In a celebrated sequence of works from the 1990s, De Concini, Kac and Procesi constructed a Poisson geometric framework for the study of the irreducible representations of big quantum groups at roots of unity. We will describe an extension of this framework to a large family of Drinfeld doubles in the setting of Nichols algebras, which includes as a special case the family of all big quantum supergroups. This is done by a new method, based on perfect pairings between restricted and non-restricted integral forms, which does not rely on any direct computations of Poisson brackets and reductions to low rank cases. We will provide an intuitive introduction to all of the above notions. This is a joint work with Nicolas Andruskiewitsch and Ivan Angiono (University of Cordoba).