A Floer-theoretic interpretation of the polynomial representation of the double affine Hecke algebra

TL;DR

This paper establishes a deep connection between wrapped Floer homology, braid skein groups, and the polynomial representation of the double affine Hecke algebra (DAHA) on a surface, revealing new algebraic-topological insights.

Contribution

It constructs an isomorphism linking Floer homology of cotangent fibers and conormal bundles to braid skein groups, and shows this yields a module over the surface Hecke algebra matching DAHA's polynomial representation.

Findings

Isomorphism between Floer homology and braid skein groups

Module structure over the surface Hecke algebra

Equivalence to DAHA polynomial representation on a torus

Abstract

We construct an isomorphism between the wrapped higher-dimensional Heegaard Floer homology of $\kappa$-tuples of cotangent fibers and $\kappa$-tuples of conormal bundles of homotopically nontrivial simple closed curves in $T^*\Sigma$ with a certain braid skein group, where $\Sigma$ is a closed oriented surface of genus $> 0$ and $\kappa$ is a positive integer. Moreover, we show this produces a (right) module over the surface Hecke algebra associated to $\Sigma$. This module structure is shown to be equivalent to the polynomial representation of DAHA in the case where $\Sigma=T^2$ and the cotangent fibers and conormal bundles of curves are both parallel copies.