  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 $\Gamma$
having the property that all projective $\mathbb{Z}\Gamma$-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.
