Rankin--Cohen brackets in Representation Theory

TL;DR

This paper explores the structure and construction of Rankin--Cohen brackets as differential operators linked to representation theory of SL(2,R), extending to higher dimensions.

Contribution

It introduces a general framework for constructing higher-dimensional analogues of Rankin--Cohen brackets from a representation-theoretic perspective.

Findings

Analyzed the combinatorial structure of Rankin--Cohen brackets.

Connected these operators to fusion rules for holomorphic discrete series.

Proposed methods for higher-dimensional generalizations.

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 Poincar\'e Institute in Paris.