Lifting projective representations of a simply connected semi-simple complex lie group Excuse the awkwardly long title.
Let $G$ be a simply connected semi-simple complex lie group.  Then I know that every smooth finite dimensional projective representation of $G$, $G\to PGL(V)$, can be lifted to an honest representation $G\to SL(V)$.
However I would like to know whether this $\textit{always}$ holds: if $G\to PGL(V)$ is a projective representation of $G$ of any dimension (finite or infinite, and no further requirements on the map), can we always lift this to an honest representation $G\to GL(V)$?
I understand this question is somehow related to group cohomology but the precise relationship is eluding me.  Any help is appreciated!
 A: Given representation $\rho:G\to\text{PGL}(V)$, pick a random lift $\widetilde{\rho(g)}$ for each $g$. This is a genuine representation of $G$ iff $\rho(1)$ is lifted to the identity and
$$\widetilde{\rho(ab)}\ =\ \widetilde{\rho(a)}\widetilde{\rho(b)}$$
for all $a,b$. The map 
$$G\times G\ \longrightarrow \ k^\times \ \ \ \text{ sending} \ \ \ a,b\ \longrightarrow \widetilde{\rho(ab)}(\widetilde{\rho(a)}\widetilde{\rho(b)})^{-1}$$
measures the failure of the lift to give a genuine representation. It's annoying to check, but it a $2$-cocycle, where $k^\times$ is the trivial $G$ module.
The $2$-coboundaries are of the form 
$$a,b\ \longrightarrow \ \lambda(ab)(\lambda(a)\lambda(b))^{-1}$$
for some function $\lambda:G\to k^\times$.
So we have a map 
$$\text{lifts}\ \longrightarrow \ H^2(G,k^\times)$$
If a lift $\widetilde{\rho}$ lands in $0$, then it differs by a coboundary from $0$, i.e. there is a function $\lambda$ with 
$$\lambda\widetilde{\rho}(ab)\left(\lambda\widetilde{\rho}(a)\lambda\widetilde{\rho}(b)\right)^{-1}\ = \ 1 $$
where $\lambda\widetilde{\rho}(a)$ means $\lambda(a)\widetilde{\rho}(a)$. That is, we can tweak the lift $\widetilde{\rho}$ to make it a homomorphism.

Conclusion: For any group $G$ with $H^2(G,k^\times)=0$, any projective representation of $G$ lifts to a genuine representation.


Note that $H^2(G,k^\times)$ corresponds to the central extensions of $G$ by $k^\times$. For instance, this implies that $H^2(\text{PGL}(V),k^\times)\ne0$, which we expect given the above. I think $H^2(G,k^\times)$ vanishes for $G$ simply connected and semisimple, but I can't find a reference.
