Rankin--Cohen brackets in Representation Theory

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 semesterRepresentation Theory and Noncommutative Geometry", held in Spring 2025 at the Henri Poincaré Institute in Paris.