  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.
