Decomposition of Green polynomials of type A and Springer modules for hooks and rectangles

This paper deals with graded representations of the symmetric group on the cohomology ring of flags fixed by a unipotent matrix. We consider a combinatorial property, called the “coincidence of dimension” of the graded representations, and give an interpretation in terms of representation theory of the symmetric group in the case where the corresponding partition of the unipotent matrix is a hook or a rectangle. The interpretation is equivalent to a recursive formula of Green polynomials at roots of unity.