Let $G$ be a compact Lie group and $V$ a finite-dimensional orthogonal $G$-representation. Write $S^V$ for the quotient $D(V)/S(V)$, where $D(V)$ and $S(V)$ are the unit disk and sphere in $V$, respectively. The space $S^V$ is known to be a finite pointed $G$-CW-complex.
Cappell and Shaneson [Ann. of Math. 113 (1981), 315-155] proved that there exists a finite group (plenty of them, in fact) and finite-dimensional orthogonal $G$-representations $V$, $W$ such that $V$ and $W$ are not equivalent as $G$-representations, but the $G$-homotopy types of $S^V$ and $S^W$ coincide.
I have found a note in another article [Smoller and Wasserman, Invent. Math. 100 (1990), 63-95] stating that this cannot happen for compact and connected Lie groups. Can anyone explain to me why this is the case?
EDIT: Retagged and added a reference as suggested by Ricardo Andrade.
