Abstract: In my talk I will consider a quantum integrable Hamiltonian system with two generic complex parameters q,t whose classical phase space is the moduli space of flat SL(2,C) connections on a genus two surface. This system and its eigenfunctions provide genus two generalization of the trigonometric Ruijsenaars-Schneider model and Macdonald polynomials, respectively. I will show that the Mapping Class Group of a genus two surface acts by automorphisms of the algebra of operators of this system. Therefore this algebra can be viewed as a genus two generalization of A_1 spherical Double Affine Hecke Algebra.

Based on arXiv:1704.02947 joint with Sh. Shakirov