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Differential symmetry breaking operators for the pair $(\operatorname{GL}_{n+1}(\mathbb{R}),\operatorname{GL}_n(\mathbb{R}))$

Jonathan Ditlevsen, Quentin Labriet

TL;DR

This work analyzes differential symmetry breaking operators between principal series of $(\mathrm{GL}_{n+1}(\mathbb{R}),\mathrm{GL}_n(\mathbb{R}))$ by employing a source operator approach to generate explicit intertwiners. For generic induction parameters, the authors construct a comprehensive basis of DSBOs as composites of source operators and restriction maps, and prove these exhaust all possibilities; these operators also appear as residues of a Meromorphic SBO family, linking algebraic and analytic viewpoints. The paper further provides a full classification in the non-generic case when $n=2$, revealing a multiplicity-two phenomenon for certain one-point supports and detailing alternative constructions of DSBO bases. Together, these results give a detailed, parameter-sensitive picture of how representations restrict along the pair $(G,H)$ and establish a residue-bridge between explicit differential operators and holomorphic families of intertwiners with explicit $e$-function factors. The findings advance the understanding of higher-rank symmetry breaking by reducing the problem to canonical source-operator data and Bernstein–Sato identities, with potential applications to explicit branching problems and automorphic contexts.$

Abstract

In this article we study differential symmetry breaking operators between principal series representations induced from minimal parabolic subgroups for the pair $(\operatorname{GL}_{n+1}(\mathbb{R}),\operatorname{GL}_n(\mathbb{R}))$. Using the source operator philosophy we construct such operators for generic induction parameters of the representations and establish that this approach yields all possible operators in this setting. We show that these differential operators occur as residues of a family of symmetry breaking operators that depends meromorphically on the parameters. Finally, in the $n=2$ case we classify and construct all differential symmetry breaking operators for any parameters, including the non-generic ones.

Differential symmetry breaking operators for the pair $(\operatorname{GL}_{n+1}(\mathbb{R}),\operatorname{GL}_n(\mathbb{R}))$

TL;DR

This work analyzes differential symmetry breaking operators between principal series of by employing a source operator approach to generate explicit intertwiners. For generic induction parameters, the authors construct a comprehensive basis of DSBOs as composites of source operators and restriction maps, and prove these exhaust all possibilities; these operators also appear as residues of a Meromorphic SBO family, linking algebraic and analytic viewpoints. The paper further provides a full classification in the non-generic case when , revealing a multiplicity-two phenomenon for certain one-point supports and detailing alternative constructions of DSBO bases. Together, these results give a detailed, parameter-sensitive picture of how representations restrict along the pair and establish a residue-bridge between explicit differential operators and holomorphic families of intertwiners with explicit -function factors. The findings advance the understanding of higher-rank symmetry breaking by reducing the problem to canonical source-operator data and Bernstein–Sato identities, with potential applications to explicit branching problems and automorphic contexts.$

Abstract

In this article we study differential symmetry breaking operators between principal series representations induced from minimal parabolic subgroups for the pair . Using the source operator philosophy we construct such operators for generic induction parameters of the representations and establish that this approach yields all possible operators in this setting. We show that these differential operators occur as residues of a family of symmetry breaking operators that depends meromorphically on the parameters. Finally, in the case we classify and construct all differential symmetry breaking operators for any parameters, including the non-generic ones.
Paper Structure (21 sections, 30 theorems, 167 equations)

This paper contains 21 sections, 30 theorems, 167 equations.

Table of Contents

  1. Introduction
  2. Results
  3. Preliminaries
  4. Structure and principal series representations for GL(n,R)
  5. The Knapp--Stein intertwining operators
  6. Symmetry breaking operators between principal series
  7. p-differential operators
  8. Differential symmetry breaking operators using the source operator
  9. Intertwining multiplication operators
  10. The source operators
  11. Differential symmetry breaking operators as residues
  12. Recollection of the holomorphic family
  13. Bernstein--Sato identities
  14. Restriction as a residue
  15. Vanishing of the source operator
  16. ...and 6 more sections

Key Result

Theorem A

Let $k\in \{0,\dots,n\}$ and $\lambda \in \mathbb{C}^n$ such that $\lambda_i-\lambda_j\notin \mathbb{Z}$ for all $i,j$. Then the dimension of the space of differential symmetry breaking operators with distributional support $x_kP_G$ is either $0$ or $1$. When it is $1$, there exists $\alpha\in \math with and

Theorems & Definitions (59)

  • Theorem A
  • Theorem B
  • Remark 2.1
  • Definition 2.2
  • Remark 2.3: KobayashiPevzner16_1
  • Proposition 2.4: see KobayashiPevzner16_1
  • Corollary 2.5
  • proof
  • Remark 2.6
  • Corollary 2.7
  • ...and 49 more