Garsia–Haiman modules for hook partitions and Green polynomials with two variables

We consider Garsia–Haiman modules for the symmetric group, a doubly graded generalization of Springer modules. Our main interest lies in singly graded submodules of a Garsia–Haiman module. We show that these submodules satisfy a certain combinatorial property, and verify that this property is implied by a behavior of Macdonald polynomials at roots of unity