  Based on previous results on the classification of finite-dimensional Nichols
algebras over dihedral groups and the characterization of simple modules of
Drinfeld doubles, we compute the irreducible characters of the Drinfeld doubles
of bosonizations of finite-dimensional Nichols algebras over the dihedral
groups $\mathbb{D}_{4t}$ with $t\geq 3$. To this end, we develop new techniques
that can be applied to Nichols algebras over any Hopf algebra. Namely, we
explain how to construct recursively irreducible representations when the
Nichols algebra is generated by a decomposable module, and show that the
highest-weight of minimum degree in a Verma module determines its socle. We
also prove that tensoring a simple module by a rigid simple module gives a
semisimple module.
