When is generic, this is a quotient of the Hecke algebra by the ideal generated by all representations whose corresponding representations are not hooks.
In fact, when the weight spaces of are irreducible representations, with arm length of the hook corresponding to weight.
Ben Webster: Does inherit a basis from the Hecke algebra in the usual way? It certainly seems that the representation has a basis on which positive integral combination of the Kazhdan-Lusztig basis vectors act with positive integral coefficients.
Also, does this picture have a categorification? One natural candidate to use in the case is the categorifications of the cell modules due to Stroppel and Mazorchuk. Also possibly of interest is their categorification of Wedderburn basis of