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On completeness of local intertwining periods

Hengfei Lu, Nadir Matringe

TL;DR

The paper develops a framework to describe invariant linear forms on induced representations in terms of intertwining periods for unimodular tempered symmetric pairs. It formulates and motivates a conjecture that, when the inducing data is square-integrable, the entire space $\mathrm{Hom}_H(I_P^G(\sigma),\mathbb{C})$ is generated by regularized (and in favorable cases normalized) intertwining periods attached to admissible parabolic orbits, providing a canonical description up to normalization. The authors verify the conjecture in several key settings, including the group case, Galois pairs $({\mathrm{GL}}_n(E),{\mathrm{U}}_n(E/F))$, inner forms of $\mathrm{GL}_n$, and rank-one cases with $G={\mathrm{SL}}_2$, and develop a geometric lemma to study the support of these periods. Their results connect the local relative Langlands program to explicit period calculations, offering practical tools for understanding distinction and multiplicities in induced representations. The work lays a path toward a general normalization framework and shows how, in important instances, distinction can be analyzed through intertwining operators and open double coset geometry.

Abstract

In this paper we study the problem of explicitly describing the space of invariant linear forms on induced distinguished representations in terms of invariant linear forms on the inducing representation. More precisely, for certain tempered reductive symmetric pairs (G,H) over a local field of characteristic zero, which we call unimodular in this paper, we study under which condition on the inducing representation, the space of H-invariant linear forms on a parabolically induced representation of G is generated by regularized intertwining periods attached to admissible parabolic orbits in G{H, as defined in the work of Matringe--Offen--Yang. We conjecture that it is the case when the inducing representation is square-integrable. Under this assumption we actually conjecture that one can replace regularized by normalized intertwining periods. We then verify the conjecture on known examples, and prove it for various pairs where G has semi-simple split rank one.

On completeness of local intertwining periods

TL;DR

The paper develops a framework to describe invariant linear forms on induced representations in terms of intertwining periods for unimodular tempered symmetric pairs. It formulates and motivates a conjecture that, when the inducing data is square-integrable, the entire space is generated by regularized (and in favorable cases normalized) intertwining periods attached to admissible parabolic orbits, providing a canonical description up to normalization. The authors verify the conjecture in several key settings, including the group case, Galois pairs , inner forms of , and rank-one cases with , and develop a geometric lemma to study the support of these periods. Their results connect the local relative Langlands program to explicit period calculations, offering practical tools for understanding distinction and multiplicities in induced representations. The work lays a path toward a general normalization framework and shows how, in important instances, distinction can be analyzed through intertwining operators and open double coset geometry.

Abstract

In this paper we study the problem of explicitly describing the space of invariant linear forms on induced distinguished representations in terms of invariant linear forms on the inducing representation. More precisely, for certain tempered reductive symmetric pairs (G,H) over a local field of characteristic zero, which we call unimodular in this paper, we study under which condition on the inducing representation, the space of H-invariant linear forms on a parabolically induced representation of G is generated by regularized intertwining periods attached to admissible parabolic orbits in G{H, as defined in the work of Matringe--Offen--Yang. We conjecture that it is the case when the inducing representation is square-integrable. Under this assumption we actually conjecture that one can replace regularized by normalized intertwining periods. We then verify the conjecture on known examples, and prove it for various pairs where G has semi-simple split rank one.

Paper Structure

This paper contains 20 sections, 22 theorems, 152 equations.

Table of Contents

  1. Unimodular and tempered symmetric pairs
  2. Description of the problem
  3. Basic properties of intertwining periods
  4. Intertwining periods and $G$-conjugacy of $H$ and $(P,M)$
  5. Intertwining periods and maximal compact subgroups
  6. Intertwining periods and $\tilde{G}$-conjugacy
  7. Intertwining periods and transitivity of parabolic induction
  8. The group case
  9. The Galois pair $({\mathrm {GL}}_n(E),{\mathrm{U}}_n(E/F))$
  10. Multiplicity one examples related to ${\mathrm {GL}}_n$ and its inner forms
  11. The Galois model of ${\mathrm {GL}}_n$ and its inner forms
  12. The twisted linear model of inner forms of ${\mathrm {GL}}_n$
  13. The geometric lemma and the support of regularized open intertwining periods
  14. The geometric lemma
  15. The support of invariant linear forms
  16. ...and 5 more sections

Key Result

Lemma 1.1

Suppose that $x\in X$ is $M$-admissible, and let $P$ be a parabolic subgroup of $G$ with $M$ as a Levi component, and $V$ as unipotent radical. Then

Theorems & Definitions (57)

  • Lemma 1.1
  • proof
  • Definition 1.2
  • Definition 1.3
  • proof
  • Definition 1.5
  • proof
  • Definition 1.7
  • Lemma 1.8
  • proof
  • ...and 47 more