Higher-dimensional Heegaard Floer homology and the polynomial representation of double affine Hecke algebras

We show that the higher-dimensional Heegaard Floer homology between tuples of cotangent fibers and the conormal bundle of a homotopically nontrivial simple closed curve on $T^2$ recovers the polynomial representation of double affine Hecke algebra of type A. We also give a topological interpretation of Cherednik’s inner product on the polynomial representation.