Positive formulae for K-types of SL₃(R)-irreps and a Blattner formula for smooth K-orbit closures

We prove a version of Blattner's conjecture, for irreducible subquotients of principal series representations with integral infinitesimal character of a real reductive Lie group whose Beilinson-Bernstein D-module is supported on a K-orbit with smooth closure. (The cases usually considered are closed orbits, or their preimages along G/B -> G/P.) We apply this to G_R = SL_3(R), where all four K-orbits on G/B have smooth closure, and refine the resulting alternating-sum formulae to ones with only positive terms.