Coincident Poisson structures on principal-bundle moduli spaces

Abstract

Consider a reductive complex algebraic group $\mathbb{G}$ equipped with an action by a linearly reductive affine group scheme $\mathbb{K}$. The extension of $\mathfrak{p}^*$ by $\mathfrak{p}$ induced by an $(\mathbb{K},\mathfrak{g})$-invariant symmetric non-degenerate bilinear form on $\mathfrak{g}:=Lie(\mathbb{G})$, for a $\mathbb{K}$-invariant parabolic $\mathbb{P}\le \mathbb{G}$, is $\mathbb{K}$-equivariantly isomorphic to the extension obtained via the standard bialgebra structure attached to a $\mathbb{K}$-invariant Cartan/Borel pair $\mathbb{H}\le \mathbb{B}\le \mathbb{P}\le \mathbb{G}$ and the same bilinear form. Associating bundle extensions on an elliptic curve $E$ to said $\mathfrak{p}$-module extensions, this identifies Poisson structures on the smooth locus of the principal-$\mathbb{P}$-bundle moduli space over $E$ respectively defined by Balduzzi (using the former extension) and Feigin-Odesskii (via the standard bialgebra structure). This in particular verifies Feigin-Odesskii's identification of the bialgebra-induced symplectic leaves with the loci of bundles mutually isomorphic after forgetting structure along $\mathbb{P}\le \mathbb{G}$.