  We extend the family non-symmetric Macdonald polynomials and define
general-basement Macdonald polynomials. We show that these also satisfy a
triangularity property with respect to the monomials bases and behave well
under the Demazure-Lusztig operators.
  The symmetric Macdonald polynomials $J_\lambda$ are expressed as a sum of
general-basement Macdonald polynomials via an explicit formula.
  By letting $q=0$, we obtain $t$-deformations of key polynomials and Demazure
atoms and we show that the Hall--Littlewood polynomials expand positively into
these. This generalizes a result by Haglund, Luoto, Mason and van Willigenburg.
  As a corollary, we prove that Schur polynomials decompose with non-negative
coefficients into $t$-deformations of general Demazure atoms and thus
generalizing the $t=0$ case which was previously known. This gives a unified
formula for the classical expansion of Schur polynomials in Hall-Littlewood
polynomials and the expansion of Schur polynomials into Demazure atoms.
