Hecke transformation on the Higgs moduli space

The semistable Higgs bundle moduli space comes naturally equipped with a C*-action and a C*-invariant proper fibration projecting onto a vector space called the Hitchin base. A C*-fixed point is called very stable if it does not admit any other nilpotent Higgs bundle in its upward flow apart from itself. The classification of connected components in the nilpotent cone containing very stable C*-fixed points is a topic of current research. In this talk I will speak about Hecke transformations of Higgs bundles and demonstrate how they can be utilized to study very stable C*-fixed points in the Higgs moduli space.