Picky elements, subnormalisers, and character correspondences
Gunter Malle
TL;DR
The paper investigates a local-global conjecture of Moretó and Rizo on character values by analyzing subnormalisers and picky elements in finite groups, with a focus on groups of Lie type and their ambient algebraic groups. It develops a unifying framework that links ${\operatorname{Sub}}_G(x)$ to parabolic and Levi subgroups, provides a complete classification of picky unipotent elements in many Lie-type families, and verifies the conjecture for a broad class of unipotent elements and several semisimple cases at the defining prime. It further extends the analysis to non-simply laced and twisted types, Suzuki and Ree groups, and algebraic-group contexts, including extensions by graph automorphisms, yielding explicit descriptions of subnormalisers and their implications for local-global character correspondences. Overall, the work advances the understanding of how local $p$-structure controls global character behavior, offering concrete classifications and verifiable instances of the Moretó–Rizo conjecture across a wide spectrum of groups.
Abstract
We gather evidence on a new local-global conjecture of Moretó and Rizo on values of irreducible characters of finite groups. For this we study subnormalisers and picky elements in finite groups of Lie type and determine them in many cases, for unipotent elements as well as for semisimple elements of prime power order. We also discuss subnormalisers of unipotent and semisimple elements in connected as well as in disconnected reductive linear algebraic groups.