  In this paper we study weighted versions of Fourier algebras of compact
quantum groups. We focus on the spectral aspects of these Banach algebras in
two different ways. We first investigate their Gelfand spectrum, which shows a
connection to the maximal classical closed subgroup and its complexification.
Secondly, we study specific finite dimensional representations coming from the
complexification of the underlying quantum group. We demonstrate that the
weighted Fourier algebras can detect the complexification structure in the
special case of $SU_q(2)$, whose complexification is the quantum Lorentz group
$SL_q(2,\mathbb{C})$.
