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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?

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

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  • $\begingroup$ Thank you for your answer. Is there any specific result when G is not simple? $\endgroup$
    – student
    Jun 15 at 15:14
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    $\begingroup$ 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). $\endgroup$ Jun 15 at 15:42

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