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We prove a version of Blattner's conjecture, for irreducible subquotients ofprincipal series representations with integral infinitesimal character of areal reductive Lie group whose Beilinson-Bernstein D-module is supported on aK-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 allfour K-orbits on G/B have smooth closure, and refine the resultingalternating-sum formulae to ones with only positive terms.

Positive formulae for K-types of SL_3(R)-irreps and a Blattner formula for smooth K-orbit closures