Frobenius map for the centers of hecke algebras

We introduce a commutative associative graded algebra structureon the direct sum Z of the centers of the Hecke algebras associated to thesymmetric groups in n letters for all n. As a natural deformation of theclassical construction of Frobenius, we establish an algebra isomorphism fromZ to the ring of symmetric functions. This isomorphism provides an identificationbetween several distinguished bases for the centers (introduced byGeck-Rouquier, Jones, Lascoux) and explicit bases of symmetric functions.