String topology prospectra and Hochschild cohomology

We study string topology for classifying spaces of connected compact Lie groups, drawing connections with Hochschild cohomology and equivariant homotopy theory. First, for a compact Lie group G, we show that the string topology prospectrum LBG^{− TBG} is equivalent to the homotopy fixed-point prospectrum for the conjugation action of G on itself, S⁰[G]^hG. Dually, we identify LBG^-ad with the homotopy orbit spectrum (DG)_hG, and study ring and co-ring structures on these spectra. Finally, we show that in homology, these products may be identified with the Gerstenhaber cup product in the Hochschild cohomology of C^*(BG) and C_*(G), respectively. These, in turn, are isomorphic via Koszul duality.