Let $G$ be a semi-simple group with maximal torus $T$ and Weyl group $W$. It looks like from some geometric considerations I can define a family $P_{\lambda,\alpha}(q,t,z)$ of $W$-invariant polynomials of $z\in T$ which depend on two additional parameters $q$ and $t$. Here $\lambda$ is a dominant weight of $G$ and $\alpha$ is a positive element of the root lattice. Moreover, when $\alpha$ tends to $\infty$, $P_{\lambda,\alpha}$ tends to the corresponding Macdonald polynomial, but I don't know how to characterize $P_{\lambda,\alpha}$ for finite $\alpha$.

Does anybody know if such polynomials exist in the literature?