Applications of the Liouville symplectic form on the cotangent bundle of a loop group

Let $G$ be a semisimple, simply connected, affine algebraic group defined over $\mathbb{C}$. Consider the Liouville symplectic structure on the total space $T^{\ast }G((t))$ of the cotangent bundle of the loop group $G((t))$, where $t$ is a formal parameter. We show that the Liouville symplectic structure on $T^{\ast }G((t))$ induces the symplectic structures on the moduli stack of framed principal Higgs $G$-bundles on a compact connected Riemann surface $X$ and also on the moduli spaces of framed $G$-connections on $X$. Similar symplectic structures – on the moduli stack of framed principal Higgs $G$-bundles, with finite order framing, and also framed connections on $X$, with finite order framing – were constructed earlier by various authors. Our results show that they all have a common origin.