Steinberg characters for Chevalley groups over finite local rings

Abstract

We construct for a Chevalley group over a finite local ring an analogue of the Steinberg character that was defined for the general linear group by P. Lees and, independently, by G. Hill. Further, we show that this analogue has a homological origin and, when irreducible, describe it in terms of a linear character of the corresponding Hecke algebra. However, we find that the analogue is reducible in general. Thus we determine its decomposition into distinct irreducible constituents and characterise these constituents using Gelfand–Graev characters.