The Glauberman character correspondence and perfect isometries for blocks of finite groups
Let \(G\) be a finite group and let \(S\) be a solvable finite group that acts on \(G\) and such that \((|G|,|S|)=1\). Let \(p\) be a prime and let \((K,O,F)\) be a \(p\)-modular system such that \(K\) is ``big enough'' for \(G\). For a central idempotent \(e\) of \(OG\), let \(R_K(G,e)\) denote the additive group of characters of \(G\) afforded by \(KGe\)-modules. Let \(\text{Irr}(G)\) denote the set of ordinary irreducible characters of \(G\) and let \(\text{Irr}_S(G)\) denote the set of \(S\)-invariant elements of \(\text{Irr}(G)\). As is well-known, in this case, the canonically defined Glauberman correspondence gives a bijection \(\pi(G,S)\colon\text{Irr}_S(G)\to\text{Irr}(C_G(S))\). The first two main results of the paper are: Theorem 1. Let \(B\) be an \(S\)-invariant \(p\)-block of \(G\) such that \(C_G(S)\) contains a defect group \(D\) of \(B\). Then \(\text{Irr}(B)\subseteq\text{Irr}_S(G)\) and there exists a unique block \(b\) of \(C_G(S)\) with defect group \(D\) such that \(\text{Irr}(b)=\pi(G,S)(\text{Irr}(B))\). Moreover there is a perfect isometry \(R\colon{\mathcal R}_K(G,B)\to{\mathcal R}_K(C_G(S),b)\) such that \(R(\chi)=\pm\pi(G,S)(\chi)\) for any \(\chi\in\text{Irr}(B)\). Theorem 2. In the situation of Theorem 1, the Brauer categories \(\text{Br}_B(G)\) and \(\text{Br}_b(C_G(S))\) are equivalent. Moreover the blocks \(B\) and \(b\) are isotypic. -- The correspondences in Theorem 2 are obtained from Theorem 1. The final Section 4 of this paper uses Shintani descent to obtain a perfect isometry. Let \(\widehat L\) be a finite field of characteristic \(\ell\), let \(\sigma\) be a Frobenius automorphism of \(\widehat L\) and let \(C_{\widehat L}(\sigma)=L\) where \([\widehat L:L]=m\). Here \(\sigma\) acts naturally on \(\text{GL}_n(\widehat L)\) with fixed points \(\text{GL}_n(L)\). Let \(B(\text{GL}_n(\widehat L))\) and \(B(\text{GL}_n(L))\) denote the principal blocks of \(\text{GL}_n(\widehat L)\) and \(\text{GL}_n(L)\) resp. Shintani descent is the key ingredient in the proof of the final result of this paper: Theorem 3. Suppose that \(p\) does not divide \(m\) and that \(\sigma\) centralizes a Sylow \(p\)-subgroup \(P\) of \(\text{GL}_n(\widehat L)\). Then \(\text{Irr}(B(\text{GL}_n(\widehat L)))\subseteq\text{Irr}_\sigma(\text{GL}_n(\widehat L))\) and there exists a perfect isometry \[ R\colon R_K(\text{GL}_n(\widehat L),B(\text{GL}_n(\widehat L)))\to R_K(\text{GL}_n(L),B(\text{GL}_n(L))) \] such that \(R(\chi)=\pm\text{Sh}(\chi)\) for any \(\chi\in\text{Irr}(B(\text{GL}_n(\widehat L)))\).
scientific article; zbMATH DE number 1340590
scientific article