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

Abstract

In this paper, we study the polynomial representation of the double affine Hecke algebra of type $(C_n^{\vee },C_n)$ 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.

45 pages

References in corpus (1)

• Non-semisimple Macdonald polynomials