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

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

2008-07-17

arxiv.org/abs/0807.2714v1

Mathematics

Representation Theory

45 pages

Scientific paper

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.

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