The number of trivial composition factors of the Steinberg module
Let G be a finite group with a split BN-pair. Let \(H=B\cap N\) and \(W=N/H\) the Weyl group of G. Let k be any field of characteristic r. The Steinberg module is the right ideal of kG spanned by the element \[ \sum_{b\in B}b\sum_{w\in W}(-1)^{l(w)}\dot w, \] where l(w) denotes the length of w, and ẇ is some arbitrary, but fixed, element of N mapping to w under the natural homomorphism. We determine the number of trivial composition factors of the Steinberg module. It turns out that this number is at most one. We apply our result to the finite groups of Lie type arising from simple algebraic groups. In particular, if \(G=G(q)\) is a (nontwisted) Chevalley group defined over the field with q elements, then the trivial module is a composition factor of the Steinberg module, if and only if r divides \(q+1\).
scientific article; zbMATH DE number 4095770