  From 1980s, it is an open problem of proposing cohomologic formula for the
basic index of a transversally elliptic basic differential operator on a vector
bundle over a foliated manifold. In 1990s, El Kacimi-Alaoui has proprosed to
use the Molino theory for study this index. Molino has proved that to every
transversally oriented Riemannien foliation, we can associate a manifold,
called basique manifold, which is \'equiped with an action of orthogonal group,
El Kacimi-Alaoui has shown how to associate a transversally elliptic basic
differential operator an operator on a vector bundle, called useful bundle,
over the basique manifold.
  The idea is to obtain the desired cohomologic formula from r\'esultats about
the operator on the useful bundle. This thesis is a first step in this
direction. While the Riemannien foliation is Killing, Goertsches et T\"oben
have remarked that there exists a naturel cohomologic isomorphism between the
equivariant basique cohomology of the Killing foliation and the equivariant
cohomology of the basique manifold.
  The principal result of this thesis is the geometric realisation of the
cohomologic isomorphism by Chern characters under some hypoth\`eses.
