Let $G$ be a finite group acting on $V$, a complex affine variety. Suppose $\pi:V\to V/G$ is the quotient map. $V/G$ is most likely singular, consider a map $i:V/G\hookrightarrow Y$ where $Y$ is affine smooth. Let $\pi'=i\circ \pi$. An example for all of the above is the nilcone into $\mathbb{A}^3$.
Consider $\pi'_*(\mathcal{O}_V)^G$ (this the the $D$-module theoretic pushforward), I want to show this is $\mathcal{O}_Y$. Because everything is affine, this is the same as asking $(\mathcal{O}_V\otimes_{D_V}\mathcal{O}_V\otimes_{\mathcal{O}_Y}D_Y)^G\cong \mathcal{O}_Y$. (Viewing $\mathcal{O}_V$ as a right module, this is just a minor thing, or just use $\Omega_V$ instead.)
If $V/G$ has only isolated singularities, then by the open-closed exact triangle, one can show this is correct.
I want to have a proof for this in general.