In low-dimensional topology, several Hecke algebras arise as braid skein algebras of Riemann surfaces. These, in turn, can be understood as the orbifold Hecke algebras of symmetric products of these Riemann surfaces. In this talk, I will explain that the Fukaya category of the cotangent bundle of a global quotient orbifold recovers the data of the orbifold Hecke algebra. In fact, the evidence indicates that the Fukaya category should recover an appropriately defined derived orbifold Hecke algebra, which I will illustrate in the case of symmetric products of the two-dimensional sphere.