Beurling-Fourier algebras of $q$-deformations of compact semisimple Lie groups and complexification

We study Beurling-Fourier algebras of $q$-deformations of compact semisimple Lie groups. In particular, we show that the space of irreducible representations of the function algebras of their Drinfeld doubles is exhausted by the irreducible representations of weighted Fourier algebras associated to a certain family of central weights.