I will start by reviewing on how A_{n-1} spherical DAHA arise in the quantization of the moduli space of SU(N) flat connections on a torus. I will then define the corresponding notion for the case of SU(2) flat connections on a genus two surface.

In particular, I will define a family of Laurent polynomials which form a basis of common eigenfunctions of certain commuting q-difference operators. This family contains A_1 Macdonald polynomials as a subclass. I will then define a genus two analogue of A_1 spherical DAHA and show that the Mapping Class Group of a genus two surface acts by outer automorphisms of this algebra.

(This is joint work with Shamil Shakirov)