On the generalized Steinberg representation (Q1604560)

Let \(G\) be a connected split reductive group defined over a finite field \(\mathbb{F}_q\). The aim of this paper is to study certain generalizations of the Steiberg representation of \(G(\mathbb{F}_q)\). The author shows that the generalized Steinberg representation \(\text{St}^{P_J}_G\) decomposes as a multiplicity free representation when \(G\) is of type \(A_n\), \(B_n\) and \(C_n\), and \(P_J\) is a maximal parabolic subgroup. He gives an explicit decomposition of \(\text{St}_G^{P_J}\) in these cases. When \(G\) is of type \(D_n\), \(E_6\) and \(E_7\), and \(P_J\) is a maximal parabolic subgroup with Abelian unipotent radical, he proves that \(\text{St}^{P_J}_G\) decomposes as a multiplicity free representation. In the case of \(E_6\) and \(E_7\), he describes the irreducible components of \(\text{St}^{P_J}_G\) using work of \textit{M. Reeder} [Indag. Math., New Ser. 9, No. 3, 431-441 (1998; Zbl 0921.22012)].