  For each indifference graph, there is an associated regular semisimple
Hessenberg variety, whose cohomology recovers the chromatic symmetric function
of the graph. The decomposition theorem applied to the forgetful map from the
regular semisimple Hessenberg variety to the projective space describes the
cohomology of the Hessenberg variety as a sum of smaller pieces. We give a
combinatorial description of the Frobenius character of each piece. This
provides a generalization of the symmetric functions attached to Stanley's
local h-polynomials of the permutahedral variety to any Hessenberg variety.
  As a consequence, we can prove that the coefficient of $e_{\lambda}$, where
$\lambda$ is any partition of length 2, in the e-expansion of the chromatic
symmetric function of any indifference graph is non-negative.
