A descent basis for the Garsia-Procesi module

We assign to each Young diagram λ a subset $B_{{\lambda }^{\prime }}^{\mathrm{maj}}$ of the collection of Garsia-Stanton descent monomials, and prove that it determines a basis of the Garsia-Procesi module $R_{\lambda }$, whose graded character is the Hall-Littlewood polynomial ${\tilde{H}}_{\lambda }[X;t]$ [14], [10], [29]. This basis is a major index analogue of the basis $B_{\lambda }\subset R_{\lambda }$ defined by certain recursions, in the same way that the descent basis is related to the Artin basis of the coinvariant algebra $R_n$, which in fact corresponds to the case when $\lambda =1^n$. 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 ${\tilde{H}}_{\lambda }[X;t]$. Despite a similar appearance, it does not appear obvious how to connect the formulas appear to the specialization of the modified Macdonald formula of Haglund, Haiman and Loehr at $q=0$.