Categorification of DAHA and Macdonald polynomials

TL;DR

This paper develops a categorification of the Double Affine Hecke Algebra and Macdonald polynomials using derived categories of graded modules over Lie superalgebras, linking algebraic relations to categorical functors.

Contribution

It introduces a new categorification framework for DAHA and Macdonald polynomials via derived categories, connecting algebraic operators to functor compositions.

Findings

Categorification of DAHA via derived categories of graded modules.

Construction of complexes whose Euler characteristics match Macdonald polynomials.

Explicit example with fsl_2 demonstrating supercharacter correspondence.

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.