  We give a cohomological treatment of a character theory for (g,K)-modules.
This leads to a nice formalism extending to large categories of not necessarily
admissible (g,K)-modules. Due to results of Hecht, Schmid and Vogan the
classical results of Harish-Chandra's global character theory extend to this
general setting. As an application we consider a general setup, for which we
show that algebraic characters answer discretely decomposable branching
problems.
