Hecke-Tyurin parametrization of the Hitchin and KZB systems

We study the parametrization of the moduli space Bun_2(C)_L of rank 2 bundles over a curve C with fixed determinant, provided by Hecke modifications at fixed points of the trivial bundle. This parametrization is closely related to the Tyurin parametrization of vector bundles over curves. We use it to parametrize the Hitchin and KZB systems, as well as lifts of the Beilinson-Drinfeld D-modules. We express a generating series for the lifts of the Beilinson-Drinfeld operators in terms of a "quantum L-operator" \ell(z). We explain the relation to earlier joint work with G. Felder, based on parametrization by flags of bundles (math/9807145) and introduce filtrations on conformal blocks, related with the Hecke modifications.