I am looking for a reference for the calculation of the negative cyclic homology of the group algebra $\mathbb{K}[\Gamma]$ of a discrete group $\Gamma$ over a field $\mathbb{K}$ of characteristic 0. (Results over $\mathbb{Z}$ are also welcome.) I’ve found this paper by Dan Burghelea: The cyclic homology of the group rings, but it calculates cyclic homology instead of negative cyclic homology.
A broader question is about results on the calculation of the $S^1$-equivariant homology theory $G^{\mathbb{T}}_*$ introduced by Jones in Cyclic homology and equivariant homology. I believe the negative cyclic homology of $\mathbb{K}[\Gamma]$ is isomorphic to the $G^\mathbb{T}$-homology of the free loop space of $B\Gamma$, as a special case.
Results on $HC_*$ and $H^\mathbb{T}_*$ seem to be much better known. How about $HC^-_*$ and $G^\mathbb{T}_*$?
