Branching of unitary \O(1,n+1)-representations with\n non-trivial ( mathfrakg,K)-cohomology

2021/06/25 by Clemens Weiske, Weiske, Clemens
Mathematics · Medicine · #22E45 (Primary) #22E46 (Secondary) #Advanced Algebra and Geometry #Advanced Neuroimaging Techniques and Applications #Advanced Operator Algebra Research #FOS: Mathematics #Mathematical Analysis and Transform Methods #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2106.13562

openalex publication_date 2021/06/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let G=\O(1,n+1) with maximal compact subgroup K and let\n\Π be a unitary irreducible representation of G with non-trivial\n( mathfrakg,K)-cohomology. Then \Π occurs inside a principal series\nrepresentation of G, induced from the \O(n)-representation\n bigwedge nolimits^p(\ℂⁿ) and characters of a minimal parabolic\nsubgroup of G at the limit of the complementary series. Considering the\nsubgroup G'=\O(1,n) of G with maximal compact subgroup K',\nwe prove branching laws and explicit Plancherel formulas for the restrictions\nto G' of all unitary representations occurring in such principal series,\nincluding the complementary series, all unitary G-representations with\nnon-trivial ( mathfrakg,K)-cohomology and further relative discrete series\nrepresentations in the cases p=0,n. Discrete spectra are constructed\nexplicitly as residues of G'-intertwining operators which resemble the\nFourier transforms on vector bundles over the Riemannian symmetric space\nG'/K'.\n

Related

• Branching laws for minimal holomorphic representations
• Branching rules for principal series representations of unramified U(1,1)
• Restriction of irreducible unitary representations of Spin(N,1) to parabolic subgroups
• Branching laws for discrete series of some affine symmetric spaces
• Translation functors, branching problems, and applications to the restriction of coherent cohomology of Shimura varieties