On blocks of finite reductive groups and twisted induction
Let \(\mathbf G\) be a connected reductive algebraic group defined over \(\mathbb{F}_q\) and \(F\colon\mathbf G\to\mathbf G\) a Frobenius morphism, so that \(G:=\mathbf G^F\) is the finite group of \(F\)-invariant points of \(\mathbf G\). Let \(\ell\) be a prime not dividing \(q\) and \(e\) the order of \(q\bmod\ell\). Let \(\mathbf L\) be an \(F\)-stable Levi subgroup of \(\mathbf G\). The Lusztig induction functor \(R^{\mathbf G}_{\mathbf L}\) takes generalized characters of \(L\) (over \(\overline{\mathbf Q}_\ell\)) into generalized characters of \(G\). The main purpose of the paper is to describe the \(\ell\)-blocks of \(G\) in terms of the functor \(R_{\mathbf L}^{\mathbf G}\). It has been proved in several cases that the \(\ell\)-blocks of \(G\) induce a partition of the set of irreducible ordinary characters of \(G\) similar to that of ordinary Harish-Chandra theory. Ordinary induction is replaced with Lusztig induction and the partition into series is given by generalized Harish-Chandra theory, first introduced by \textit{P.~Fong} and \textit{B.~Srinivasan} for unipotent characters [in J. Algebra 104, 301-309 (1986; Zbl 0606.20035)]. Rational Levi subgroups of rational parabolic subgroups are replaced by \(e\)-split Levi subgroups of \(\mathbf G\) (the order of \(G\) is a polynomial in \(q\), and one can talk of the ``polynomial order'' of the pair \((\mathbf G,F)\); the centralizer in \(\mathbf G\) of an \(F\)-stable torus whose polynomial order is a power of the \(e\)th cyclotomic polynomial is called an \(e\)-split Levi subgroup of \(\mathbf G\)), cuspidal irreducible characters are replaced by \(e\)-cuspidal irreducible characters (an irreducible character \(\chi\) of \(G\) is said to be \(e\)-cuspidal if \(^*R^{\mathbf G}_{\mathbf L}(\chi)=0\) for every proper \(e\)-split Levi subgroup \(\mathbf L\) of \(\mathbf G\), where \(^*R^{\mathbf G}_{\mathbf L}\) denotes the adjoint of the functor \(R^{\mathbf G}_{\mathbf L}\)). The \(\ell\)-blocks of \(G\) have been described in special cases by means of Lusztig induction (see, e.g., \textit{P.~Fong} and \textit{B.~Srinivasan} [J. Reine Angew. Math. 396, 122-191 (1989; Zbl 0656.20039), Invent. Math. 69, 109-153 (1982; Zbl 0507.20007)] for classical groups, and \textit{M.~Cabanes} and \textit{M.~Enguehard} [Invent. Math. 117, No. 1, 149-164 (1994; Zbl 0817.20046)] for unipotent blocks). In the paper under review the authors give such a description for arbitrary \(\mathbf G\), with light restriction on \(\ell\). If \(\mathbf T\) is an \(F\)-stable maximal torus of \(\mathbf G\), let \(\mathcal E(G,\ell')\) denote the set of irreducible constituents of the Deligne-Lusztig characters \(R^{\mathbf G}_{\mathbf T}(\theta)\), where \(\theta\) is a character of \(T\) of order prime to \(\ell\). Let \((\mathbf L,\zeta)\) be a pair consisting of an \(e\)-split Levi subgroup \(\mathbf L\) and an \(e\)-cuspidal irreducible character \(\zeta\) in \(\mathcal E(L,\ell')\). Assuming \(\ell\geq 7\) if some component of \(\mathbf G\) is of type \(E_8\) and \(\ell\geq 5\) otherwise, the authors prove that all irreducible constituents of \(R_{\mathbf L}^{\mathbf G}(\zeta)\) belong to the same \(\ell\)-block of \(G\) and that this defines a bijection between the \(G\)-conjugacy classes of pairs \((\mathbf L,\zeta)\), where \(\mathbf L\) is an \(e\)-split Levi subgroup of \(\mathbf G\) and \(\zeta\) is an \(e\)-cuspidal character which belongs to \(\mathcal E(L,\ell')\) and the \(\ell\)-blocks of \(G\). Let \(\leq_e\) be the relation defined by \((\mathbf L,\zeta)\leq_e(\mathbf M,\xi)\) if \(\mathbf L,\mathbf M\) are \(e\)-split Levi subgroups with \(\mathbf L\subseteq\mathbf M\), \(\zeta\), \(\xi\) are irreducible characters of \(L\), \(M\), respectively, and \(\xi\) is a constituent of \(R^{\mathbf M}_{\mathbf L}(\zeta)\). The authors prove moreover that, if a block \(b\) of \(G\) corresponds to the pair \((\mathbf L,\zeta)\) by the above bijection, then the irreducible characters in \(\text{Irr}(b)\cap\mathcal E(G,\ell')\) are those characters \(\chi\) such that \((\mathbf L,\zeta)\ll_e(\mathbf G,\chi)\), where \(\ll_e\) is the transitive closure of \(\leq_e\).
scientific article; zbMATH DE number 1340110
scientific article