[Règles de branchement pour les groupes de Lie semi-simples et les noyaux reproduisants]
Pour un groupe de Lie semi-simple G satisfaisant la condition de rang, la famille de représentations irréductibles unitaires la plus fondamentale est la série discrète trouvée par Harish-Chandra. Dans cet article, nous étudions quelques règles de branchement pour ces séries restreintes à un sous-groupe H de G du même type, en combinant les résultats classiques avec des travaux récents de T. Kobayashi. Nous analysons des cas où des opérateurs de brisure de symétrie sont des opérateurs différentiels ; en particulier, nous prouvons dans le cas dit admissible que tout opérateur de brisure de symétries H-équivariant est un opérateur différentiel. Nous prouvons la propriété de décomposabilité discrète sous la condition de cuspidalité de Harish-Chandra sur les noyaux reproduisants.
For a semisimple Lie group G satisfying the equal rank condition, the most basic family of unitary irreducible representations is the discrete series found by Harish-Chandra. In this paper, we study some of the branching laws for these when restricted to a subgroup H of the same type by combining the classical results with the recent work of T. Kobayashi. We analyze aspects of having differential operators being symmetry-breaking operators; in particular, we prove in the so-called admissible case that every symmetry breaking (H-map) operator is a differential operator. We prove discrete decomposability under Harish-Chandra's condition of cusp form on the reproducing kernels. Our techniques are based on realizing discrete series representations as kernels of elliptic invariant differential operators.
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@article{CRMATH_2019__357_9_697_0,
author = {Bent {\O}rsted and Jorge A. Vargas},
title = {Branching problems for semisimple {Lie} groups and reproducing kernels},
journal = {Comptes Rendus. Math\'ematique},
pages = {697--707},
publisher = {Elsevier},
volume = {357},
number = {9},
year = {2019},
doi = {10.1016/j.crma.2019.09.004},
language = {en},
}
Bent Ørsted; Jorge A. Vargas. Branching problems for semisimple Lie groups and reproducing kernels. Comptes Rendus. Mathématique, Volume 357 (2019) no. 9, pp. 697-707. doi : 10.1016/j.crma.2019.09.004. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2019.09.004/
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