A commutative algebra on degenerate CP¹ and Macdonald polynomials

We introduce a unital associative algebra A over degenerate CP^1. We show that A is a commutative algebra and whose Poincar'e series is given by the number of partitions. Thereby we can regard A as a smooth degeneration limit of the elliptic algebra introduced by one of the authors and Odesskii. Then we study the commutative family of the Macdonald difference operators acting on the space of symmetric functions. A canonical basis is proposed for this family by using A and the Heisenberg representation of the commutative family studied by one of the authors. It is found that the Ding-Iohara algebra provides us with an algebraic framework for the free filed construction. An elliptic deformation of our construction is discussed, showing connections with the Drinfeld quasi-Hopf twisting a la Babelon Bernard Billey, the Ruijsenaars difference operator and the operator M(q,t_1,t_2) of Okounkov-Pandharipande.