The canonical compactification of a finite group of Lie type

Abstract

Let $G$ be a finite group of Lie type. We construct a finite monoid $\mathcal {M}$ having $G$ as the group of units. $\mathcal {M}$ has properties analogous to the canonical compactification of a reductive group. The complex representation theory of $\mathcal {M}$ yields Harish-Chandra’s philosophy of cuspidal representations of $G$. The main purpose of this paper is to determine the irreducible modular representations of $\mathcal {M}$. We then show that all the irreducible modular representations of $G$ come (via the 1942 work of Clifford) from the one-dimensional representations of the maximal subgroups of $\mathcal {M}$. This yields a semigroup approach to the modular representation theory of $G$, via the full rank factorizations of the ’sandwich matrices’ of $\mathcal {M}$. We then determine the irreducible modular representations of any finite monoid of Lie type.