Let $G$ be a compact group and $\rho: G \to End(U)$ its linear representation in a finite dimensional vector space $U$. Fix $V \subset U$ - a subspace invariant under $\rho(G)$. Then it is well known that $V$ has some invairant complement.
What are the sufficient conditions (on $G$, $\rho$ or $V$) to ensure this complement is unique?
Stated another way, starting with any scalar product on $U$, invariant complements to $V$ can be found by averaging it w.r.t. to a Haar measure on $G$ and taking $V^\perp$. In this case my question becomes