  In joint work with J.-S. Koskivirta, we had previously introduced the notion
of "quasi-constant" character (of a maximal torus of a connected reductive
group over a field); we showed that over an algebraically closed field it
naturally unifies the notions "minuscule" and "co-minuscule". In this note we
describe a characterization of quasi-constant fundamental weights in terms of
the Weyl group of a maximal Levi subgroup. Equivalently, purely in the language
of root systems, the result characterizes special and co-special vertices of
Dynkin diagrams in terms of Weyl groups of maximal sub-root systems.
