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In this paper, we study the polynomial representation of the double affineHecke algebra of type $(C^\vee_n, C_n)$ for specialized parameters. Inductivelyand combinatorially, we give a linear basis of the representation in terms oflinear combinations of non-symmetric Koornwinder polynomials. The basisconsists of generalized eigenfunctions with respect to $q$-Dunkl-Cherednikoperators $\hat{Y}_i$, and it gives a way to cancel out poles of non-symmetricKoornwinder polynomials. We examine irreducibility and $Y$-semisimplicity ofthe representation for the specialized parameters. For some cases, we give acharacterization of the subrepresentations by vanishing conditions for Laurentpolynomials.

The polynomial representation of the double affine Hecke algebra of type $(C^\vee_n, C_n)$ for specialized parameters