  Associated to any compact quantum group $G\subset U_N^+$ is a canonical
family of group dual subgroups $\widehat{\Gamma}_Q\subset G$, parametrized by
unitaries $Q\in U_N$, playing the role of "maximal tori" for $G$. We present
here a series of conjectures, relating the various algebraic and analytic
properties of $G$ to those of the family $\{\widehat{\Gamma}_Q|Q\in U_N\}$.
