Subrepresentations in the Polynomial Representation of the Double Affine Hecke Algebra of type $G{L}_n$ at $t^{k+1}{q}^{r-1}=1$

Abstract

We study a Laurent polynomial representation $V$ of the double affine Hecke algebra of type $G{L}_n$ for specialized parameters $t^{k+1}{q}^{r-1}=1$. We define a series of subrepresentations of $V$ by using a vanishing condition. For some cases, we give an explicit basis of the subrepresentation in terms of nonsymmetric Macdonald polynomials. These results are nonsymmetric versions of \cite{FJMM} and \cite{KMSV}.

21 pages

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