Branching of unitary $\mathrm{O}(1,n+1)$-representations with non-trivial $(\mathfrak{g},K)$-cohomology

Weiske, Clemens

doi:10.5802/aif.3622

Branching of unitary

O (1, n + 1)

-representations with non-trivial

(𝔤, K)

-cohomology

Weiske, Clemens ¹

¹ Mathematical Sciences, Chalmers University of Technology, SE-412 96 Göteborg (Sweden)

Annales de l'Institut Fourier, Volume 74 (2024) no. 6, pp. 2331-2377.

Abstract
Résumé

Let $G = O (1, n + 1)$ with maximal compact subgroup $K$ and let $Π$ be a unitary irreducible representation of $G$ with non-trivial $(𝔤, K)$ -cohomology. Then $Π$ occurs inside a principal series representation of $G$ , induced from the $O (n)$ -representation $⋀^{p} (ℂ^{n})$ and characters of a minimal parabolic subgroup of $G$ at the limit of the complementary series. Considering the subgroup $G^{'} = O (1, n)$ of $G$ with maximal compact subgroup $K^{'}$ , we prove branching laws and explicit Plancherel formulas for the restrictions to $G^{'}$ of all unitary representations occurring in such principal series, including the complementary series, all unitary $G$ -representations with non-trivial $(𝔤, K)$ -cohomology and further relative discrete series representations in the cases $p = 0, n$ . Discrete spectra are constructed explicitly as residues of $G^{'}$ -intertwining operators which resemble the Fourier transforms on vector bundles over the Riemannian symmetric space $G^{'} / K^{'}$ .

Soient $G = O (1, n + 1)$ , $K$ un sous-groupe compact maximal de $G$ et $Π$ une représentation unitaire irréductible de $G$ possédant une $(𝔤, K)$ -cohomologie non triviale. Alors $Π$ apparaît comme une sous-représentation d’une série principale de $G$ , induite depuis la représentation de $O (n)$ sur $⋀^{p} (ℂ^{n})$ et un caractère d’un sous-groupe parabolique maximal de $G$ à la limite de la série complémentaire. En considérant le sous-groupe $G^{'} = O (1, n)$ de $G$ ayant un sous-groupe compact maximal $K^{'}$ , nous prouvons des lois de branchement et des formules de Plancherel explicites pour la restriction à $G^{'}$ de toutes les représentations unitaires apparaissant dans de telles séries principales. Ceci inclut la série complémentaire, toutes les représentations unitaires de $G$ ayant une $(𝔤, K)$ -cohomologie non triviale, et d’autres représentations de la série discrète relative dans les cas $p = 0, n$ . Les spectres discrets sont construits explicitement en tant que résidus d’opérateurs d’entrelacement qui ressemblent à la transformée de Fourier pour des fibrés vectoriels sur l’espace symétrique riemannien $G^{'} / K^{'}$ .

Received: 2022-05-18
Revised: 2022-12-01
Accepted: 2022-12-16
Published online: 2024-10-11

DOI: 10.5802/aif.3622

Classification: 22E45, 22E46
Keywords: Real reductive groups, unitary representations, branching laws, direct integral, symmetry breaking operators.
Mot clés : Groupes réductif réels, représentations unitaires, lois de branchement, intégrale directe, opérateurs de brisures de symétries.

Author's affiliations:

Weiske, Clemens ¹

¹ Mathematical Sciences, Chalmers University of Technology, SE-412 96 Göteborg (Sweden)

License:

CC-BY-ND 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     author = {Weiske, Clemens},
     title = {Branching of unitary $\mathrm{O}(1,n+1)$-representations with non-trivial $(\mathfrak{g},K)$-cohomology},
     journal = {Annales de l'Institut Fourier},
     pages = {2331--2377},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {74},
     number = {6},
     year = {2024},
     doi = {10.5802/aif.3622},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3622/}
}

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Weiske, Clemens. Branching of unitary $\mathrm{O}(1,n+1)$-representations with non-trivial $(\mathfrak{g},K)$-cohomology. Annales de l'Institut Fourier, Volume 74 (2024) no. 6, pp. 2331-2377. doi : 10.5802/aif.3622. https://aif.centre-mersenne.org/articles/10.5802/aif.3622/

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