Resumen de Genus two generalization of A1 spherical DAHA

S. Arthamonov, Sh. Shakirov

• We consider a system of three commuting difference operators in three variables x12,x13,x23 with two generic complex parameters q, t. This system and its eigenfunctions generalize the trigonometric A1 Ruijsenaars-Schneider model and A1 Macdonald polynomials, respectively. The principal object of study in this paper is the algebra generated by these difference operators together with operators of multiplication by xij+x−1ij . We represent the Dehn twists by automorphisms of this algebra and prove that these automorphisms satisfy all relations of the mapping class group of the closed genus two surface. Therefore we argue from topological perspective this algebra is a genus two generalization of A1 spherical DAHA.