Let $\mathcal{C}$ be a category with a finite group action $G$, There is a notion called G-equivariant category, denoted by $\mathcal{C}^G$. In the paper Kuznetsov's Fano threefold conjecture via K3 categories and enhanced group actions by Alex Perry and Arend Bayer section 3.3, they call $\mathcal{C}^G$ an invariant category under $G$ and they also defined what is called coinvariant category, denoted by $\mathcal{C}_G$ in the language of infinity category.
They gave an example of $G$-invariant category $D_\text{perf}(X)^G\simeq D_\text{perf}([X/G])$. I have several questions on co-invariant category $\mathcal{C}_G$.
Are there any other reference on the construction of $G$ co-invariant category? For example, for $G$-invariant category $\mathcal{C}^G$, we have a concrete description of objects in $\mathcal{C}^G$ but also the morphism between two objects in this category.
If $\mathcal{C}=D_\text{perf}(X)$, what does $G$-coinvariant category $\mathcal{C}_G$ look like?
Let $Y$ be a smooth cubic threefold with a semi-orthogonal decomposition $$D^b(Y)=\langle\mathcal{A}_Y,\mathcal{O}_Y,\mathcal{O}_Y(1)\rangle,$$ where $\mathcal{A}_Y$ is the non-trivial component. Assume that there is a geometric involution $\tau$ defined on $Y$. Then by some well known results of Alexey Elagin, the $\tau$-invariant category $D^b(Y)^{\tau}$ admits the semi-orthogonal decomposition $$D^b(Y)^{\tau}=\langle\mathcal{A}_Y^{\tau},\langle\mathcal{O}_Y\rangle^{\tau},\langle\mathcal{O}_Y(1)\rangle^{\tau}\rangle,$$ and we know $\langle\mathcal{O}_Y\rangle^{\tau}=\langle\mathcal{O}_Y\otimes\chi_0,\mathcal{O}_Y\otimes\chi_1\rangle$, where $\chi_0,\chi_1$ are character of $\langle\tau\rangle=\mathbb{Z}_2$.
I was wondering for $\tau$-coinvariant category $D^b(Y)_\tau$, do we also have semi-orthogonal decomposition? What does it look like?
- Let $Y$ be a smooth quartic double solid, which is a double cover of $\mathbb{P}^3$ branched over a smooth K3 surface $S$ of degree 4, then by results of Kuznetsov–Perry that $\mathcal{A}_Y^{\tau}\simeq D^b(S)$,
What does $\tau$-coinvariant category $\mathcal{A}_Y$ look like?
