Schur function analogs for a filtration of the symmetric function space

We consider a filtration of the symmetric function space given by Λ (k) t , the linear span of Hall-Littlewood polynomials indexed by partitions whose first part is not larger than k. We introduce symmetric functions called the k-Schur functions, providing an analog for the Schur functions in the subspaces Λ (k) t . We prove several properties for the k-Schur functions including that they form a basis for these subspaces that reduces to the Schur basis when k is large. We also show that the connection coefficients for the k-Schur function basis with the Macdonald polynomials belonging to Λ (k) t are polynomials in q and t with integral coefficients. In fact, we conjecture that these integral coefficients are actually positive, and give several other conjectures generalizing Schur function theory.