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

Abstract

Lees, and independently Hill, described an irreducible character of GL n (Z/p h Z) for h > 1, which is an analogue of the Steinberg character in the h = 1 case. We give a new construction of this analogue. In particular, we prove that it is induced from the Steinberg character of an appropriate parabolic subgroup. Further, we use this to show that it has a characterisation identical to that given by Curtis to the Steinberg character.  2004 Elsevier Inc. All rights reserved.

Steinberg characters for Chevalley groups over finite local rings

“…When Σ = A n , this reduces to the definition given in [2]. Further, Suzuki [24] showed that every subgroup of G containing B is of this form provided that κ has odd characteristic; q = 3 for types A 3 , B n , C n , D n and F 4 ; and char κ = 3 for type G 2 .…”

“…As in [2], the analogue of the Steinberg character will be described as an alternating sum of permutation characters over certain "parabolic" subgroups. This is similar to both Steinberg's original construction [20] for the general linear group and Curtis' later definition [5] for finite groups with BN-pair.…”

“…Similarly, H J = H α : α ∈ J is a standard parabolic subgroup for each J ⊆ S and the expression for St is exactly Curtis' formula [5] for the Steinberg character of the Chevalley group over the residue field. (ii) If > 1 and Σ = A n then the parabolic subgroups H J are the same as the subgroups described in [2], at least modulo their centres. In this case St is identical to the analogue considered in [2] and by Lees in [15].…”

“…(ii) If > 1 and Σ = A n then the parabolic subgroups H J are the same as the subgroups described in [2], at least modulo their centres. In this case St is identical to the analogue considered in [2] and by Lees in [15].…”

“…Hill [11] independently described the same analogue as the unique common constituent of the permutation character (1 B ) G over the subgroup B of upper triangular matrices and a version of the Gelfand-Graev character Γ 0 (see [4]). Our approach will be to adapt the construction and techniques used for GL n (R) in [2].…”