Moduli spaces of framed $G$--Higgs bundles and symplectic geometry

Let $X$ be a compact connected Riemann surface, $D\subset X$ a reduced effective divisor, $G$ a connected complex reductive affine algebraic group and $H_x⊊G_x$ a Zariski closed subgroup for every $x\in D$. A framed principal $G$–bundle is a pair $(E_G,\varphi )$, where $E_G$ is a holomorphic principal $G$–bundle on $X$ and $\varphi$ assigns to each $x, \in, D$apointofthequotientspace$(E_G)x/H_x$\mathrm{.}Aframed$G$–Higgs bundle is a framed principal $G$–bundle $(E_G,\varphi )$ together with a section $\theta, \in, H^0(X,, $\text{ad}$(E_G)\otimes K_X\otimes{\mathcal O}X(D))$suchthat$\theta(x)$iscompatiblewiththeframing$\phi$forevery$x, \in, D$. We construct a holomorphic symplectic structure on the moduli space ${\mathcal{M}}_{FH}(G)$ of stable framed $G$–Higgs bundles. Moreover, we prove that the natural morphism from ${\mathcal{M}}_{FH}(G)$ to the moduli space ${\mathcal{M}}_H(G)$ of $D$-twisted $G$–Higgs bundles $(E_G,\theta )$ that forgets the framing, is Poisson. These results generalize \cite{BLP} where $(G,\{H_x{\}}_{x\in D})$ is taken to be $($\text{GL}$(r,{\mathbb C}),, {$\text{I}${r\times r}}{x\in D})$\mathrm{.}WealsoinvestigatetheHitchinsystemfor$$\mathcal{M}${FH}(G)$anditsrelationshipwiththatfor$$\mathcal{M}${H}(G)$.