  The smooth action of a compact Lie group on a compact manifold can be
resolved to an iterated space, as made explicit by Pierre Albin and the second
author. On the resolution the lifted action has fixed isotropy type
corresponding to the open stratum and also in an iterated sense, with
connecting equivariant fibrations over the boundary hypersurfaces covering the
resolutions of the other strata. This structure descends to a resolution of the
quotient as a stratified space. For an Abelian group action the equivariant
K-theory can then be described in terms of bundles over the bases `dressed' by
the representations of the isotropy types with morphisms covering the
connecting maps. A similar model is given here covering the non-Abelian case.
Now the reduced objects are torsion-twisted bundles over finite covers of the
bases, corresponding to the projective action of the normalizers on the
representations of the isotropy groups, again with morphisms over all the
boundaries. This leads to a closely related iterated deRham model for
equivariant cohomology and, now with values in forms twisted by flat bundles of
representation rings over the bases, for delocalized equivariant cohomology. We
show, as envisioned by Baum, Brylinksi and MacPherson, that the usual
equivariant Chern character, mapping to equivariant cohomology, factors through
a natural Chern character from equivariant K-theory to delocalized equivariant
cohomology with the latter giving an Atiyah-Hirzebruch isomorphism.
