A generalized character related to the local structure and representation theory of a finite group

TL;DR

This paper defines and studies the generalized character $\Psi_{1,p,G}$, which vanishes on $p$-singular elements and records the number of $p$-elements in centralizers of $p$-regular elements. It develops root-counting interpretations, block-theoretic decompositions, and local-structure arguments to determine when $\Psi_{1,p,G}$ is a genuine character, conjecturally realized by a projective $RG$-module, and proves extensive cases including all primes for $G\cong PSL(2,q)$ and related groups. The work connects $\Psi_{1,p,G}$ to truncated conjugation modules, BN-pairs, and Brauer theory, providing explicit constructions, reductions, and positivity criteria that yield both new characters and new projective modules in many structural settings. The results have broad implications for representation theory and Brauer theory, linking local subgroup structure to global character properties and offering a framework for extending the conjecture to broader classes of finite groups.

Abstract

We consider the generalized character $Ψ_{1,p,G}$ of a finite group $G$ which vanishes on all $p$-singular elements of $G$ and whose value at each $p$-regular $y \in G$ is the number of $p$-elements of $C_{G}(y)$. We conjecture that this is always a character, and may be afforded by a projective $RG$-module, where $R$ is an appropriate complete discrete valuation ring whose residue field has characteristic $p$. We examine a number of case where this is the case, and consider consequences for the representation theory and character theory of $G$ when this conjecture is known to hold. In particular, we prove, among other things, that the conjecture is valid for all primes $p$ in the case that $G \cong {\rm PSL}(2,q)$ or ${\rm SL}(2,q)$ for every prime power $q$.