Hitchin map on even very stable upward flows

We define even very stable Higgs bundles and study the Hitchin map restricted to their upward flows. In the ${\mathrm{\text{GL}}}_n$ case, we classify the type $(1,\dots ,1)$ examples, and find that they are governed by a root system formed by the roots of even height. We discuss how the spectrum of equivariant cohomology of real and quaternionic Grassmannians, $4n$-spheres and the real Cayley plane appear to describe the Hitchin map on even cominuscule upward flows. The even upward flows in question are the same as upward flows in Higgs bundle moduli spaces for quasi-split inner real forms. The latter spaces have been pioneered by Oscar García-Prada and his collaborators.