Equivariant sheaves on loop spaces

Let $X$ be an affine, smooth, and Noetherian scheme over $\mathbb{C}$ acted on by an affine algebraic group $G$. Applying the technique developed in $\href{https://doi.org/10.48550/arXiv.1807.03266}{[3, }\href{ https://doi.org/10.48550/arXiv.1812.03583}{4]}$, we define a dg‑model for the derived category of dg‑modules over the dg‑algebra of differential forms $\Omega_X$ on $X$ equivariant with respect to the action of a derived group scheme $(G, \Omega_G)$. We compare the obtained dg‑category with the one considered in $\href{https://doi.org/10.48550/arXiv.1510.07472}{[2]}$ given by coherent sheaves on the derived Hamiltonian reduction of $T^\ast X$.