$\mathbb{Z}/m\mathbb{Z}$-graded Lie algebras and perverse sheaves, III: graded double affine Hecke algebra

TL;DR

This paper constructs geometric representations of graded double affine Hecke algebras (DAHA) associated with $ ext{Z}/m ext{Z}$-graded Lie algebras, extending previous work to more general blocks and parameters.

Contribution

It introduces a geometric method to build representations of graded DAHA from $ ext{Z}/m ext{Z}$-graded Lie algebras and their nilpotent cones, generalizing prior principal block results.

Findings

Constructed representations of graded DAHA from geometric data.

Extended previous results to non-principal blocks.

Connected algebraic structures with geometric and representation-theoretic frameworks.

Abstract

In this paper we construct representations of certain graded double affine Hecke algebras (DAHA) with possibly unequal parameters from geometry. More precisely, starting with a simple Lie algebra $\mathfrak{g}$ together with a $\mathbb{Z}/m\mathbb{Z}$-grading $\oplus_{i}\mathfrak{g}_{i}$ and a block of $G_{\underline{0}}$-equivariant complexes on the nilpotent cone of $\mathfrak{g}_{\underline{1}}$ as introduced in \cite{LY1}, we attach a graded DAHA and construct its action on the direct sum of spiral inductions in that block. This generalizes results of Vasserot \cite{V} and Oblomkov-Yun \cite{OY} which correspond to the case of the principal block.