Computing K-type multiplicities in standard representations (after Vogan)

The first part of these notes is an updating (and correction) of [Khat] and is devoted to a paremetrization of the irreducible representations of the (generally disconnected) maximal compact subgroup of a real group in Harish-Chandra’s class. The second part describes how to use that paremetrization of the first part to compute K-type multiplicities in standards modules. (By Frobenius reciprocity, this is equivalent to Blattner’s formula and branching from K to K ∩M which is complicated significantly by the disconnectedness of the groups in question.) An interlude between the first and second parts describes which K-types are relevant for determining unitarity for irreducible Hermitian (g,K) modules. All of this material is intended as a report on ideas of David Vogan.