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Inversion of Rankin-Cohen operators via holographic transform. (Inversion des opérateurs de Rankin-Cohen par transformation holographique.) (English. French summary) Zbl 1478.22009

Let \(G'\) be a subgroup of a group \(G\). An irreducible representation \(\pi\) of \(G\) on a vector space \(V\) is also a representation of \(G'\) by restriction. For each irreducible representation \((\rho,W)\) of \(G'\), one can associate a continuous linear map \(V\rightarrow W\), intertwining \(\pi_{\mid_{G'}}\) and \(\rho\), known as a symmetry breaking operator. More generally, one can consider a family of symmetry breaking operators \(\{R_\ell\}_{\ell\in\Lambda}\): \[R_{\ell}:V\rightarrow W_\ell\] associated with a family of \(G'\)-representations \(\{(\rho_\ell,W_\ell)\}_{\ell\in\Lambda}\). The paper under review addresses the following two problems.
Problem A: Can one recover an element \(v\) of \(V\) from its symmetry breaking transform \(R(v)=\{R_\ell(v)\}_{\ell\in\Lambda}\) ?
Problem B: Find a closed formula for the norm of an element \(v\) in \(V\) in terms of its symmetry breaking transform \(\{R_\ell(v)\}_{\ell\in\Lambda}\) ?
The strategy is based on the F-method developed by the authors. It deals with the Fourier transform of geometric realizations of representations and involves certain families \(\{ P_\ell\}\) of polynomials. Problems A and B are considered in the particular cases of Rankin-Cohen transform and holomorphic Juhl transform, where the polynomials \(P_\ell\)’s coincide with Jacobi polynomials and Gegenbauer polynomials.
The paper offers a thorough study of symmetry breaking operators leading to a precise relationship between special polynomials and branching rules for infinite-dimension representations of real reductive groups. Several examples and tables are provided to illustrate the issues and the various results.
Reviewer: Salah Mehdi (Metz)

MSC:

22E45 Representations of Lie and linear algebraic groups over real fields: analytic methods
22E46 Semisimple Lie groups and their representations
32M15 Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects)
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
33C80 Connections of hypergeometric functions with groups and algebras, and related topics
43A85 Harmonic analysis on homogeneous spaces
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