$\g g$-module, with the action $$\label{xcdtTT}
(x\cdot T)u:=\pi^{\prime}(x)Tu-(-1)^{|T||x|}T\pi(x)u\,\$$ for $x\in\g g$, $T\in\Hom_\C%
(U,U^{\prime })$, and $u\in U$. The special case $U^*:=\Hom_\C(U,\C^{1|0})$, where $\C^{1|0}$ is the trivial $%
\g g$-module, is the *contragredient* $\g g$-module. Moreover, the map is $\g g$-equivariant. Set

$$\Hom_\g g(U,U^{\prime }):= \Hom_\C(U,U^{\prime})^{\g g} \ \text{ and }\ \mathrm{%
End}_\g g(U):=\mathrm{End}_\C(U)^\g g.$$

The category of $\g g$-modules is a symmetric monoidal category, and $$\Mor_{\g g\text{-mod}}(U,U^{\prime })\cong \Hom_\g g(U,U^{\prime })_\eev
.$$

Fix a $\Ztwo$-graded vector space $W$. For every homogeneous $w\in W$, let $\partial_w$ be the superderivation of $\sP(W)$ with parity $|w|$ that is defined uniquely by $$\label{pww*}
\partial_w(w^*):=(-1)^{|w|}\lag w^*,w\rag\text { for every } w^*\in W^*\cong %
\sP^1(W).$$ Thus, $\partial_w(a_1a_2)=(\partial_w a_1)a_2+(-1)^{|w|\cdot|a_1|}a_1\partial_wa_2$ for every $a_1,a_2\in\sP(W)$. For every $b\in \sS(W)$ we define $\partial_b\in%
\mathrm{End}_\C(\sP(W))$ as follows. First we set $$\partial_{w_1\cdots w_r}:=\partial_{w_1}\cdots\partial_{w_r} \text{ for homogeneous }%
w_1,\ldots,w_r\in W,$$ and then we extend the definition of $\partial_{b}$ to all $b\in \sS(W)$ by linearity.

Let $\sPD(W)$ be the associative superalgebra of polynomial-coefficient differential operators on $W$. More explicitly, $\sPD(W)$ is the subalgebra of $\mathrm{End}_\C(\sP(W))$ spanned by elements of the form $a\partial_b$, where $a\in \sP(W)$ and $b\in\sS(W)$. If $W$ is a $\g g$-module, then $\sPD%
(W)$ is a $\g g$-invariant subspace of $\mathrm{End}_\C(\sP(W))$ with respect 