Explicit determination of the Picard group of moduli spaces of semistable \(G\)-bundles on curves (Q2487134)

Let \(G\) be a connected, simply-connected, simple affine algebraic group, \(C\) a smooth irreducible projective curve of genus \(g\geq1\) over the complex numbers and \(M(G)\) the moduli space of S-equivalence classes of semistable principal \(G\)-bundles on \(C\). It was proved by \textit{S.~Kumar} and \textit{M.~S.~Narasimhan} [Math. Ann. 308, No. 1, 155--173 (1997; Zbl 0884.14004)] that \(\text{Pic}\,M(G)\) is isomorphic to \(\mathbb Z\), generalising the corresponding result for \(G= \text{SL}(n)\) proved by \textit{J.-M.~Drezet} and \textit{M.~S.~Narasimhan} [Invent. Math 97, No. 1, 53--94 (1989; Zbl 0689.14012)]. However a precise generator of \(\text{Pic}\,M(G)\) is known only for the classical groups and for \(G_2\). In this paper, the authors obtain an explicit generator of \(\text{Pic}\,M(G)\) for all \(G\).