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

What are the sufficient conditions that for any choice of the initial scalar product, the resulting $V^\perp$ is unique.

There are examples when the complement is unique. For example, consider $\mathbb{R}^3$ and the action of $SO(2)$ given by the rotation around the $z$-axis. Then if $V$ is a $z$-axis it has a unique complement.

Particular example I am trying to understand is the following: consider the group $U(n)$ acting on $\mathbb{R}^{2n}$ in a standard way. This action induces a representation $\rho$ in $U=Hom(\mathbb{R}^{2n} \wedge \mathbb{R}^{2n}, \mathbb{R}^{2n})$ given by $$ (g S) (u\wedge v) = g^{-1}S (gu \wedge gv) $$ Consider a map $A: Hom(\mathbb{R}^{2n}, \mathfrak{u}(n)) \to Hom(\mathbb{R}^{2n} \wedge \mathbb{R}^{2n}, \mathbb{R}^{2n})$ defined by $$ (AS)(u \wedge v) = S(u)v - S(v)u $$ and define $V = A(Hom(\mathbb{R}^{2n}, \mathfrak{u}(n)))$.

For what $n$, there is a unique invariant complement to $V$?

Such construction appears in the Cartan's method of equivalence. Each $U(n)$-invariant complement to $V$ corresponds to a certain $U(n)$-invariant linear connection.

Thanks,