Quasi-constant characters: Motivation, classification and applications

TL;DR

This paper introduces and classifies quasi-constant characters in reductive groups, demonstrating their relevance to Hodge line bundles and group-theoretical invariants, extending previous work on Shimura varieties.

Contribution

It generalizes the notion of minuscule characters to quasi-constant characters, proves their properties for Hodge line bundles, and classifies all such characters in reductive groups over any field.

Findings

The Hodge line bundle character is always quasi-constant.

Quasi-constant cocharacters enable construction of group-theoretical Hasse invariants.

Classification of quasi-constant characters in reductive groups over arbitrary fields.

Abstract

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.