An Elliptic Generalization of $A_1$ Spherical DAHA at $K=2$

Abstract: We construct an algebra that is an elliptic generalization of $A_1$ spherical DAHA acting on its finite-dimensional module at $t=-q^{-K/2}$ with $K=2$. We prove that $PSL(2,\mathbb Z)$ acts by automorphisms of the algebra we constructed, and provide an explicit representation of automorphisms and algebra operators alike by $3\times 3$ matrices of elliptic functions. A relation of this construction to the K-theory character of affine Laumon space is conjectured. We point out two potential applications, respectively to $SL(3,\mathbb Z)$ symmetry of Felder-Varchenko functions and to new elliptic invariants of torus knots and Seifert manifolds.