$${(X_1+\xi_1,X_2+\xi_2)}= {\tfrac{1}{2}}\big({\langle \xi_1 | 
X_2\rangle}+{\langle \xi_2 |  X_1\rangle}\big) ,$$ where $X_i\in{T_{{\mathbb{C}}}{\boldsymbol{X}}}$ and $\xi_i\in{T^*_{{\mathbb{C}}}{\boldsymbol{X}}}$. The corresponding Clifford bundle ${\mathcal{C}({\boldsymbol{E}})}$ can be identified with the vector bundle $(\wedge^\bullet{T_{{\mathbb{C}}}{\boldsymbol{X}}})\otimes(\wedge^\bullet{T^*_{{\mathbb{C}}}{\boldsymbol{X}}})$, under which the Clifford action of ${\mathcal{C}({\boldsymbol{E}})}$ on the spinor bundle $$\wedge^\bullet{T^*_{{\mathbb{C}}}{\boldsymbol{X}}}=\bigoplus_{p,q}{(T^{p,q}{\boldsymbol{X}})^*}$$ is given by $$(W\otimes \xi ){\cdot}{\lambda}=
(-1)^{\frac{w(w-1)}{2}}{\iota_{W}}(\xi \wedge {\lambda}) .$$ Here $W\in \wedge^w {T_{{\mathbb{C}}}{\boldsymbol{X}}}$, $\xi , {\lambda}\in \wedge^\bullet
{T^*_{{\mathbb{C}}}{\boldsymbol{X}}}$, and the symbol ${\iota_{W}}$ denotes the standard contraction $${\langle {\iota_{W}}\xi  | X\rangle}={\langle \xi  | W\wedge X\rangle} ,$$ for $\xi\in\wedge^p{T^*_{{\mathbb{C}}}{\boldsymbol{X}}}$ and $X\in\wedge^{p-w}{T_{{\mathbb{C}}}{\boldsymbol{X}}}$ with $p\geq w$.

Let $({\boldsymbol{X}},H)$ be an extended Poisson manifold of complex dimension $n$. Then ${{A}^*}_H$ is a Lie algebroid and the **Evens-Lu-Weinstein module** [@MR1726784] of ${{A}^*}_H$ is the complex line bundle $${Q_{{{A}^*}_H}}=\wedge^{2n}{{A}^*}_H\otimes\wedge^{2n}{T^*_{{\mathbb{C}}}{\boldsymbol{X}}}.$$ The representation of ${{A}^*}_H$ on ${Q_{{{A}^*}_H}}$ is given by $$\begin{gathered}
\label{Eqn:RepELW}
{\nabla^H}_{\alpha}(\alpha_1\wedge\cdots\wedge\alpha_{2n}\otimes\mu)
=\sum_{i=1}^{2n}\big(\alpha_1\wedge\cdots\wedge{[\alpha,\alpha_i]_*^H}
\wedge\cdots\wedge\alpha_{2n}\otimes\mu\big) \\
+\alpha_1\wedge\cdots\wedge\alpha_{2n}\otimes{L_{{{{a}_*}^H}(\alpha)}}\mu,\end{gathered}$$ where $\alpha,~\alpha_1,\cdots ,\alpha_{2n}\in{\Gamma({{A}^*}_H)}$, $\mu\in {\Gamma(\wedge^{2n} {T^*_{{\mathbb{C}}}{\boldsymbol{X}}})}$.

A simple computation yields that ${Q_{{{A}^*}_H}}\cong\wedge^{n}{{(T^{1,0}{\boldsymbol{X}})^*}}\otimes\wedge^{n}{{(T^{1,0}{\boldsymbol{X}})^*}}$. Accordingly, $${\mathscr{L}}={Q^{{\tfrac{1}{2}}}_{{{A}^*}_H}}\cong\wedge^n{{(T^{1,0}{\boldsymbol{X}})^*}}={(T^{n,0}{\boldsymbol{X}})^*}$$ is also an ${{A}^*}_H$-module and we use ${\nabla^H}$ again to denote the representation. Equivalently, we have an operator $$\label{Eqt:DDH} 
{{\mathcal{D}}^H}:~{\Gamma({\mathscr{L}})}\to{\Gamma({A}\otimes{\mathscr{L}})},$$ such that $${\iota_{\alpha}}{{\mathcal{D}}^H}s={\nabla^H}_{\alpha}s,\quad\forall\alpha\in{\Gamma({{A}^*})}, s\in{\Gamma({\mathscr{L}})} ,$$ which allows us to define a differential operator $${{\breve{{d}}}_*^H}:{\Gamma(\wedge^k{A}\otimes{\mathscr{L}})}\to{\Gamma(\wedge^{k+1}{A}\otimes{\mathscr{L}})}$$ by $$\label{Eqt:bdeesHuotimess}
{{\breve{{d}}}_*^H}(u\otimes s)=({\bar{\partial}^H}u)\otimes s+(-1)^{k}u\wedge{{\mathcal{D}}^H}s,$$ for all $u\in{\Gamma(\wedge^k{A})}$ and $s\in{\Gamma({\mathscr{L}})}$.

The following lemma is needed later.

The relation $${\tau}(X \otimes s) = X{\cdot}s ,$$ where 