Elliptic Sklyanin integrable systems for arbitrary reductive groups

We present the analogue, for an arbitrary complex reductive group G, of the elliptic integrable systems of Sklyanin. The Sklyanin integrable systems were originally constructed on symplectic leaves, of a quadratic Poisson structure, on a loop group of type A. The phase space, of our integrable systems, is a group-like analogue of the Hitchin system over an elliptic curve Σ. The phase space is the moduli space of pairs (P, φ), where P is a principal G-bundle on Σ, and φ is a meromorphic section of the adjoint group bundle. The Poisson structure, on the moduli space, is related to the Poisson structures on loop groupoids, constructed by Etingof and Varchenko, using Felder’s elliptic solutions of the Classical Dynamical Yang-Baxter Equation. e-print archive: http://xxx.lanl.gov/math.AG/0203031 The first author of this article would like to thank NSERC and FCAR for their support Partially supported by NSF grant number DMS-9802532 874 Elliptic Sklyanin integrable systems . . .