BRANCHING OF UNITARY O(1, ņ + 1)-REPRESENTATIONS WITH NON-TRIVIAL (g, Ķ)-COHOMOLOGY

Journal article, 2024

— 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 Vp(Cn) 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′.