Categorification of DAHA and Macdonald polynomials

Abstract

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.