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Catalan functions, the graded Euler characteristics of certain vector bundles on the flag variety, are a rich class of symmetric functions which include k -Schur functions and parabolic Hall-Littlewood polynomials. We prove that Catalan functions indexed by partition weight are the characters of U q (sl ˆ ℓ) -generalized Demazure crystals as studied by Lakshmibai-Littelmann-Magyar and Naoi. We obtain Schur positive formulas for these functions, settling conjectures of Chen-Haiman and Shimozono-Weyman. Our approach more generally gives key positive formulas for graded Euler characteristics of certain vector bundles on Schubert varieties by matching them to characters of generalized Demazure crystals. [ABSTRACT FROM AUTHOR]

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 from [24] ; (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, solving a substantial special case of a problem of Broer and Shimozono-Weyman on parabolic Hall-Littlewood polynomials [37]. In addition, we prove the conjecture that the k -Schur functions defined via k -split polynomials [25] agree with those defined in terms of strong tableaux [21]. [ABSTRACT FROM AUTHOR]