and is given in terms of an integral over the *Heisenberg cosphere bundle* $S^*_HM$ over $M$. A variant of this cocycle may also be obtained by taking the boundary of the localized operator trace under the extension $$0 \longrightarrow \Psi_{H,c}^{-1}(M) \rtimes G \longrightarrow \Psi_{H,c}^{0}(M) \rtimes G \longrightarrow {C^{\infty}_c}(S^{*}_H M) \rtimes G \longrightarrow 0\ ,$$ leading to a cocycle $\partial[\tau]\in {\mathrm{HP}}^1 ({C^{\infty}_c}(S^{*}_H M) \rtimes G)$ over the algebra of Heisenberg principal symbols, which may easily be related to $\phi$. The main result of this paper (Theorem \[ttodd\]) is a geometric realization of this cocycle :

\[intro thm equivariant index\] Let $M$ be a foliated manifold and $G$ be a discrete group of foliated diffeomorphisms. Let $EG$ be the universal bundle over the classifying space $BG$ de $G$. Let $$0 \longrightarrow \Psi^{-1}_{H,c}(M)\rtimes G \longrightarrow \Psi^0_{H,c}(M)\rtimes G \longrightarrow C^{\infty}_c (S^*_HM) \rtimes G \longrightarrow 0$$ be the equivariant Heisenberg pseudodifferential extension. Then, the image of the localized trace at the unit $\partial[\tau]\in {\mathrm{HP}}^1 ({C^{\infty}_c}(S^{*}_H M) \rtimes G)$ by excision is given by $$\partial([\tau]) = \Phi({\mathrm{Td}}(TM\otimes{\mathbb{C}}))$$ where $\Phi: H^{\mathrm{ev}}(EG\times_{G} S^*_HM) \to HP^1(C^{\infty}_c(S^*_HM)\rtimes G)$ is Connes’ characteristic map from equivariant cohomology to cyclic cohomology, and ${\mathrm{Td}}(TM\otimes{\mathbb{C}}) $ is the equivariant Todd class of the complexified tangent 