cyclic 1-cocycle over the completed tensor algebra of ${\mathcal{A}}={C^{\infty}_c}(S_H^*M)\rtimes G$, $$(\phi\circ\sigma_*)({\hat{a}}_0,{\hat{a}}_1) = {\mathop{\ooalign{$\displaystyle{\int}$\cr$-$}}}\big(\sigma({\hat{a}}_0) [\log \Delta^{1/4}, \sigma({\hat{a}}_1)]\big)_{[1]} \ ,\qquad \forall \ {\hat{a}}_0,{\hat{a}}_1\in {\widehat{T}}{\mathcal{A}}\ ,$$ for any choice of sub-Laplacian $\Delta$ and linear splitting $\sigma:{\mathcal{A}}\to \Psi_{H,c}^0(M)\rtimes G$.

We have so far obtained a local index formula for a $G$-Heisenberg elliptic operator, in the sense that it is not sensitive under perturbations by elements in $\Psi_c^{-\infty}(M) \rtimes G$. However, the formula is really hard to compute explicitely, because we have to calculate the term of degree $-v-2h$ in the expansion of the symbol $aU_g [\log \Delta^{1/4}, bU_{g^{-1}}]$, and adding the fact that the symbolic calculus is not commutative, the task is even more difficult.\
The general strategy to handle this problem is the construction of a cohomologous $(b,B)$-cocycle, in which the leading symbol is exactly of degree $-v-2h$. Then, we will just have to take the principal symbol to obtain a formula as an integral over the Heisenberg cosphere bundle. This will be possible through an “algebraic JLO formula” whose entries are formal (Heisenberg) symbols. This construction includes the following main steps:

-   An algebra of operators acting on symbols,

-   A related notion of “heat kernel”,

-   A related notion of trace,

