Monoids and Cuspidal Group Characters

Abstract

Let M be a finite monoid with unit group G. By the work of Munn and Ponizovski, the irreducible complex representations of M are classified according to which J-class (apex) they come from. Consider the irreducible representations of M with apex \(\neq G\). These representations restrict to representations of G, whose components we view as coming from J-classes below G. The remaining irreducible representations (and their characters) of G are called cuspidal. We show that an irreducible character \(\chi\) of G is cuspidal if and only if \(\chi(\sum V(e)) =0\) for all idempotents \(e \neq 1\), where \(V(e) = \{x \in G \mid exe = e\}\).