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

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 $T^{\ast }\mathrm{\Sigma }$ with a certain braid skein group, where $\mathrm{\Sigma }$ 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 $\mathrm{\Sigma }$. This module structure is shown to be equivalent to the polynomial representation of DAHA in the case where $\mathrm{\Sigma }=T^2$ and the cotangent fibers and conormal bundles of curves are both parallel copies.