T^{*}_x M$, the metric is the duality bracket. Let us also recall the commutation relations of symbols : in the coordinate system $(x,p)$ over $T^{*}U$, we have : $$[x^{i},p_j] = -{\textup{\textbf{i}}}\delta^i_{j}, \quad [x^{i},x^{j}] = [p_i, p_j] = 0$$ Thus, every element $a \in {\mathcal{S}}_H(M,E)$ is locally a formal series $$a(x,p,\psi,\overline{\psi}) = \sum_{j=0} a_{m-j}(x,p,\psi,\overline{\psi})$$ where the functions $a_{m-j}$ are Heisenberg-homogeneous in $p$ and polynomial in the variables $\psi$ and $\overline{\psi}$.

Wodzicki residue on $\Psi_H(M)$
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Let $M$ be a foliated manifold. By choosing a Riemannian metric on $M$, one can construct a sub-elliptic sub-laplacian $\Delta$ as in the flat example \[subelliptic laplacian\]. Its complex powers $\Delta^{-z}$ are defined as properly-supported Heisenberg pseudodifferential operators, using the parametrix $(\lambda - \Delta)^{-1}$ and an appropriate Cauchy integral $$\Delta^{-z} = \dfrac{1}{2 \pi {\textup{\textbf{i}}}} \int \lambda^{-z} (\lambda - \Delta)^{-1} \, d\lambda$$ where the contour is a vertical line pointing downwards.\

\[CMzeta\] *(Connes - Moscovici, [@CM1995])* Let $M$ be a foliated manifold of dimension $n$, $v$ be the dimensions of the leaves, $h$ their codimension, and $P \in \Psi^m_{H,c}(M)$ be a compactly-supported Heisenberg pseudodifferential operator of order $m \in {\mathbb{R}}$. Then, for any sub-elliptic sub-laplacian $\Delta$, the zeta function $$\zeta_P(z) = {\mathrm{Tr}}(P \Delta^{-z/4})$$ is holomorphic on the 