  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 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 the quantum affine algebra (it is a geometric realization of the center).
Then the symplectic leaves, and corresponding equivalence classes of central
characters, are parameterized by certain G-bundles on an elliptic curve.
