Schur function analogs for a filtration of the symmetric function space

We consider a filtration of the symmetric function space given by $\Lambda^{(k)}_t$, 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 $\Lambda^{(k)}_t$. 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 $\Lambda^{(k)}_t$ 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.