Spectral data for higher direct images of parabolic Higgs bundles

Mochizuki's Kobayashi-Hitchin correspondence allows us to represent local systems on 
quasiprojective varieties by parabolic logarithmic Higgs bundles. The Sabbah-Mochizuki
theory generalizing Saito's mixed Hodge modules then provides a framework for the 6-functor 
formalism on these objects. However, this setup involves working with lambda-connections for
all values of lambda. In our work with Donagi and Pantev, we made explicit the 
formula for the higher direct image, looking solely at the Dolbeault fiber (lambda = 0) of 
Higgs bundles. In generic cases this yields a nice formula for the spectral variety
of the higher direct image Higgs bundle. We'll then look at how this applies to the
question of calculating Hecke correspondences in the Donagi-Pantev program for constructing
Hecke eigensheaves in geometric Langlands.