A support theorem for the Hitchin fibration: the case of $GL_n$ and $K_C$

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.