## T. Kobayashi and M. Pevzner, *Inversion of Rankin-Cohen operators via
holographic transform*, Ann. Inst. Fourier (Grenoble) **70** (2020),
no. 5, 2131-2190.
DOI: 10.5802/aif.3386.
arXiv: 1812.09733..

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 formulæ 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.

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© Toshiyuki Kobayashi