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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?

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Are there some difference operators which annihilate $P_{\lambda,\alpha}$ and deform the Macdonald operators? –  Pavel Safronov Apr 30 '12 at 21:11
That's precisely what I don't know. In my situation I have a family of representations of $G\times {\mathbb C}^*\times{\mathbb C}^*$ depending on $\lambda$ and $\alpha$ such that their characters become Macdonald polynomials when $\alpha=\infty$. But otherwise I don't know anything about them. –  Alexander Braverman May 1 '12 at 21:13

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