Distinguishing finite group characters and refined local-global phenomena

TL;DR

This paper investigates bounds on when primitive irreducible characters of finite groups agree, providing new results for small degrees and insights into automorphic representations, with implications for local-global phenomena.

Contribution

It establishes sharp bounds for primitive characters of degrees 2 and 3, and extends the analysis to specific families of groups, enhancing understanding of local-global principles.

Findings

Sharp bounds for primitive characters of degree 2 and 3

Character agreement bounds for PSL(2,q) and SL(2,q)

Insights into strong multiplicity one phenomena for automorphic representations

Abstract

Serre obtained a sharp bound on how often two irreducible degree $n$ complex characters of a finite group can agree, which tells us how many local factors determine an Artin $L$-function. We consider the more delicate question of finding a sharp bound when these objects are primitive, and answer these questions for $n=2,3$. This provides some insight on refined strong multiplicity one phenomena for automorphic representations of GL$(n)$. For general $n$, we also answer the character question for the families PSL$(2,q)$ and SL$(2,q)$.