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