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