Geometry of the analytic loop group

We introduce and study a notion of analytic loop group with a Riemann-Hilbert factorization relevant for the representation theory of quantum affine algebras at roots of unity Uϵ(gˆ) with non-trivial central charge. We introduce a Poisson structure and study properties of its Poisson dual group. We prove that the Hopf-Poisson structure is isomorphic to the semi-classical limit of the center of Uϵ(gˆ) (it is a geometric realization of the center). Then the symplectic leaves, and corresponding equivalence classes of central characters of Uϵ(gˆ), are parameterized by certain G-bundles on an elliptic curve.