Beurling-Fourier algebras of compact quantum groups: characters and finite dimensional representations

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