THE STEINBERG LATTICE OF A FINITE CHEVALLEY GROUP AND ITS MODULAR REDUCTION

Abstract

Let $p$ be a prime and let $q = p^a$, where $a$ is a positive integer. Let $G = G({\mathbb{F}}_q)$ be a Chevalley group over $\mathbb{ F}_q$, with associated system of roots $\Phi$ and Weyl group $W$. Steinberg showed in 1957 that $G$ has an irreducible complex representation whose degree equals the $p$-part of $|G|$ [11]. This representation, now known as the Steinberg representation, has remarkable properties, which reflect the structure of $G$, and there have been many research papers devoted to its study. The module constructed in [11] is in fact a right ideal in the integral group ring $\mathbb{ Z}G$ of $G$, and is thus a $\mathbb{ Z}G$-lattice, which we propose to call the Steinberg lattice of $G$. It should be noted that lattices not integrally isomorphic to the Steinberg lattice may also afford the Steinberg representation, and such lattices may differ considerably in their properties compared with the Steinberg lattice.