I'll discuss some work in progress in collaboration with M.A. de Cataldo and J. Heinloth.
Let C be a nonsingular projective curve of genus >1, and let n and d be two coprime integers.
Given the moduli space M of stable Higgs bundles
(E,\Phi), where E is a vector bundle on C of rank n and degree d,
and \Phi: E \to E\otimes K_C is the Higgs field,
there is an associated Hitchin fibration \chi: M \to A,
where A is the affine space parameterizing the spectral curves C_a \subset T^\vee C associated to Higgs bundles.
The map \chi is projective, hence, by the decomposition theorem of Beilinson Bernstein Deligne and Gabber,
R\chi_* Q_M splits in a direct sum of (shifted) intersection cohomology complexes.
We determine completely the supports of those intersection cohomology complexes which intersect the locus
A_red parameterizing reduced spectral curves.
We show in particular that, for every partition \lambda=(\lambda_1, ... , \lambda_r) of n, there is a string of summands appearing in the decomposition of R\chi_* Q_M which is supported on
the locus A_\lambda
where the spectral curve splits into r reducible components mapping to C with degrees \lambda_1, ... , \lambda_r.
The proof relies on the technique of "higher discriminants", developed in collaboration with V. Shende, and on some results on the universal family of compactified jacobians, due to V. Shende, F. Viviani and myself.