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Gamma factors and root numbers of pairs for the Galois and the linear model

Nadir Matringe

TL;DR

The paper generalizes Ok’s relative local converse theorem to square-integrable representations in the Galois and linear models for ${GL}_n$ over a non-Archimedean field, using harmonic analysis on Harish-Chandra Schwartz spaces of symmetric spaces. It develops an abstract Fourier inversion framework for symmetric spaces, proves triviality of central gamma and root numbers at $s=1/2$ for distinguished pairs, and establishes a relative converse theorem by extending Ok's Whittaker inversion and stability methods to the Schwartz setting. The results yield both local and global consequences, including new proofs of Prasad–Ramakrishnan-type conjectures and the taming of root numbers for distinguished representations, with an emphasis on tempered and square-integrable cases. The approach blends Bernstein–Plancherel inversion with stable Fourier-analytic techniques on unipotent radicals, providing a robust toolkit for relative converse problems and their Langlands-parameter reformulations.

Abstract

Using harmonic analysis on Harish-Chandra Schwartz spaces of various spherical spaces, we extend a relative local converse theorem of Youngbin Ok for the Galois model of p-adic GLn, from the class of cuspidal representations to that of square integrable representations, which is its optimal form. We also prove a variant of this result for linear models by the same method. The above statements are luckily non empty as we verify triviality results for gamma and epsilon factors of pairs of distinguished representations at the central value s=1/2. Along the way, we offer a new proof of conjectures of D. Prasad and D. Ramakrishnan on local components of symplectic cuspidal automorphic representations, and root numbers of pairs of symplectic representations.

Gamma factors and root numbers of pairs for the Galois and the linear model

TL;DR

The paper generalizes Ok’s relative local converse theorem to square-integrable representations in the Galois and linear models for over a non-Archimedean field, using harmonic analysis on Harish-Chandra Schwartz spaces of symmetric spaces. It develops an abstract Fourier inversion framework for symmetric spaces, proves triviality of central gamma and root numbers at for distinguished pairs, and establishes a relative converse theorem by extending Ok's Whittaker inversion and stability methods to the Schwartz setting. The results yield both local and global consequences, including new proofs of Prasad–Ramakrishnan-type conjectures and the taming of root numbers for distinguished representations, with an emphasis on tempered and square-integrable cases. The approach blends Bernstein–Plancherel inversion with stable Fourier-analytic techniques on unipotent radicals, providing a robust toolkit for relative converse problems and their Langlands-parameter reformulations.

Abstract

Using harmonic analysis on Harish-Chandra Schwartz spaces of various spherical spaces, we extend a relative local converse theorem of Youngbin Ok for the Galois model of p-adic GLn, from the class of cuspidal representations to that of square integrable representations, which is its optimal form. We also prove a variant of this result for linear models by the same method. The above statements are luckily non empty as we verify triviality results for gamma and epsilon factors of pairs of distinguished representations at the central value s=1/2. Along the way, we offer a new proof of conjectures of D. Prasad and D. Ramakrishnan on local components of symplectic cuspidal automorphic representations, and root numbers of pairs of symplectic representations.
Paper Structure (20 sections, 27 theorems, 134 equations)

This paper contains 20 sections, 27 theorems, 134 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Locally profinite groups
  4. Non Archimedean reductive groups
  5. Some properties of symmetric spaces
  6. Specific notations for symmetric pairs attached to GLn
  7. The Harish-Chandra Schwartz space of a symmetric space
  8. Representations
  9. Tempered symmetric pairs
  10. The Whittaker Harish-Chandra Schwartz space
  11. The abstract Fourier inversion formula for symmetric spaces
  12. Gamma factors and root numbers of pairs of distinguished representations, and a global application
  13. Partial Fourier coefficient of square integrable matrix coefficients
  14. Stable integrals of smooth functions on some unipotent groups
  15. Partial Fourier coefficients of smooth functions
  16. ...and 5 more sections

Key Result

Theorem 1

Let $a:=\frac{2}{[E:F]}\in \{1,2\}$, $\psi:E\to {\mathbb {C}}^\times$ be a non trivial character of $E$ trivial on $F$ when $[E:F]=2$, and $\pi$ be a square integrable representation of $G_{an}$ for $n\geq 2$. If the Rankin-Selberg central gamma value $\gamma(1/2,\pi,\pi',\psi)$ is equal to one for

Theorems & Definitions (52)

  • Theorem : RLC(n,n-1)
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • Corollary 2.3
  • Proposition 2.4
  • Proposition 2.5
  • Remark 2.6
  • Lemma 2.7
  • Theorem 3.1
  • ...and 42 more