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Intertwining periods, L-functions and local-global principles for distinction of automorphic representations

Nadir Matringe, Omer Offen, Chang Yang

TL;DR

The paper establishes a local-global principle for distinguishing automorphic representations of GL over division algebras via three period types (Galois, linear, twisted linear). It fuses a novel local analysis of intertwining periods with Maass–Selberg relations to relate pole structures of global L-functions to the non-vanishing of periods, and extends the framework to inner forms in the Jacquet–Langlands setting. The main contributions include a comprehensive criterion linking local distinction, a local obstruction (compatibility), and poles of a product L-function $\mathcal{L}(s,\pi,\theta)$ across three symmetric-pair contexts, plus new instances of twisted linear period results and a partial resolution of the Guo–Jacquet conjecture. The techniques unify local and global harmonic analysis on symmetric spaces, leveraging parabolic orbit analyses, the geometric lemma, and transfer principles to derive precise pole-order equalities and compatibilities with functorial transfers. The findings have significant implications for understanding period integrals, L-value criteria, and functorial transfers in the relative Langlands program, especially for inner forms of general linear groups.

Abstract

We provide a criterion for non-vanishing of period integrals on automorphic representations of a general linear group over a division algebra. We consider three different periods: linear periods, twisted-linear periods and Galois periods. Our criterion is a local-global principle, which is stated in terms of local distinction, a further local obstruction, and poles of certain global L-functions associated to the underlying involution via the Jacquet-Langlands correspondence. Our local-global principle follows from a new method, relying on the Maass-Selberg relations and a careful analysis of singularities of local and global intertwining periods. Our results generalize to inner forms, known results for split general linear groups. Moreover, our result for twisted linear periods is new even in the split situation. As a consequence of our local-global principle, we complete the proof of one direction of the Guo-Jacquet conjecture.

Intertwining periods, L-functions and local-global principles for distinction of automorphic representations

TL;DR

The paper establishes a local-global principle for distinguishing automorphic representations of GL over division algebras via three period types (Galois, linear, twisted linear). It fuses a novel local analysis of intertwining periods with Maass–Selberg relations to relate pole structures of global L-functions to the non-vanishing of periods, and extends the framework to inner forms in the Jacquet–Langlands setting. The main contributions include a comprehensive criterion linking local distinction, a local obstruction (compatibility), and poles of a product L-function across three symmetric-pair contexts, plus new instances of twisted linear period results and a partial resolution of the Guo–Jacquet conjecture. The techniques unify local and global harmonic analysis on symmetric spaces, leveraging parabolic orbit analyses, the geometric lemma, and transfer principles to derive precise pole-order equalities and compatibilities with functorial transfers. The findings have significant implications for understanding period integrals, L-value criteria, and functorial transfers in the relative Langlands program, especially for inner forms of general linear groups.

Abstract

We provide a criterion for non-vanishing of period integrals on automorphic representations of a general linear group over a division algebra. We consider three different periods: linear periods, twisted-linear periods and Galois periods. Our criterion is a local-global principle, which is stated in terms of local distinction, a further local obstruction, and poles of certain global L-functions associated to the underlying involution via the Jacquet-Langlands correspondence. Our local-global principle follows from a new method, relying on the Maass-Selberg relations and a careful analysis of singularities of local and global intertwining periods. Our results generalize to inner forms, known results for split general linear groups. Moreover, our result for twisted linear periods is new even in the split situation. As a consequence of our local-global principle, we complete the proof of one direction of the Guo-Jacquet conjecture.

Paper Structure

This paper contains 60 sections, 66 theorems, 358 equations.

Table of Contents

  1. Introduction
  2. The main result: Galois periods
  3. The main result: Linear periods
  4. The main result: Twisted linear periods
  5. Our technique of proof and the main local result
  6. Notation and preliminaries
  7. Generalities on reductive groups
  8. Representations
  9. Parabolic induction
  10. Intertwining operators
  11. Symmetric pairs-generalities
  12. Parabolic orbits
  13. The geometric lemma
  14. Symmetric pairs for inner forms of $\operatorname{GL}$
  15. Explicit families of symmetric spaces for inner forms of $\operatorname{GL}$
  16. ...and 45 more sections

Key Result

Theorem 1.1

Let $\pi$ be an irreducible, cuspidal automorphic representation of $G(\mathbb{A})$ such that ${\mathrm {JL}}(\pi)$ is also cuspidal. If $d$ is odd then the following are equivalent: When these conditions are satisfied the pole of $L(s,{\mathrm {JL}}(\pi),{\mathrm {As}}^+)$ at $s=1$ is simple.

Theorems & Definitions (138)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Corollary 1.5
  • proof
  • Theorem 1.6
  • Theorem 1.7
  • Definition 2.1
  • Remark 2.2
  • ...and 128 more