Homotopy Representations and the Picard Group of the Equivariant Stable Homotopy Category

If $G$ is a finite group or a torus, it is known that there is an isomorphism between the Grothendieck group of homotopy representations and that of generalized homotopy representations for $G$. We prove that there is such an isomorphism when $G$ is a compact Lie group with component group $\mathrm{\Gamma }$ having the property that all projective $\mathbb{Z}\mathrm{\Gamma }$-modules are stably free. This resolves a conjecture of Fausk, Lewis, and May for such $G$, giving a better description of the Picard group of the homotopy category of $G$-spectra.