We describe a categorification of the Double Affine Hecke Algebra (${ \mathcal{ H}\kern -.4em\mathcal{ H}}$) associated with an affine Lie algebra $\widehat{ \mathfrak{ g}}$, including a categorification of the polynomial representation and Macdonald polynomials. Our categorification results are presented in the derived setting, focusing on the derived category of graded modules over the Lie superalgebra ${ \mathfrak I}[\xi]$, where ${ \mathfrak I} \subset \widehat{ \mathfrak{ g}}$ is the Iwahori subalgebra of the affine Lie algebra and $\xi$ is a formal odd variable. First, we show that the compositions of induction and restriction functors associated with minimal parabolic subalgebras ${ \mathfrak{ p}}_{ i}$ categorify the Demazure operators $T_i + 1 \in { \mathcal{ H}\kern -.4em\mathcal{ H}}$, ensuring that all algebraic relations of $T_i$ have categorical interpretations. Second, for each dominant weight $\lambda$ we introduce a complex ${ \mathbb{ EM}}_{ \lambda}$ of ${ \mathfrak{ I}}[\xi]$-modules and a complex ${ \mathbb{ PM}}_{ \lambda}$ of ${ \mathfrak{ g}}[z,\xi]$-modules, whose Euler characteristics are equal to nonsymmetric $E_{ \lambda}$ and symmetric $P_{ \lambda}$ Macdonald polynomials respectively. 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 certain normalizations of nonsymmetric Macdonald polynomials.
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