Inversion of Rankin-Cohen operators via Holographic Transform

Abstract

The analysis of branching problems for restriction of representations brings the concept of symmetry breaking transform and holographic transform. Symmetry breaking operators decrease the number of variables in geometric models, whereas holographic operators increase it. Various expansions in classical analysis can be interpreted as particular occurrences of these transforms. From this perspective, we investigate two remarkable families of differential operators: the Rankin-Cohen operators and the holomorphic Juhl conformally covariant operators. Then we establish for the corresponding symmetry breaking transforms the Parseval-Plancherel type theorems and find explicit inversion formul{\ae} with integral expression of holographic operators. The proof uses the F-method which provides a duality between symmetry breaking operators in the holomorphic model and holographic operators in the $L^2$-model, leading us to deep links between special orthogonal polynomials and branching laws for infinite-dimensional representations of real reductive Lie groups.