Branching of unitary O(1,n+1)-representations with non-trivial (𝔤,K)-cohomology
[Branchement de représentations cohomologiques unitaires O(1,n+1) avec une cohomologie (𝔤,K) non triviale]
Weiske, Clemens1
1 Mathematical Sciences, Chalmers University of Technology, SE-412 96 Göteborg (Sweden)
Annales de l'Institut Fourier, Online first, 47 p.

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 ′ .

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 ′ .

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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.
Weiske, Clemens 1

1 Mathematical Sciences, Chalmers University of Technology, SE-412 96 Göteborg (Sweden) 
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     title = {Branching of unitary $\mathrm{O}(1,n+1)$-representations with non-trivial $(\mathfrak{g},K)$-cohomology},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
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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, Online first, 47 p.

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