Functors between equivariant derived category and derived category of the quotient

Question

Suppose $M$ is a quasi-projective variety, $G$ is a finite group acting on $M$. Let $X$ be the quotient $M/G$ (we assume $X$ to be singular) and $\pi: M\to X$ be the natural projection.

We have $(\pi)^{G}_{*}\circ \pi^{*}=id$, and in the case when $G$ is acting freely, the two functors are left and right adjoints of each other.

My question is:

1.What is the adjoint relation of the two functors in general?

2.I heard that since $X$ is singular, $L\pi^*$ would not be a map from $D^b(X)$ to $D^{b}(Coh_G(M))$. And this has to do with perfect complexes. Can someone explain this a little more and maybe provide some references?

Thanks!

Answer 1

The functors are still adjoint and the same relation holds. For details, see for example http://arxiv.org/pdf/1406.4409.pdf (especially the proof of Theorem 3.0.14) and references therein.