On a symmetry of M\"uger's centralizer for the Drinfeld double of a semisimple Hopf algebra

In this paper we prove a formula that relates M\"uger's centralizer in the category of representations of a factorizable Hopf algebra to the notion of Hopf kernel of a representation of the dual Hopf algebra. Using this relation we obtain a complete description for M\"uger's centralizer of some fusion subcategories of the fusion category of finite dimensional representations of a Drinfeld double of a semisimple Hopf algebra.