  We define and study the existence of very stable Higgs bundles on Riemann
surfaces, how it implies a precise formula for the multiplicity of the very
stable components of the global nilpotent cone and its relationship to mirror
symmetry. The main ingredients are the Bialynicki-Birula theory of ${\mathbb
C}^*$-actions on semiprojective varieties, ${\mathbb C}^*$ characters of
indices of ${\mathbb C}^*$-equivariant coherent sheaves, Hecke transformation
for Higgs bundles, relative Fourier-Mukai transform along the Hitchin
fibration, hyperholomorphic structures on universal bundles and cominuscule
Higgs bundles.
