\longrightarrow S_{H,c}^{0}(M) \rtimes G \longrightarrow 0,$$ where $\Psi^{-\infty}_c(M)\subset \Psi_{H,c}^{0}(M)$ is the two-sided ideal of smoothing operators in the algebra of order $\leq 0$ Heisenberg pseudodifferential operators, and $S_{H,c}^{0}(M) = \Psi_{H,c}^{0}(M)/\Psi^{-\infty}_c(M)$ is the quotient algebra of formal Heisenberg symbols. Let ${\mathrm{Tr}}_{[1]}$ be the trace on $\Psi_{c}^{-\infty}(M) \rtimes G$ obtained from the usual trace on $\Psi_{c}^{-\infty}(M)$ by localization at the unit of $G$ : $${\mathrm{Tr}}_{[1]}\left(\sum_{g \in G} P_g U_g \right) = {\mathrm{Tr}}(P_1)$$ We will mainly be interested in the image of ${\mathrm{Tr}}_{[1]}$ under the excision map in periodic cyclic cohomology $\partial: {\mathrm{HP}}^0(\Psi_{c}^{-\infty}(M) \rtimes G) \to {\mathrm{HP}}^1({\mathcal{S}}_{H,c}(M) \rtimes G)$.

\[intro thm equivariant Radul cocycle\] The boundary of the localized operator trace $\partial [{\mathrm{Tr}}_{[1]}] \in {\mathrm{HP}}^1({\mathcal{S}}_{H,c}(M) \rtimes G)$ is represented by the *equivariant Radul cocycle* $$\phi(a_0, a_1) = {\mathop{\ooalign{$\displaystyle{\int}$\cr$-$}}}\big(a_0 [\log \Delta_H^{1/4}, a_1]\big)_{[1]}$$ where $\Delta_H^{1/4}$ is the sub-elliptic sub-laplacian (Example \[subelliptic laplacian\]) associated to $M$, the subscript $_{[1]}$ denotes the term localized at the unit, and the integral denotes the Connes-Moscovici residue over the algebra of formal symbols $S^0_H(M)$.

We refer to section \[sindex\] for a description of the Connes-Moscovici residue. The cocycle $\phi$ may be viewed as a *local* formula in the sense that it only involves the formal Heisenberg symbol of pseudodifferential operators, 