Catalan functions and $k$-Schur positivity

摘要

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.