  Singular nonsymmetric Macdonald polynomials are constructed by use of the
representation theory of the Hecke algebras of the symmetric groups. These
polynomials are labeled by quasistaircase partitions and are associated to
special parameter values $(q,t)$. For $N$ variables, there are singular
polynomials for any pair of positive integers $m$ and $n$, with $2\leq n\leq
N$, and parameters values $(q,t)$ satisfying $q^{a}t^{b}=1$ exactly when $a=rm$
and $b=rn$, for some integer $r$. The coefficients of nonsymmetric Macdonald
polynomials with respect to the basis of monomials $\big\{ x^{\alpha}\big\}$
are rational functions of $q$ and $t$. In this paper, we present the
construction of subspaces of singular nonsymmetric Macdonald polynomials
specialized to particular values of $(q,t)$. The key part of this construction
is to show the coefficients have no poles at the special values of $(q,t)$.
Moreover, this subspace of singular Macdonald polynomials for the special
values of the parameters is an irreducible module for the Hecke algebra of type
$A_{N-1}$.
