Given a topological representation of a noncompact real Lie group like $SL_3(\mathbb R)$, one classically constructs an algebraic replacement called a $(\mathfrak g,K)$-module, and from there a geometric replacement called a $\mathcal D_{G/B}$-module, which is supported on a $K$-orbit closure on $G/B$ (of which there are finitely many). Given the audience, I'll recall this story in some detail.

When the $K$-orbit closure is smooth (and the "infinitesimal character" is integral), I'll use equivariant localization to compute the $K$-multiplicities in the representation. This generalizes Blattner's conjecture.

Then I'll refine this alternating sum to a combinatorial formula, in the case of the $SO(3)$-multiplicities in the four types of $SL(3,\mathbb R)$-irreps.