A descent basis for the Garsia-Procesi module

Abstract

We assign to each Young diagram $\lambda$ a subset $\mathcal{B}_{\lambda'}$ 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 $\tilde{H}_{\lambda}[X;t]$. This basis is a major index analogue of the basis $\mathcal{B}_\lambda \subset R_\lambda$ 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 $R_n$, 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 these formulas appear to the specialization of the modified Macdonald formula of Haglund, Haiman and Loehr at $q=0$.