  We prove that graded $k$-Schur functions are $G$-equivariant Euler
characteristics of vector bundles on the flag variety, settling a conjecture of
Chen-Haiman. We expose a new miraculous shift invariance property of the graded
$k$-Schur functions and resolve the Schur positivity and $k$-branching
conjectures in the strongest possible terms by providing direct combinatorial
formulas using strong marked tableaux.
