The n-cohomology of limits of discrete series

AbstractLet G be a semisimple Lie group which has a compact Cartan subgroup H, let K be a maximal compact subgroup of G containing H, and let n be the sum of the negative root spaces of G corresponding to an arbitrary choice of a positive root system of (G, H). We compute the n-cohomology of the K-finite vectors in a limit of a discrete series representation π of G. In the special case when π is a discrete series representation our result reduces to the result of W. Schmid. In the special case when π is a holomorphic limit we interpret its multiplicity in L2(ΓβG) cohomologically, where Γ is a discrete subgroup of G. In the general case we present a conjecture for this multiplicity