Integrability of the moduli stack of parahoric Higgs bundles

Abstract: The cotangent bundle of the stack of principal G-bundles on a curve was shown by Hitchin to be an algebraically integrable system. Motivated by the Langlands program we consider the following generalisation: take the cotangent bundle of the stack of torsors over a non-constant group scheme. Do we still get an integrable system? We show that this is the case for the so-called parahoric group schemes. A special cases of this is the cotangent bundle of the stack of parabolic G-bundles. Determining the base of the integrable system turns out to be deeply related to the Kazhdan-Lusztig map, which sends nilpotent orbits to conjugacy classes in the Weyl group. This is Joint work with Masoud Kamgarpour and Rohith Varma.