The down operator and expansions of near rectangular k-Schur functions

We prove that the Lam–Shimozono “down operator” on the affine Weyl group induces a derivation of the affine Fomin–Stanley subalgebra. We use this to verify a conjecture of Berg, Bergeron, Pon and Zabrocki describing the expansion of non-commutative k-Schur functions of “near rectangles” in the affine nilCoxeter algebra. Consequently, we obtain a combinatorial interpretation of the corresponding k-Littlewood–Richardson coefficients.