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?



Picard group of moduli of principal bundles
Theorem A in:


S. Kumar and M. S. Narasimhan. *Picard group of the moduli spaces of G-bundles.* Math. Ann., 308(1):155-173, 1997,


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 $\mathrm{Pic}(M)\cong \mathbb{Z}.$


