Shintani descent and perfect isometries for blocks of finite general linear groups.
Let \(q\) be a prime power and let \(n,d\) be positive integers. Let \(\mathcal F\) denote the algebraic closure of the field of \(q\) elements and let \(V\) be an \(n\)-dimensional vector space over \(\mathcal F\). Set \({\mathcal G}=\text{GL}_n({\mathcal F})=\text{GL}(V,{\mathcal F})\) (the general linear group of \(V\) over \(\mathcal F\)). Let \(F\colon{\mathcal G}\to{\mathcal G}\) denote the standard Frobenius morphism on \(\mathcal G\) corresponding to an \({\mathcal F}_q\)-rational structure for \(\mathcal G\) such that \({\mathcal G}^F\cong\text{GL}_n(q)\). Clearly \({\mathcal G}^{F^d}\cong\text{GL}_n(q^d)\). Let \(\sigma\colon{\mathcal G}^{F^d}\to{\mathcal G}^{F^d}\) be the automorphism of \({\mathcal G}^{F^d}\) obtained by restriction of \(F\). Set \(\Gamma={\mathcal G}^{F^d}\rtimes\langle\sigma\rangle\) and let \(K\) denote a field of characteristic zero that is a splitting field for \({\mathcal G}^F\) and \({\mathcal G}^{F^d}\). Let \({\mathcal C}({\mathcal G}^F)\) denote the set of \(K\)-valued class functions on \({\mathcal G}^F\). Note that the subset \({\mathcal G}^{F^d}\sigma\) of \(\Gamma\) is invariant under conjugation by elements of \(\Gamma\). Two elements \(g,h\) of \({\mathcal G}^{F^d}\) are called \(\sigma\)-conjugate if \(h=xg^\sigma x^{-1}\) for some \(x\in{\mathcal G}^{F^d}\). There is a well-defined bijective map, \(N_{F^d/F}\), that sends \(\sigma\)-conjugacy classes of \({\mathcal G}^{F^d}\) into conjugacy classes of \({\mathcal G}^F\). Also, if \(g,h\in{\mathcal G}^{F^d}\), then \(g\sigma\) and \(h\sigma\) are conjugate in \(\Gamma\) if and only if \(g\) and \(h\) are \(\sigma\)-conjugate in \({\mathcal G}^{F^d}\). The set of functions on \({\mathcal G}^{F^d}\sigma\) obtained by restriction of \(K\)-valued class functions on \(\Gamma\) is denoted by \({\mathcal C}({\mathcal G}^{F^d}\sigma)\). The map \(\text{Sh}_{F^d/F}\colon{\mathcal C}({\mathcal G}^{F^d}\sigma)\to{\mathcal C}({\mathcal G}^F)\) defined by: \(\text{Sh}_{F^d/F}(f)(N_{F^d/F}(g))= f(g\sigma)\) for all \(f\in{\mathcal C}({\mathcal G}^{F^d}\sigma)\) and all \(g\in{\mathcal G}^F\) is a (well-defined) bijection. If \(\gamma\) is a class function on \(\Gamma\), then \(\text{Sh}_{F^d/F}(\gamma)\) will mean \(\text{Sh}_{F^d/F}(\text{Res}^\Gamma_{{\mathcal G}^{F^d}\sigma}(\gamma))\). The main result of this paper employs an important result of T. Shintani and C. Bonnafé: Theorem 1.1. Suppose that \(\chi\) is a \(\sigma\)-stable irreducible character of \({\mathcal G}^{F^d}\). Then there is a unique extension of \(\chi\) to a class function \(\widetilde\chi\) of \(\Gamma\) such that \(\text{Sh}_{F^d/F}(\widetilde\chi)\in\text{Irr}_K({\mathcal G}^F)\). Also the map \(\chi\mapsto\text{Sh}_{F^d/F}(\widetilde\chi)\) is a bijection between the set of \(\sigma\)-stable characters of \({\mathcal G}^{F^d}\) and the set \(\text{Irr}_K({\mathcal G}^F)\). Let \(\ell\) be a prime not dividing \(q\) and let \((K,{\mathcal O},k)\) be a splitting \(\ell\)-modular system for the finite groups considered above. The main result of this paper is: Theorem 1.2. Assume that \(d\) is relatively prime to both \(\ell\) and the order of \(q\) (modulo \(\ell\)). Let \(c\) be a unipotent block of \(\mathcal{OG}^F\) and let \(b\) be a unipotent block of \(G^{F^d}\) such that \(b\) and \(c\) have the same core. Then all elements of \(\text{Irr}_K(b)\) are \(\sigma\)-stable and for each \(\chi\in\text{Irr}_K(b)\), there is a \(\delta_\chi\in\{1,-1\}\) such that the map \(\chi\mapsto\delta_\chi\text{Sh}_{F^d/F}(\widetilde\chi)\) is a perfect isometry between the blocks \(b\) and \(c\). Also if \(d\) is odd, then \(\delta_\chi\) can be chosen to be 1 for all \(\chi\in\text{Irr}_K(b)\).
scientific article; zbMATH DE number 2116870