For a transversally elliptic basic differential operator on a vector bundle over a foliated manifold, it is an open problem from 1980s of proposing cohomological  formula for its basic index. In 1990s, El Kacimi-Alaoui proprosed to use the Molino theory, that to every transversally oriented Riemannien foliation, we can associate a manifold, called basic manifold, equipped with an action of orthogonal group. El Kacimi-Alaoui then showed how to associate a transversally elliptic basic differential operator to a vector bundle, called useful bundle, over the basique manifold.

The idea is to obtain the desired cohomological formula from results about the operator on the useful bundle. Our work is a first step in this direction. When the Riemannien foliation is Killing, Goertsches et Töben have remarked that there exists a natural cohomological isomorphism between the equivariant basic cohomology of the Killing foliation and the equivariant cohomology of the basic manifold.

The principal result of this talk is the geometric realization of the cohomological isomorphism by Chern characters under some hypothesis.  

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