Let $G$ a topological group, $H$ a complex Hilbert space and $U(H)$ the group of unitary operators. A unitary group representation is a group homomorphism $\phi:G\longrightarrow U(H)$ continuos respect the strong operator topology. $\phi$ is continuos if only if the induced action $G\curvearrowright H$ is continuos.
Q1: Why the strong operator topology is the right topology for the definition? (Instead of the weak operator topology, the norm topology or the compact-open). If moreover $G$ is a Lie group, we could restrict to $\phi$ such that the action $G\curvearrowright H$ is smooth. What implies this restriction?
I read that the continuos (with SOT) group homomorphism $\phi:\mathbb R \longrightarrow U(H)$ are in bijection with densely defined skew-selfadjoint operators $A$ in $H$. In particular we can define the exponential map $ \exp(A):=V(1)$.
Every $\phi:G \longrightarrow U(H)$ unitary representation of a Lie group induce the "differential" map $d\phi:Lie(G) \longrightarrow \mathcal S(H)$, with range $\mathcal S(H)$ the densely defined skew-selfadjoint operators in $H$, defined by: $$ \phi(e^{tX})=\exp(td\phi(X)) $$
Q2.1: $\mathcal S(H)$ forms a Lie algebra with the brackets $[A,B]:=AB-BA$. Is it true that $d\phi$ is a Lie algebra homomorphism?
Q2.2: If $G$ is a simply-connected Lie group, does every Lie algebra homomorphism $d\phi:Lie(G) \longrightarrow \mathcal S(H)$ come from the differential of a unitary representation? What can we say if we change the topology in $U(H)$ or if we restrict to smooth (unitary) actions $G\curvearrowright H$?
Q3: Are there any good reference for these questions and more topics on group representations?