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Rankin--Cohen brackets in Representation Theory

Toshiyuki Kobayashi, Michael Pevzner

TL;DR

This work analyzes Rankin--Cohen brackets as holomorphic differential symmetry breaking operators and situates them within a representation theoretic framework for branching problems. It develops the F-method together with the Duality, Localness, and Extension theorems to classify and construct differential SBOs, unveiling deep connections to Jacobi polynomials and higher dimensional Hermitian symmetric spaces. The authors provide explicit higher dimensional generalizations of RC brackets, reveal multiplicity phenomena at singular parameters, and introduce generating operators to bridge discrete operator families with continuous parameter families, highlighting a unifying approach for branching in real reductive groups. The study thus bridges modular form theory, holomorphic discrete series, and differential operators in a coherent algebraic analytic framework with potential for broad higher rank applications.

Abstract

The Rankin--Cohen brackets provide a basic example of ``non-elementary" differential symmetry breaking operators. They can be interpreted as bi-differential operators remarkable for reflecting the structure of fusion rules for holomorphic discrete series representations of the Lie group $SL(2,\mathbb R)$ and are intimately connected to classical special polynomials. In this introductory article, we explore the combinatorial structure of these operators and discuss a general framework for constructing their higher-dimensional analogues from the representation-theoretic perspective on branching problems. The exposition is based on lectures delivered by the authors during the thematic semester ``Representation Theory and Noncommutative Geometry", held in Spring 2025 at the Henri Poincaré Institute in Paris.

Rankin--Cohen brackets in Representation Theory

TL;DR

This work analyzes Rankin--Cohen brackets as holomorphic differential symmetry breaking operators and situates them within a representation theoretic framework for branching problems. It develops the F-method together with the Duality, Localness, and Extension theorems to classify and construct differential SBOs, unveiling deep connections to Jacobi polynomials and higher dimensional Hermitian symmetric spaces. The authors provide explicit higher dimensional generalizations of RC brackets, reveal multiplicity phenomena at singular parameters, and introduce generating operators to bridge discrete operator families with continuous parameter families, highlighting a unifying approach for branching in real reductive groups. The study thus bridges modular form theory, holomorphic discrete series, and differential operators in a coherent algebraic analytic framework with potential for broad higher rank applications.

Abstract

The Rankin--Cohen brackets provide a basic example of ``non-elementary" differential symmetry breaking operators. They can be interpreted as bi-differential operators remarkable for reflecting the structure of fusion rules for holomorphic discrete series representations of the Lie group and are intimately connected to classical special polynomials. In this introductory article, we explore the combinatorial structure of these operators and discuss a general framework for constructing their higher-dimensional analogues from the representation-theoretic perspective on branching problems. The exposition is based on lectures delivered by the authors during the thematic semester ``Representation Theory and Noncommutative Geometry", held in Spring 2025 at the Henri Poincaré Institute in Paris.
Paper Structure (12 sections, 11 theorems, 85 equations, 1 table)

This paper contains 12 sections, 11 theorems, 85 equations, 1 table.

Table of Contents

  1. Introduction
  2. Holomorphic modular forms
  3. Holomorphic discrete series representations
  4. Weight Watchers
  5. Connection to orthogonal polynomials
  6. Symmetry breaking Paradigm
  7. The big three
  8. F-method
  9. Various examples
  10. Blow-up of Multiplicities
  11. Higher-dimensional examples
  12. Further Perspectives

Key Result

Theorem 2.1

Let $H' \subset H$ be (possibly disconnected) closed subgroups of a Lie group $G$ with Lie algebras $\mathfrak{h}' \subset \mathfrak{h}$, respectively. Suppose that $V$ and $W$ are finite-dimensional representations of $H$ and $H'$, respectively, and denote by $V^\vee$ and $W^\vee$ corresponding con Then there is a natural linear isomorphism: or equivalently, For $\varphi \in \operatorname{Hom}_

Theorems & Definitions (17)

  • Definition 1.1
  • Definition 1.2
  • Theorem 2.1: Duality Theorem
  • Example 2.2
  • Corollary 2.3
  • Theorem 2.4: Duality theorem in the holomorphic setting
  • Corollary 2.5: KP16a
  • Theorem 2.6: Localness Theorem
  • Theorem 2.7: Extension Theorem
  • Remark 2.8
  • ...and 7 more