Classical BV formalism for group actions

We study the derived critical locus of a function $f:[X/G]\to \mathbb{A}_{\mathbb{K}}^1$ on quotient stack smooth affine scheme $X$ by action group $G$. It is shown that $\mathrm{dCrit}(f) \simeq [Z/G]$ for $Z$, whose dg-algebra functions described explicitly. Our results generalize classical BV formalism in finite dimensions from Lie algebra to actions.