Semen Artamonov (Rutgers), Representation theory of genus two A_1 spherical DAHA

Spherical Double Affine Hecke Algebra can be viewed as a noncommutative (q,t)-deformation of the SL(N,C) character variety of the fundamental group of a torus. This deformation inherits major topological property from its commutative counterpart, namely Mapping Class Group of a torus SL(2,Z) acts by atomorphisms of DAHA. In my talk I will define a genus two analogue of A_1 spherical DAHA and show that the Mapping Class Group of a closed genus two surface acts by automorphisms of such algebra. I will then show that for special values of parameters q,t satisfying q^n t^2=1 for some nonnegative integer n this algebra admits finite dimensional representations. I will conclude with discussion of potential applications to TQFT and knot theory.