Poisson structures on certain moduli spaces for bundles on a surface

Let $\mathrm{\Sigma }$ be a closed surface, $G$ a compact Lie group, with Lie algebra $g$, and $\xi \text{\hspace{0.17em}⁣}:P\to \mathrm{\Sigma }$ a principal $G$-bundle. In earlier work we have shown that the moduli space $N(\xi )$ of central Yang- Mills connections, for appropriate additional data, is stratified by smooth symplectic manifolds and that the holonomy yields a diffeomorphism from $N(\xi )$ onto a certain representation space $\text{roman}{Rep}_{\xi }(\mathrm{\Gamma },G)$, with reference to suitable smooth structures $C^{\mathrm{\infty }}(N(\xi ))$ and $C^{\mathrm{\infty }}(\text{roman}{Rep}_{\xi }(\mathrm{\Gamma },G))$ where $\mathrm{\Gamma }$ denotes the universal central extension of the fundamental group of $\mathrm{\Sigma }$. Given an invariant symmetric bilinear form on $g^{\ast }$, we construct here Poisson structures on $C^{\mathrm{\infty }}(N(\xi ))$ and $C^{\mathrm{\infty }}(\text{roman}{Rep}_{\xi }(\mathrm{\Gamma },G))$ in such a way that the mentioned diffeomorphism identifies them. When the form on $g^{\ast }$ is non-degenerate the Poisson structures are compatible with the stratifications where $\text{roman}{Rep}_{\xi }(\mathrm{\Gamma },G)$ is endowed with the corresponding stratification and, furthermore, yield structures of a {\it stratified symplectic space/}, preserved by the induced action of the mapping class group of $\mathrm{\Sigma }$.