the formalism in [@Per2012; @Per2014] is to be *invariant under diffeomorphisms*.\
These steps being accomplished, we compute the JLO cocycle for two different Dirac operators, the first one gives the Radul cocycle of the pseudodifferential extension, the second one gives the Poincaré dual of the equivariant Todd class. The usual homotopy arguments from the original JLO formula apply verbatim there, thus giving the theorem.\
We then apply Theorem \[intro thm equivariant index\] to the Connes-Moscovici index problem for *transversally* hypoelliptic operators on foliations. After reduction to a complete transversal $W$, the holonomy groupoid of a given foliation is Morita equivalent to an étale groupoid $W\rtimes G$ where $G\subset {\mathrm{Diff}}(W)$ is a discrete (pseudo)group of diffeomorphisms. The main object of study is therefore the crossed product algebra ${C^{\infty}_c}(W) \rtimes G$ where, for notational simplicity, we treat $G$ as a group. A first problem occuring is that $G$ does not preserve any measure on $W$ in general, and even less a Riemannian metric, so that $G$-invariant elliptic differential operators do not exist even at the leading symbol level. The idea of Connes in [@C83] is to pass by a Thom isomorphism to the bundle of Riemannian metrics $M$ over $W$. This fibration 