# Picard group of moduli of principal bundles

I am looking for the Picard group of the moduli space of principal $$G$$-bundles for a connected reductive complex algebraic group $$G$$.

Is it isomorphic to $$\mathbb{Z}$$? If not, what can we say when $$G=\mathrm{Sp}(2n,\mathbb{C})?$$

Is there any reference for this?

shows that when $$G$$ is a simple simply-connected connected complex affine algebraic group, $$C$$ is a complex smooth irreducible projective curve of genus at least 2, and $$M$$ is the moduli space of semistable principal $$G$$-bundles on $$C$$, then $$\mathrm{Pic}(M)\cong \mathbb{Z}.$$
• I do not know of any references that specifically address the non-simple case. I suspect that if $G$ is complex, reductive (and connected) then the above result can be used to determine what happens when the derived subgroup $DG$ is simply-connected. That intuition comes from my paper with C. Manon that does the computation for $G$-character varieties of punctured surfaces: arxiv.org/pdf/1504.01210.pdf (corollary 2.4). But of course that is a different moduli space (although a related one). Jun 15 at 15:42