Complete integrability of the parahoric Hitchin system

Let $\mathcal{G}$ be a parahoric group scheme over a complex projective curve $X$ of genus greater than one. Let ${\mathrm{B}\mathrm{u}\mathrm{n}}_{\mathcal{G}}$ denote the moduli stack of $\mathcal{G}$-torsors on $X$. We prove several results concerning the Hitchin map on $T^{\ast }\text{\hspace{0.17em}⁣}{\mathrm{B}\mathrm{u}\mathrm{n}}_{\mathcal{G}}$. We first show that the parahoric analogue of the global nilpotent cone is isotropic and use this to prove that ${\mathrm{B}\mathrm{u}\mathrm{n}}_{\mathcal{G}}$ is “very good” in the sense of Beilinson-Drinfeld. We then prove that the parahoric Hitchin map is a Poisson map whose generic fibres are abelian varieties. Together, these results imply that the parahoric Hitchin map is a completely integrable system.