We study various aspects of the multiplicities of K-types for discrete series representations of a real reductive group G, where K is a maximal compact subgroup sharing the same maximal torus. First, we give a geometric interpretation of the Blattner formula based on D-modules associated to closed K-orbits on the flag variety G/B. Then we specialize to the case K = U(p) \times U(q), G = U(p,q), where we investigate positive formulas for the multiplicities. We give one such formula, in the case of one and two noncompact simple roots, based on the Gan-Gross-Prasad conjecture by considering a crystal structure on combinatorial patterns resembling Gelfand-Tsetlin patterns. Then we interpret the K-types as N_K-invariants in a ring of matrix coefficients, and use SAGBI basis theory to state a conjecture towards polyhedral formulas for the general case and prove it for U(2,2).