Boundaries of K-types in discrete series

A fundamental problem about irreducible representations of a reductive Lie group G is understanding their restriction to a maximal compact subgroup K.
In certain important cases, known as the discrete series, the Blattner formula gives the multiplicity of any given irreducible K-representation (or K-type) as
an alternating sum. It is not immediately clear from this formula which K-types, indexed by their highest weights, have non-zero multiplicity. Evidence suggests that the collection is very close to a set of lattice points in a convex polyhedron. I shall describe a
recursive algorithm for finding the boundary facets of this polyhedron.