  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.
