  We describe a categorification of the Double Affine Hecke Algebra
${\mathcal{H}\kern -.4em\mathcal{H}}$ associated with an affine Lie algebra
$\widehat{\mathfrak{g}}$, a categorification of the polynomial representation
and a categorification of Macdonald polynomials.
  All categorification results are given in the derived setting. That is, we
consider the derived category associated with graded modules over the Lie
superalgera ${\mathfrak I}[\xi]$, where ${\mathfrak
I}\subset\widehat{\mathfrak{g}}$ is the Iwahori subalgebra of the affine Lie
algebra and $\xi$ is a formal odd variable. The Euler characteristic of graded
characters of a complex of ${\mathfrak I}[\xi]$-modules is considered as an
element of a polynomial representation. First, we show that the compositions of
induction and restriction functors associated with minimal parabolic
subalgebras ${\mathfrak{p}}_{i}$ categorify Demazure operators
$T_i+1\in{\mathcal{H}\kern -.4em\mathcal{H}}$, meaning that all algebraic
relations of $T_i$ have categorical meanings. Second, we describe a natural
collection of complexes ${\mathbb{EM}}_{\lambda}$ of ${\mathfrak
I}[\xi]$-modules whose Euler characteristic is equal to nonsymmetric Macdonald
polynomials $E_\lambda$ for dominant $\lambda$ and a natural collection of
complexes of $\mathfrak{g}[z,\xi]$-modules ${\mathbb{PM}}_{\lambda}$ whose
Euler characteristic is equal to the symmetric Macdonald polynomial
$P_{\lambda}$.
  We illustrate our theory with the example $\mathfrak{g}=\mathfrak{sl}_2$
where we construct the cyclic representations of Lie superalgebra ${\mathfrak
I}[\xi]$ such that their supercharacters coincide with renormalizations of
nonsymmetric Macdonald polynomials.
