  We define a subclass of Hessenberg varieties called abelian Hessenberg
varieties, inspired by the theory of abelian ideals in a Lie algebra developed
by Kostant and Peterson. We give an inductive formula for the
$S_n$-representation on the cohomology of an abelian regular semisimple
Hessenberg variety with respect to the action defined by Tymoczko. Our result
implies that a graded version of the Stanley-Stembridge conjecture holds in the
abelian case, and generalizes results obtained by Shareshian-Wachs and Teff.
Our proof uses previous work of Stanley, Gasharov, Shareshian-Wachs, and
Brosnan-Chow, as well as results of the second author on the geometry and
combinatorics of Hessenberg varieties. As part of our arguments, we obtain
inductive formulas for the Poincar\'e polynomials of regular abelian Hessenberg
varieties.
