Complete Integrability of the Parahoric Hitchin System

Let G be a parahoric group scheme over a complex projective curve X of genus greater than one. Let BUNG denote the moduli stack of G-torsors on X. We prove several results concerning the Hitchin map on T*BunG. We first show that the parahoric analogue of the global nilpotent cone is isotropic and use this to prove that BUNG 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.