Suppose that $(M,E)$ is a compact contact manifold, and that a compact Lie
group $G$ acts on $M$ transverse to the contact distribution $E$. In
arxiv:0712.2431v4, we defined a $G$-transversally elliptic Dirac operator
$\dirac$, constructed using a Hermitian metric $h$ and connection $\nabla$ on
the symplectic vector bundle $E\to M$, whose equivariant index is well-defined
as a generalized function on $G$, and gave a formula for its index. By analogy
with the geometric quantization of symplectic manifolds, the virtual Hilbert
space $[\ker \dirac] - [\ker \dirac^*]$ can be interpreted as the
``quantization'' of the contact manifold $(M,E)$; the character of the
corresponding virtual $G$-representation is then given by the equivariant index
of $\dirac$. In this article, we discuss the extent to which the analogy with
the geometric quantization of symplectic manifolds can be carried out.