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?



Norm continuous infinite dimenisonal representation of a Lie group
Your question also makes sense for a Banach space and that is where the following results also apply. The Hilbert space case is then only a particular case where you can get some more information about the infinitesimal generators.


So let $G$ be a connected Lie group (the discrete pieces are sort of uninteresting...)
For a Banach space $V$ a group morphism $U\colon G \longrightarrow B(V)$ into the (invertible) bounded operators on $V$ is norm continuous iff it is of the form
\begin{equation}
U\_{\exp(t\xi)} = \sum\_{r=0}^\infty \frac{t^n}{n!} A\_\xi^n
\end{equation}
for all $\xi \in \mathfrak{g}$ (the Lie algebra of $G$) and with a continuous operator $A\_\xi$, where the exponential series converges in the operator norm. The map $\xi \mapsto A\_\xi$ is then a Lie algebra representation by continuous operators on $V$.


The proof of this fact is essentially as in finite dimensions so you can take you favorite differential geometry book/Lie group book and copy the proof. Note that such a representation automatically will also converge for complex $t$ and hence gives a representation of the complexified Lie group on the nose.


The reason why this is sort of representation is rather un-interesting is that in "real life" they simply don't show up. The simnplest example is perhaps the translation group acting on $L^2(\mathbb{R})$ for which the above statement is false. In "real life" the generators $A\_\xi$ of the representation will typically be unbounded operators and the exponential series does not converge in a norm sense. Nevertheless, a big part of harmonic analysis deals with the question on which vectors of $V$ one can have a Lie algebra representation such that the exponential series will converge after applied to such a vector (then in the norm sense of $V$).


