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

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 $T^{\ast }\Sigma$ 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.

30 pages. Comments are welcome! arXiv admin note: text overlap with arXiv:2202.05593 by other authors