Let $\mathfrak{g}$ be a simple (complex) Lie algebra. Given an invariant bilinear form $\kappa : \mathfrak{g} \otimes \mathfrak{g} \to \mathbb{C}$, we can form the central extension $\hat{\mathfrak{g}}_{\kappa}$ of the loop algebra $\mathfrak{g}((z))= \mathfrak{g} \otimes \mathbb{C}((z))$. At least when $\kappa$ is an integral multiple of the basic form, this Lie algebra should integrate to a central extension $\hat{G}_{\kappa}$ of the loop group $G((z))$, where $G$ is the simply connected semisimple group with Lie algebra $\mathfrak{g}$.
Now suppose we have a smooth representation $V$ of $\hat{\mathfrak{g}}_{\kappa}$, where the generator of the central $\mathbb{C}$ acts by $1$. Now some people say that $V$ is integrable if there exists a representation $V'$ of $\hat{G}_{\kappa}$ such that $V$ is the differential of $V'$. Others say that $V$ is integrable if for every positive root $\alpha$ of $\mathfrak{g}$ and integer $n$, the elements $e_{\alpha} \otimes z^n$ act locally nilpotently (here $e_{\alpha}$ is the generator of the corresponding root space).
Do these two notions of integrability coincide?
We can add a further constraint into the picture. The multiplicative group $\mathbb{C}^{\times}$ acts on $\hat{G}_{\kappa}$ by loop rotation, so we can form the semidirect product $\hat{G}^{aff}_{\kappa}$ and consider positive energy representations of this group (i.e. representations such that the loop rotation $\mathbb{C}^{\times}$ has only nonnegative nonzero weight spaces). Is there any connection between these representations and the integrable representations of $\hat{\mathfrak{g}}_{\kappa}$? Looking through the literature it seems that integrable representations of $\hat{\mathfrak{g}}_{\kappa}$ and positive energy representations of $\hat{G}^{aff}_{\kappa}$ behave similarly, although I haven't been able to find a precise statement relating all these things.
Assuming the two notions of integrability coincide, can we always lift an integrable representation of $\hat{\mathfrak{g}}_{\kappa}$ to a positive energy representation of $\hat{G}^{aff}_{\kappa}$?
