  We compute the supports of the perverse cohomology sheaves of the Hitchin
fibration for $GL_n$ over the locus of reduced spectral curves. In contrast to
the case of meromorphic Higgs fields we find additional supports at the loci of
reducible spectral curves. Their contribution to the global cohomology is
governed by a finite twist of Hitchin fibrations for Levi subgroups. The
corresponding summands give non-trivial contributions to the cohomology of the
moduli spaces for every $n \geq 3$. A key ingredient is a restriction result
for intersection cohomology sheaves that allows us to compare the fibration to
the one defined over the versal deformations of spectral curves.
