Hecke algebras, difference operators, and quasi-symmetric functions

We define a new action of the symmetric group and its Hecke algebra on poly nomial rings whose invariants are exactly the quasi-symmetric polynomials. We interpret this construction in terms of a Demazure character Formula for the irreducible polynomial modules of a degenerate quantum group. We use t he action of the generic Hecke algebras to define quasi-symmetric and nonco mmutative analogues of Hall-Littlewood functions. We show that these genera lized functions share many combinatorial properties with the classical ones .