Twisted Cherednik spectrum as a $q,t$-deformation

The common eigenfunctions of the twisted Cherednik operators can be first analyzed in the limit of $q⟶1$. Then, the polynomial eigenfunctions form a simple set originating from the symmetric ground state of non-vanishing degree and excitations over it, described by non-symmetric polynomials of higher degrees and enumerated by weak compositions. This pattern is inherited by the full spectrum at $q\mathrm{\ne }1$, which can be considered as a deformation. The whole story looks like a typical NP problem: the Cherednik equations are difficult to solve, but easy to check the solution once it is somehow found.