Let $G$ be a Lie group and $\mathfrak{g}$ be its Lie algebra. Denote the universal enveloping algebra by $\mathcal{U}(\mathfrak{g})$.
When studying finite-dimensional unitary representations $(\pi,V)$ of $G$, we have that if $G$ is connected then $(\pi,V)$ is irreducible if and only if the Lie algebra representation $(\pi_*,V)$ of $\mathfrak{g}$ is irreducible. This is also equivalent to the extension $(\tilde{\pi}_*,V)$ being irreducible as an associative algebra representation of $\mathcal{U}(\mathfrak{g})$.
I am looking for a reference which explains how this can be extended to the infinite dimensional case. How does one 'differentiate' the representation (can we even do this in a sensible way?) and is there an analog to the above equivalences of irreducibility?