Given a Lie group G and an infinite dimensional Hilbert space $\mathcal{H}$. In the literature I have only encountered the two following notions of a representation $\pi$ of G on $\mathcal{H}$ :

1) $\pi$ : G $\to$ $B(\mathcal{H})$ is a group homomorphism (consider only the invertible bounded linear operators...) and $G \times \mathcal{H} \to \mathcal{H}$ is continuous.

2) $\pi$ : G $\to$ $B(\mathcal{H})$ is a continuous group homomorphism, where we consider the strong operator topology on $B(\mathcal{H})$.

I got the notion, that nobody seems interested in the case, when $\pi$ : G $\to$ $B(\mathcal{H})$ is a continuous group homomorphism with the usual operator topology on $B(\mathcal{H})$. If my impression is not correct, could you please give me some references that deal with this case or some examples? If my impression is correct, what are the reasons that one neglects this case?