An algebraic model for free rational G-spectra for connected compact Lie groups G

We show that the category of free rational G-spectra for a connected compact Lie group G is Quillen equivalent to the category of torsion differential graded modules over the polynomial cohomology ring on the classifying space, H*(BG). The ingredients are enriched Morita equivalences, functors making rational spectra algebraic, and Koszul duality and thick subcategory arguments based on the simplicity of the derived category of a polynomial ring.