An analogue of Whittaker reduction for group-valued moment maps

We construct an analogue of Whittaker reduction for Poisson actions of a semisimple complex Poisson-Lie group G. The reduction takes place along a class of transversal slices to unipotent orbits in G, which are generalizations of the Steinberg cross-section and are indexed by conjugacy classes in the Weyl group. We give an interpretation of these reductions in the framework of Dirac geometry, and we use this to describe their symplectic leaves.