The Glauberman correspondent of a nilpotent block of a finite group.
Let \(p\) be a prime integer, let \(G\) be a finite group, let \((\mathcal O,K,k=\mathcal O/J(\mathcal O))\) be a \(p\)-modular system such that \(k\) is an algebraically closed field of characteristic \(p\), \(K\) is a field of characteristic zero that contains a primitive \(|G|\)-th root of unity. Also let \(A\) be a solvable subgroup of \(\Aut(G)\) and set \(C=C_G(A)\). The Glauberman correspondence is a natural bijection \(\pi(G,A)\colon\text{Irr}_K(G)^A\to\text{Irr}_K(C)\). The main result of this paper is: Theorem 1. Assume that \(B\) is an \(A\)-stable, nilpotent block of \(\mathcal OG\) and let \(b\) be the Glauberman-Watanabe corresponding block of \(\mathcal OC\). Then \(b\) is also nilpotent and there is a basic Morita equivalence between \(B\) and \(b\) that induces the Glauberman correspondence \(\pi(G,A)\) sending \(\text{Irr}_K(B)\) onto \(\text{Irr}_K(b)\).
scientific article; zbMATH DE number 5509684