Modular Harish-Chandra theory. II
Let \(\mathbf G\) be a connected, reductive algebraic group defined over \({\mathbf F}_q\), and \(F : {\mathbf G} \to {\mathbf G}\) a Frobenius morphism so that the group \(G = {\mathbf G}^F\) of \(F\)-fixed points of \(\mathbf G\) is a finite reductive group. In an earlier paper [Part I, Math. Z. 211, 49-71 (1992; Zbl 0798.20034)] the authors studied the modular Harish- Chandra theory of \(G\), where \(\mathbf G\) has a connected center, with respect to certain ``regular'' cuspidal characters of subgroups of the form \(L = {\mathbf L}^F\) where \({\mathbf L} \subset {\mathbf P}\) are an \(F\)- stable Levi subgroup and an \(F\)-stable parabolic subgroup, respectively. In this paper they extend their results to arbitrary \(\mathbf G\) by embedding \(\mathbf G\) in a group \(\widetilde {\mathbf G}\) with a connected center. Let \((k, O, K)\) be a split \(r\)-modular system where \(r\) does not divide \(q\). The regular cuspidal characters of \(G\), which are constituents of Gelfand-Graev representations, are shown to be restrictions to \(G\) of regular cuspidal characters of \(\widetilde {G}\). If \(L\) is a subgroup of \(G\) as above then a regular cuspidal character \(\mu\) of \(L\) is associated with a semisimple conjugacy class \((s)\) in the dual group \(L^*\) by Deligne-Lusztig theory. The authors construct a canonical \(O\)-lattice \(D_O (\mu)\) for the \({\mathbf C} L\)-module affording \(\mu\). Assuming a certain condition on \((s)\) they study the endomorphism ring \(H_O (\mu) = \text{End}_{OG} (R^G_L (D_O (\mu)))\) where \(R^G_L\) stands for Harish-Chandra induction and extend the results of their earlier paper (op. cit.). In particular, it is shown that \(\text{End}_{RG} (R^G_L (D_O (\mu))) \cong R \otimes_O H_O (\mu)\) for all \(R \in \{k, O, K\}\), and that the \(r\)- decomposition matrix of \(H_O (\mu)\) is a submatrix of the \(r\)- decomposition matrix of \(G\).
scientific article; zbMATH DE number 743527
scientific article