Invariant vector of a unitary representation

For a unitary representation $\pi:G\to\mathcal U(H)$ , an invariant vector is $v\in H$ with $\pi(g)v=v$ for every $g\in G$ . The invariant subspace is

H^G=\bigcap_{g\in G}\ker(\pi(g)-I).

It is closed, and its orthogonal complement is invariant.

Fixed vector versus invariant subspace

An invariant line may transform by a nontrivial character and contain no nonzero invariant vector. For Koopman representations, constants always form a trivial subrepresentation; ergodicity says the orthogonal complement of constants contains no invariant vector.