  Let $G$ be a connected, linear, real reductive Lie group with compact centre.
Let $K<G$ be maximal compact. For a tempered representation $\pi$ of $G$, we
realise the restriction $\pi|_K$ 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 $\pi|_K$. In a companion paper, we use
this realisation of $\pi|_K$ to give a geometric expression for the
multiplicities of the $K$-types of $\pi$, in the spirit of the quantisation
commutes with reduction principle. This generalises work by Paradan for the
discrete series to arbitrary tempered representations.
