Dirac cohomology and the bottom layer K-types

Let G be a connected real reductive Lie group with a maximal compact subgroup K corresponding to a Cartan involution Θ of G. Let q=l u be a θ-stable parabolic subalgebra of the complexified Lie algebra g of G, where θ=dΘ. Let L be the centralizer of q in G. We show that, under certain dominance assumptions, cohomological induction with respect to q takes irreducible unitary (l, L ∩ K)-modules with nonzero Dirac cohomology to irreducible unitary (g,K)-modules which also have nonzero Dirac cohomology