intended to serve as the main local ingredient in Beyond Endoscopy, his proposed strategy for proving the Principle of Functoriality. Working over a local field of characteristic zero and assuming a hypothesis on the local Langlands correspondence for p-adic groups, we prove the existence of continuous stable transfer operators between spaces of stable orbital integrals of Harish-Chandra Schwartz functions, test functions, and K-finite test functions. This is achieved via stable Paley--Wiener theorems for each of the three types of function spaces. The stable Paley--Wiener theorem for Harish-Chandra Schwartz functions is new and includes the result that stable tempered characters span a weak-* dense subspace of the space of stable tempered distributions, a result previously unknown for p-adic groups.
Paper Structure (53 sections, 39 theorems, 305 equations)

This paper contains 53 sections, 39 theorems, 305 equations.

Table of Contents

  1. Introduction
  2. Notation and conventions
  3. Parabolic and Levi subgroups
  4. Maximal compact subgroups
  5. Weyl groups
  6. Group norms
  7. Universal enveloping algebras
  8. Measures
  9. Spaces of functions and distributions
  10. Invariant Harmonic Analysis
  11. Invariant distributions
  12. Orbital integrals.
  13. Invariance.
  14. Representations and characters.
  15. Parabolic descent.
  16. ...and 38 more sections

Key Result

Theorem 1.1

If $F$ is non-archimedean, we assume hypothesis for $H$ and $G$. There exists a continuous linear operator whose transpose satisfies for all $\phi\in\Phi_\mathrm{temp}(H)$. Moreover, $\mathcal{T}_\xi$ restricts to continuous linear operators $\mathcal{S}_c(G)\to\mathcal{S}_c(H)$ and $\mathcal{S}_f(G)\to\mathcal{S}_f(H)$.

Theorems & Definitions (58)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 3.1
  • Theorem 3.2
  • Lemma 3.3
  • Corollary 3.4
  • Lemma 3.5
  • proof
  • Theorem 3.6
  • Theorem 3.7
  • ...and 48 more