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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.

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  • $\begingroup$ Perhaps one could use the decomposition theorem? One can write $i_*IC_{V/G} \overset{\oplus}{\hookrightarrow} R\pi'_*\mathbb{C}$ upto your preferred shift. Since $V/G$ is cohen-macaulay, by Serre duality $\mathcal{O}_{V/G}$ is the underlying left D-module for the perverse sheaf $IC_{V/G}$ (is this $\mathbb{C}_{V/G}$?). Hence, $i_*\mathcal{O}_{V/G} \overset{\oplus}{\hookrightarrow} \pi'_*\mathcal{O}_{V}$ is also a D-module summand. Does this make sense? $\endgroup$
    – guest0803
    Oct 9, 2020 at 5:25
  • $\begingroup$ @guest0803, sorry i actually dont know what decomposition theorem you are talking about, I have to admit my knowledge of algebraic geometry is not very good. Can you possibly spell out more detail? $\endgroup$ Oct 13, 2020 at 0:02

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