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 ${\mathfrak I}[ξ]$, where ${\mathfrak I} \subset \widehat{\mathfrak{g}}$ is the Iwahori subalgebra of the affine Lie algebra and $ξ$ 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 $λ$ we introduce a complex ${\mathbb{EM}}_λ$ of ${\mathfrak{I}}[ξ]$-modules and a complex ${\mathbb{PM}}_λ$ of ${\mathfrak{g}}[z,ξ]$-modules, whose Euler characteristics are equal to nonsymmetric $E_λ$ and symmetric $P_λ$ Macdonald polynomials respectively. We illustrate our theory with the example $\mathfrak{g}=\mathfrak{sl}_2$ where we construct the cyclic representations of Lie superalgebra ${\mathfrak{I}}[ξ]$ such that their supercharacters coincide with certain normalizations of nonsymmetric Macdonald polynomials.

Theorems & Definitions (112)

• Theorem 1
• Theorem 2: Theorem \ref{['thm::DAHA']}
• Theorem 3: Theorem \ref{['thm::Y_eigen']}
• Theorem 4: Theorem \ref{['thm::PJ::categorify']}
• Conjecture 5
• Conjecture 6
• Remark 1.5
• proof
• Theorem 1.11: Cherednik Ch1 § 1
• Lemma 2.6