# Notions of integrability for affine Lie algebras and positive energy representations

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 $$\hat{G}^{aff}_{\kappa}$$?