Is there an example of $G$, $\rho$ as below?

• $G$ is a locally compact group.

• $\rho$ is a continuous representation of $G$ on a Hilbert space $V$. This means that we have a continuous homomorphism from $G$ to the group of bounded linear operators on $V$ with bounded inverse, such that $G \times V \rightarrow V$ is continuous.

• Schur's lemma fails for $\rho$.

(Asked first on MSE: https://math.stackexchange.com/questions/3096704/failure-of-schurs-lemma-for-topological-group-representations)

Is there an example of $G$, $\rho$ as below?

• $G$ is a locally compact group.

• $\rho$ is an irreducible continuous representation of $G$ on a complex Hilbert space $V$. This means that we have a continuous homomorphism from $G$ to the group of bounded linear operators on $V$ with bounded inverse, such that $G \times V \rightarrow V$ is continuous. And there are no nontrivial closed invariant subspaces.

• Schur's lemma fails for $\rho$.

(Asked first on MSE: https://math.stackexchange.com/questions/3096704/failure-of-schurs-lemma-for-topological-group-representations)

Added later: For concreteness, I will record below one version of Schur's lemma that I have in mind. (But I'm not too picky about this.)

Schur's lemma: Every bounded operator commuting with $\rho(G)$ is a scalar.