  For a Lie groupoid G with a twisting (a PU(H)-principal bundle over G), we
use the (geometric) deformation quantization techniques supplied by Connes
tangent groupoids to define an analytic index morphism in twisted K-theory. In
the case the twisting is trivial we recover the analytic index morphism of the
groupoid.
  For a smooth foliated manifold with twistings on the holonomy groupoid we
prove the twisted analog of Connes-Skandalis longitudinal index theorem. When
the foliation is given by fibers of a fibration, our index coincides with the
one recently introduced by Mathai-Melrose-Singer.
  We construct the pushforward map in twisted K-theory associated to any smooth
(generalized) map $f:W\longrightarrow M/F$ and a twisting $\sigma$ on the
holonomy groupoid $M/F$, next we use the longitudinal index theorem to prove
the functoriality of this construction. We generalize in this way the wrong way
functoriality results of Connes-Skandalis when the twisting is trivial and of
Carey-Wang for manifolds.
