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There are a fair number of papers by Beauville, Laszlo, Sorger, Kumar and others on the geometry of $LG/L^+G = G((z))/G[[z]]$ where $G$ is a simply connected and simple group over $\mathbb{C}$. In particular it is known that $Pic(LG/L^+G)$ is equal to $\mathbb{Z}$. The pull back to $LG$ of a generator $1 \in Pic(LG/L^+G)$ is nontrivial and can be chosen so it corresponds to the basic central extension of $LG$. So $Pic(LG)$ is at least $\mathbb{Z}$ and I think that should be everything.

A divisor for the basic line bundle can be taken to be the complement of $L^-G\cdot L^+G$ where $L^-G = G[z^{-1}]$ so I think the claim would follow if $Pic(L^-G \cdot L^+G) = 0$; but I haven't seen this question addressed in my limited knowledge of the literature.

So is it true that $Pic(LG) = \mathbb{Z}$ and what's a reference that discusses this?

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