Quantum cohomology and the $k$-Schur basis

We prove that structure constants related to Hecke algebras at roots of unity are special cases of $k$-Littlewood-Richardson coefficients associated to a product of $k$-Schur functions. As a consequence, both the 3-point Gromov-Witten invariants appearing in the quantum cohomology of the Grassmannian, and the fusion coefficients for the WZW conformal field theories associated to $\widehat{su}(\ell)$ are shown to be $k$-Littlewood-Richardson coefficients. From this, Mark Shimozono conjectured that the $k$-Schur functions form the Schubert basis for the homology of the loop Grassmannian, whereas $k$-Schur coproducts correspond to the integral cohomology of the loop Grassmannian. We introduce dual $k$-Schur functions defined on weights of $k$-tableaux that, given Shimozono's conjecture, form the Schubert basis for the cohomology of the loop Grassmannian. We derive several properties of these functions that extend those of skew Schur functions.