Complete integrability of the parahoric Hitchin system

Abstract

Let $\mathcal{G}$ be a parahoric group scheme over a complex projective curve $X$ of genus greater than one. Let ${\mathrm{Bun}}_{\mathcal{G}}$ denote the moduli stack of $\mathcal{G}$-torsors on $X$. We prove several results concerning the Hitchin map on $T^{\ast }{\mathrm{Bun}}_{\mathcal{G}}$. We first show that the parahoric analogue of the global nilpotent cone is isotropic and use this to prove that ${\mathrm{Bun}}_{\mathcal{G}}$ 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.

The previous version of this paper has been split into two separate papers. This version is the first of the two papers and concerns complete integrability of the parahoric Hitchin system. In the forthcoming second paper, we study the base of the parahoric Hitchin map and its local counterpart