I understand that when $G$ is a compact, connected, simple and simply connected Lie group, there is a universal central extension of the loop group $LG$, which is denoted as $\widetilde{LG}$.
My question is regarding Proposition (9.3.9) in 'Loop Groups' by Andrew Pressley and Graeme Segal. I do not understand all the details in this proposition, but it seems like they are trying to say that, when $G$ is also simply laced, all irreducible representations of $\widetilde{LG}$ are obtainable from irreducible representations of $\widetilde{LG}$ of level one. So it seems to me that irreducible representations of $\widetilde{LG}$ effectively always have level one. Is this correct?