Higher-dimensional Heegaard Floer homology and Hecke algebras

Given a closed oriented surface $\Sigma$ of genus greater than 0, we construct a map $\mathcal{F}$ from the higher-dimensional Heegaard Floer homology of the cotangent fibers of $T^*\Sigma$ to the Hecke algebra associated to $\Sigma$ and show that $\mathcal{F}$ is an isomorphism of algebras. We also establish analogous results for punctured surfaces.