  The characters of Kazhdan--Lusztig elements of the Hecke algebra over $S_n$
(and in particular, the chromatic symmetric function of indifference graphs)
are completely encoded in the (intersection) cohomology of certain subvarieties
of the flag variety. Considering the forgetful map to some partial flag
variety, the decomposition theorem tells us that this cohomology splits as a
sum of intersection cohomology groups with coefficients in some local systems
of subvarieties of the partial flag variety. We prove that these local systems
correspond to representations of subgroups of $S_n$. An explicit
characterization of such representations would provide a recursive formula for
the computation of such characters/chromatic symmetric functions, which could
settle Haiman's conjecture about the positivity of the monomial characters of
Kazhdan--Lusztig elements and Stanley--Stembridge conjecture about
$e$-positivity of chromatic symmetric function of indifference graphs. We also
find a connection between the character of certain homology groups of
subvarieties of the partial flag varieties and the Grojnowski--Haiman hybrid
basis of the Hecke algebra.
