An Elliptic Generalization of A1 Spherical DAHA at K = 2

Abstract We construct an algebra that is an elliptic generalization of A₁ spherical DAHA acting on its finite-dimensional module at t=-q^-K/2 with K=2. We prove that PSL (2, Z) acts by automorphisms of the algebra we constructed, and provide an explicit representation of automorphisms and algebra operators alike by 3 3 matrices with matrix elements given by products 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, Z) symmetry of Felder–Varchenko functions and to new elliptic invariants of torus knots and Seifert manifolds.