Quasi-constant fundamental weights in terms of Levi Weyl groups

TL;DR

This paper characterizes quasi-constant fundamental weights using Levi Weyl groups, linking special vertices of Dynkin diagrams to sub-root system Weyl groups, thus unifying minuscule and co-minuscule weights.

Contribution

It provides a new characterization of quasi-constant fundamental weights via Levi subgroup Weyl groups, connecting root system vertices to sub-root systems.

Findings

Characterization of quasi-constant fundamental weights in terms of Levi Weyl groups.

Identification of special and co-special vertices of Dynkin diagrams.

Unification of minuscule and co-minuscule weights.

Abstract

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.