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,



Invariant complement to invariant subspace. 
This happens if and only if $U$ is expressible as a sum of pairwise non-isomorphic irreducible $G$-modules. If, for example $U \cong V \oplus V$ for an irreducible $G$-module $V,$ then the natural invariant submodule $ \{(v,0): v \in V \}$ has at least two complements: one is the natural choice $\{(0,v): v \in V \}.$ Another is $\{(v,v): v \in V \}.$ On the other hand, if $U \cong \bigoplus\_{i=1}^{n} V\_i,$ where $V\_i \not \cong V\_j$ for $i \neq j,$ then the only $G$-submodules of $U$ are of the form $V\_I = \bigoplus\_{i \in I} V\_i$ for a subset $I$ of $\{1,2,\ldots ,n\},$ and each such $G$-submodule $V\_I$ has the unique complement $V\_{I^{\prime}},$ where $I^{\prime} = \{ 1,2,\ldots ,n \} \backslash I.$


