A COMMUTATIVE ALGEBRA ON DEGENERATE CP 1 AND MACDONALD POLYNOMIALS

Abstract

Abstract. We introduce a unital associative algebra A over degenerate CP 1. We show that A is a commutative algebra and whose Poincaré 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 [FO]. 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 [S2]. It is found that the Ding-Iohara algebra [DI] 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 [D] a la Babelon Bernard Billey [BBB], the Ruijsenaars difference operator [R] and the operator M(q, t1, t2) of Okounkov-Pandharipande [OP]. 1.