Quasi-constant characters: Motivation, classification and applications

In our previous paper "Strata Hasse invariants, Hecke algebras and Galois representations", initially motivated by questions about the Hodge line bundle of a Hodge-type Shimura variety, we singled out a generalization of the notion of {\em minuscule character} which we termed {\em quasi-constant}. Here we prove that the character of the Hodge line bundle is always quasi-constant. Furthermore, we classify the quasi-constant characters of an arbitrary connected, reductive group over an arbitrary field. As an application, we observe that, if $\mu$ is a quasi-constant cocharacter of an ${\mathbf F}_p$-group $G$, then our construction of group-theoretical Hasse invariants in loc. cit. applies to the stack $G\mbox{-Zip}^{\mu}$, without any restrictions on $p$, even if the pair $(G, \mu)$ is not of Hodge type and even if $\mu$ is not minuscule. We conclude with a more speculative discussion of some further motivation for considering quasi-constant cocharacters in the setting of our program outlined in loc cit.