$k$-Schur expansions of Catalan functions

We make a broad conjecture about the $k$-Schur positivity of Catalan functions, symmetric functions which generalize the (parabolic) Hall-Littlewood polynomials. We resolve the conjecture with positive combinatorial formulas in cases which address the $k$-Schur expansion of (1) Hall-Littlewood polynomials, proving the $q=0$ case of the strengthened Macdonald positivity conjecture of Lapointe, Lascoux, and Morse; (2) the product of a Schur function and a $k$-Schur function when the indexing partitions concatenate to a partition, describing a class of Gromov-Witten invariants for the quantum cohomology of complete flag varieties; (3) $k$-split polynomials, proving a substantial case of a problem of Broer and Shimozono-Weyman on parabolic Hall-Littlewood polynomials. In addition, we prove the conjecture that $k$-Schur functions defined in terms of $k$-split polynomials agree with strong tableau $k$-Schur functions.