A Construction of Generalized Harish-Chandra Modules with Arbitrary   Minimal k-Type
  Let g be a semisimple complex Lie algebra and k in g be any algebraic subalgebra reductive in g. For any simple finite dimensional k-module V, we construct simple (g; k)-modules M with finite dimensional k-isotypic components such that V is a k-submodule of M and the Vogan norm of any simple k-submodule V' of M; V' not isomorphic to V, is greater than the Vogan norm of V . The (g; k)-modules M are subquotients of the fundamental series of (g; k)-modules introduced in [PZ2]. 
Ivan Penkov, Gregg Zuckerman