Embeddings of Harish-Chandra modules, 𝔫-homology and the composition series problem: the case of real rank one

Abstract

Let G G be a connected semisimple matrix group of real rank one. Fix a minimal parabolic subgroup P = M A N P = MAN and form the (normalized) principal series representations I P G ( U ) I_P^G(U) . In the case of regular infinitesimal character, we explicitly determine (in terms of Langlands’ classification) all irreducible submodules and quotients of I P G ( U ) I_P^G(U) . As a corollary, all embeddings of an irreducible Harish-Chandra module into principal series are computed. The number of such embeddings is always less than or equal to three. These computations are equivalent to the determination of zero n {\mathfrak {n}} -homology.