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

Quasi-constant fundamental weights in terms of Levi Weyl groups