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More precisely, there are two ways to think of D-modules on a stack $X/G$ (aka $G$-equivariant D-modules on $X$). … the second step, fixing the action of the Lie algebra (or equivalently the formal group) tells you which sheaves on $X_{dR}/G$ actually come from $X_{dR}/G_{dR}$, ie are strongly equivariant.. …

The idea that the equivariant version of the derived category consists simply of complexes with equivariance structure is correct, once you work in a refined enough setting such as differential graded, … But at this enhanced level, the equivariant derived category is indeed just the derived invariants of the group acting on the derived category of X. …

The equivariant derived category in this case is equivalent to modules for the homology of the circle, ie exterior algebra on a generator in degree -1. … The (dg-enhanced) equivariant derived category of a point is equivalent to modules for the dg algebra $C_*(G)$ of chains on $G$ under convolution (the ``topological group algebra"). …

The simplicial variety you write is a presentation of the stack $X/G$ as a simplicial sheaf, and the motive you obtain is just be the motive of $X/G$, no? Of course you might use this as the definitio …

Until the relevant experts arrive, I can mention that one relevant key phrase is "Waldhausen A-theory" (or K-theory of spaces, cf. Waldhausen's paper with this title), which is the K-theory of $\Omega …