A geometric realisation of tempered representations restricted to maximal compact subgroups

Let G be a connected, linear, real reductive Lie group with compact centre. Let K< G be maximal compact. For a tempered representation π of G, we realise the restriction π| _K as the K-equivariant index of a Dirac operator on a homogeneous space of the form G/H, for a Cartan subgroup H< G. (The result in fact applies to every standard representation.) Such a space can be identified with a coadjoint orbit of G, so that we obtain an explicit version of Kirillov’s orbit method for π| _K. In a companion paper, we use this realisation of π| _K to give a geometric expression for the multiplicities of the K-types of π, in the spirit of the quantisation commutes with reduction principle. This generalises work by Paradan for the discrete series to arbitrary tempered representations.