Branching of unitary O(1,n+1)-representations with non-trivial (đ”€,K)-cohomology
Annales de l'Institut Fourier, Online first, 47 p.

Let G=O(1,n+1) with maximal compact subgroup K and let Π be a unitary irreducible representation of G with non-trivial (đ”€,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 â€Č =O(1,n) of G with maximal compact subgroup K â€Č , we prove branching laws and explicit Plancherel formulas for the restrictions to G â€Č of all unitary representations occurring in such principal series, including the complementary series, all unitary G-representations with non-trivial (đ”€,K)-cohomology and further relative discrete series representations in the cases p=0,n. Discrete spectra are constructed explicitly as residues of G â€Č -intertwining operators which resemble the Fourier transforms on vector bundles over the Riemannian symmetric space G â€Č /K â€Č .

Soient G=O(1,n+1), K un sous-groupe compact maximal de G et Π une reprĂ©sentation unitaire irrĂ©ductible de G possĂ©dant une (đ”€,K)-cohomologie non triviale. Alors Π apparaĂźt comme une sous-reprĂ©sentation d’une sĂ©rie principale de G, induite depuis la reprĂ©sentation de O(n) sur ⋀ p (ℂ n ) et un caractĂšre d’un sous-groupe parabolique maximal de G Ă  la limite de la sĂ©rie complĂ©mentaire. En considĂ©rant le sous-groupe G â€Č =O(1,n) de G ayant un sous-groupe compact maximal K â€Č , nous prouvons des lois de branchement et des formules de Plancherel explicites pour la restriction Ă  G â€Č de toutes les reprĂ©sentations unitaires apparaissant dans de telles sĂ©ries principales. Ceci inclut la sĂ©rie complĂ©mentaire, toutes les reprĂ©sentations unitaires de G ayant une (đ”€,K)-cohomologie non triviale, et d’autres reprĂ©sentations de la sĂ©rie discrĂšte relative dans les cas p=0,n. Les spectres discrets sont construits explicitement en tant que rĂ©sidus d’opĂ©rateurs d’entrelacement qui ressemblent Ă  la transformĂ©e de Fourier pour des fibrĂ©s vectoriels sur l’espace symĂ©trique riemannien G â€Č /K â€Č .

Online First:
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)
     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},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     year = {2024},
     doi = {10.5802/aif.3622},
     language = {en},
     note = {Online first},
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PB  - Association des Annales de l’institut Fourier
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%0 Unpublished Work
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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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