A non-commutative generalization of $k$-Schur functions
  We introduce non-commutative analogues of $k$-Schur functions of Lapointe-Lascoux and Morse. We give an explicit formulas for the expansions of non-commutive functions with one and two parameters in terms of these new functions. These results are similar to the conjectures existing in the commutative case. 
N. Bergeron, F. Descouens, M. Zabrocki