The extended Steinberg character
Suppose \(G\) is a finite group with a split BN-pair of characteristic \(p\) and \(\text{rank } n \geq 1\), and it is a normal subgroup of some finite group \(\widehat {G}\). Let \(\chi = \text{St}_G\) be the Steinberg character of \(G\). It is well known that \(\chi\) is the unique irreducible character of \(G\) satisfying \((\chi_B, 1_B) = 1\) and \((\chi_P, 1_P) = 0\) for all subgroups \(P \gneqq B\) of \(G\). The aim of this paper is to evaluate \(\chi = \text{St}_{\widehat {G}}\), including signs. The authors show the following main theorems: Theorem A. There is a unique irreducible character \(\widehat {\chi}\) of \(\widehat {G}\) extending \(\chi\) and satisfying \((\widehat {\chi}_{\widehat {B}}, 1_{\widehat {B}}) = 1\). Theorem B. The \(\mathbb{Q} G\)-module \(H_{n - 1} (\Delta)\) affords \(\text{St}_{\widehat {G}}\) for \(n > 1\) and affords \(1_{\widehat {G}} + \text{St}_{\widehat {G}}\) for \(n = 1\). In any case, \(1_{\widehat{G}} + (-1)^{n - 1} \text{St}_{\widehat{G}}\) is the Lefschetz character of the homology representation on the \(\widehat{G}\)-complex \(\Delta = \Delta(G)\) over the rationals. Theorem C. Suppose \(G\) satisfies the commutator relations. Let \(x \in \widehat{G}\). Either \(\text{St}_{\widehat {G}} (\widehat {x}) = 0\), or \(\Delta_x\) is a Moufang complex, say of rank \(n_x\), and \(\text{St}_{\widehat {G}} (x) = (-1)^{n + n_x} |C_G (x) |_p\). We have \(\text{St}_{\widehat{G}} (x) = 0\) precisely when the \(p\)-part of \(x\) does not lie in a defect group of the \(p\)-block of \(\widehat {G}\) containing \(\text{St}_{\widehat {G}}\).
scientific article; zbMATH DE number 783928
scientific article