  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
$U_q(\widehat{\mathfrak{sl}}_\ell)$-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.
