Let $G$ be a group. A unitary representation of $G$ on a Hilbert space $H$ is called primary if for any invariant subspace $V\subseteq H$ one has that either:
- $V\in \{0, H\}$.
- There is a non-zero intertwining operator $T:V\to V^\perp$.
As an example: If $M$ is an irreducible representation of $G$, $\omega$ a cardinal then then $M^\omega$ is primary. In fact if $H$ contains a single irreducible representation $M$ and $H$ is primary then $H=M^\omega$ for some $\omega$.
Primary representations that are not a multiple of an irreducible are thus not allowed to contain any irreducible sub-representations, which motivates my question:
Whats an example of a primary representation that is not a multiple of an irreducible representation?
It may be helpful to note that a representation is primary if and only if the von Neumann algebra generated by its image is a factor, ie has trivial center.