A descent basis for the Garsia-Procesi module

We assign to each Young diagram λ a subset Bλ' of the collection of Garsia-Stanton descent monomials, and prove that it determines a basis of the Garsia-Procesi module R_λ, whose graded character is the Hall-Littlewood polynomial H̃λ[X;t]. This basis is a major index analogue of the basis B_λ ⊂ R_λ defined by certain recursions in due to Garsia and Procesi, in the same way that the descent basis is related to the Artin basis of the coinvariant algebra Rₙ, which in fact corresponds to the case when λ=1ⁿ. By anti-symmetrizing a subset of this basis with respect to the corresponding Young subgroup under the Springer action, we obtain a basis in the parabolic case, as well as a corresponding formula for the expansion of H̃λ[X;t]. Despite a similar appearance, it does not appear obvious how to connect these formulas appear to the specialization of the modified Macdonald formula of Haglund, Haiman and Loehr at $q=0$.