Hecke algebras, difference operators, and quasi-symmetric functions

We define a new action of the symmetric group and its Hecke algebra on polynomial 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 the action of the generic Hecke algebras to define quasi-symmetric and noncommutative analogues of Hall-Littlewood functions. We show that these generalized functions share many combinatorial properties with the classical ones.