Irreducible representations of subgroups of finite index in Baumslag–Solitar groups

We describe all finite-dimensional irreducible linear representations over the field of complex numbers for an arbitrary subgroup of finite index of a Baumslag–Solitar group $BS(p,q)=\langle a,t\mid t^{-1}a^pt=a^q\rangle$ for coprime $p$ and $q$. We find necessary and sufficient conditions for the equivalence of these representations, using only standard facts of linear algebra and the description of subgroups of finite index in Baumslag–Solitar groups.