Let $V$ be a $\mathbb{C}$-vectorial space of dimension $n$ and $V^*$ the complex dual space.
I would like to understand the following isomorphism $$D_{V^* \leftarrow V \times V^*} \overset{L}\otimes_{\mathcal{O}_{V \times V^*}} D_{V \times V^* \rightarrow V} \simeq D_{V \times V^*} \otimes det(V^*).$$ Here we use the standard notations for $D$-modules. In this case we work with Weyl algebras and $det(V^*)$ is the $n-th$ exterior product of $V^*$. The maps involved in the transfer modules are of course the projections.
Thank you for any help.