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On the definition of stable transfer factors

Tian An Wong

TL;DR

The paper develops a general framework for stable transfer in the group-case, constructing stable geometric and spectral transfer factors for general quasisplit connected reductive groups over local fields under the refined local Langlands correspondence. It introduces mesoscopic data as a flexible, beyond-endoscopy apparatus, builds stable kernels S(δ,φ) and S′(δ′,φ′) and derives adjoint relations that connect stable orbital integrals and stable characters, yielding an explicit formula for the stable geometric transfer factor Θ_{ ilde{ξ}′}(δ′,δ). From these geometric ingredients, stable spectral transfer factors Θ_{ ilde{ξ}′}(φ′,φ) are defined, and, assuming surjectivity of a transfer map 𝔗^{𝔽} (unconditional in Archimedean settings and conditional non-Archimedean), show that geometric and spectral transfers agree, culminating in a proof of the stable transfer conjecture in the provided framework. The work also develops transfer spaces and surjectivity machinery, aiming toward the primitisation of the stable trace formula and a robust local foundation for beyond-endoscopy in the group setting.

Abstract

We construct stable geometric and spectral transfer factors for a general reductive group and develop some of their basic properties, assuming the refined local Langlands correspondence. Using our definition of stable geometric transfer factors, we show that the stable transfer conjecture for orbital integrals implies the stable transfer of characters and vice versa. The latter is also implied by local Langlands, and in particular this establishes archimedean stable geometric transfer. Finally, we show how the stable geometric transfer factors can be used to define stable spectral transfer factors.

On the definition of stable transfer factors

TL;DR

The paper develops a general framework for stable transfer in the group-case, constructing stable geometric and spectral transfer factors for general quasisplit connected reductive groups over local fields under the refined local Langlands correspondence. It introduces mesoscopic data as a flexible, beyond-endoscopy apparatus, builds stable kernels S(δ,φ) and S′(δ′,φ′) and derives adjoint relations that connect stable orbital integrals and stable characters, yielding an explicit formula for the stable geometric transfer factor Θ_{ ilde{ξ}′}(δ′,δ). From these geometric ingredients, stable spectral transfer factors Θ_{ ilde{ξ}′}(φ′,φ) are defined, and, assuming surjectivity of a transfer map 𝔗^{𝔽} (unconditional in Archimedean settings and conditional non-Archimedean), show that geometric and spectral transfers agree, culminating in a proof of the stable transfer conjecture in the provided framework. The work also develops transfer spaces and surjectivity machinery, aiming toward the primitisation of the stable trace formula and a robust local foundation for beyond-endoscopy in the group setting.

Abstract

We construct stable geometric and spectral transfer factors for a general reductive group and develop some of their basic properties, assuming the refined local Langlands correspondence. Using our definition of stable geometric transfer factors, we show that the stable transfer conjecture for orbital integrals implies the stable transfer of characters and vice versa. The latter is also implied by local Langlands, and in particular this establishes archimedean stable geometric transfer. Finally, we show how the stable geometric transfer factors can be used to define stable spectral transfer factors.
Paper Structure (37 sections, 18 theorems, 237 equations)

This paper contains 37 sections, 18 theorems, 237 equations.

Table of Contents

  1. Introduction
  2. The stable transfer of distributions
  3. This paper
  4. Stable geometric transfer factors
  5. Stable spectral transfer factors
  6. Mesoscopic data: A simplified case
  7. Preliminaries
  8. $K$-groups
  9. Stable conjugacy classes
  10. Mesoscopic datum
  11. Images of semisimple elements
  12. Auxiliary data
  13. Stable kernels and adjoint relations
  14. Stable orbital integrals
  15. Conjugacy classes again
  16. ...and 22 more sections

Key Result

Lemma 3.1

Let $F$ be an archimedean local field. Then there exist smooth functions $S(\phi,\delta)$ and $S(\delta,\phi)$ of $\phi\in\Phi(G,\zeta)$ and $\delta\in\Delta(G)$, which are respectively $\zeta$ and $\zeta^{-1}$-equivariant under translation of $\delta$ by $Z(F)$, such that for any $f\in {{\mathcal{H}}}(G,\zeta)$.

Theorems & Definitions (45)

  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Proposition 4.1
  • proof
  • Remark 4.2
  • ...and 35 more