Complete Integrability of the Parahoric Hitchin System

Abstract

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.

Wild Character Varieties, Meromorphic Hitchin Systems and Dynkin Diagrams

“…We leave this as a question rather than a conjecture as we've not looked into it. This is compatible with [9] though, which came to light during conference (although they use a more stringent definition). 3 The terminology for parahoric bundles is similar to that for parabolic bundles: A quasi-parahoric bundle is a torsor for a parahoric group scheme G → Σ, as in [86].…”

On the image of the parabolic Hitchin map

“…is an equality, it suffices to show both sides have the same dimension. From [BKV17], we have dim(A G,P ) = dim(Bun G,P ) = dim(Bun G ) + dim(G/P ) = dim(G)(g − 1) + dim(G/P ).…”

“…Our notation is motivated by the fact that p is a parahoric subalgebra of g(K) and p ⊥ is its annihilator under the canonical non-degenerate pairing on g(K); cf. [BKV17].…”

“…established in [BKV17] (for arbitrary G and P ). Comparing Equation (1) with the dimension of Γ(X, Ω d i ((d i − m i ).x)) establishes the theorem.…”

Moduli spaces of anti-invariant vector bundles and twisted conformal blocks

“…When the type is τ trivial, that's τ i = +1 for all i, the corresponding moduli stack is denoted simply by S U σ,− X (r). These moduli stacks correspond to moduli stacks M Y (G) of parahoric G−torsors over the quotient curve Y , for some twisted parahoric Bruhat-Tits group schemes G. Parahoric G−torsors have attracted the attention of many mathematician recently (see [PR08a], [Hei10], [BS14], [BKV16]), since they can be considered as a generalization of parabolic G−bundles. Their moduli spaces has been constructed in the untwisted case by Balaji and Seshadri [BS14].…”