  We study K-theory classes of Hamiltonian loop group spaces represented by
admissible Fredholm complexes. We prove various equivariant index formulae in
this context. In a sequel to this article we show that, when specialized to a
family of non-local elliptic boundary value problems over the moduli space of
framed flat connections on a surface, one obtains a gauge theory analogue of
the Teleman-Woodward index formula.
