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

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