Picard group and fundamental group of the moduli of Higgs bundles on curves

Let $X$ be an irreducible smooth projective curve of genus $g\ge 2$ over $\mathbb{C}$. Let $G$ be a connected reductive affine algebraic group over $\mathbb{C}$. Let ${\mathrm{M}}_{G,\mathrm{H}\mathrm{i}\mathrm{g}\mathrm{g}\mathrm{s}}^{\delta }$ be the moduli space of semistable principal $G$–Higgs bundles on $X$ of topological type $\delta \in \pi_1(G)$. In this article, we compute the fundamental group and Picard group of ${\mathrm{M}}_{G,\mathrm{H}\mathrm{i}\mathrm{g}\mathrm{g}\mathrm{s}}^{\delta }$.