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

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