An analogue of the Steinberg character for the general linear group over the integers modulo a prime power. (Q1888817)

Lees, and independently Hill, described an irreducible character of \(G=\text{GL}_n(\mathbb{Z}/p^h\mathbb{Z})\) for \(h>1\), which is an analogue of the Steinberg character in the \(h=1\) case. More generally, let \(\mathfrak o\) be a discrete valuation ring with finite residue field and maximal ideal \(\mathfrak p\). The author gives a new construction of the analogue of the Steinberg character when \(G=\text{GL}_n({\mathfrak o}/{\mathfrak p}^h)\). It employs an explicit idempotent in the group ring \(\mathbb{C} G\). He then checks some of its properties.