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Counting \(p'\)-characters in finite reductive groups. (English) Zbl 1251.20017

Let \(G\) be a finite group and let \(p\) be a prime divisor of the order \(|G|\) of \(G\). Let \(\text{Irr}(G)\) denote the set of irreducible characters of \(G\), and let \(\text{Irr}_{p'}(G)\) be the subset of irreducible characters with degree prime to \(p\). Denote by \(|\text{Irr}_{p'}(G)|\) the cardinality of \(\text{Irr}_{p'}(G)\).
John McKay has conjectured the equality \[ |\text{Irr}_{p'}(G)|=|\text{Irr}_{p'}(\text{Nor}_G(P))|, \] for any fixed \(p\)-Sylow subgroup of \(G\), where \(\text{Nor}_G(P)\) denotes the normalizer of \(P\) in \(G\).
While it is proved for some groups, it remains open in general. Recently, Isaacs, Malle and Navarro reduced this conjecture to a new question, the so-called inductive McKay condition, which concerns properties of perfect central extensions of finite simple groups.
Let \(Z\) denote the centre of \(G\), and let \(\nu\) be a fixed linear character of \(Z\). Let \(\text{Irr}_{p'}(G|\nu)\) denote the subset of characters \(\chi\in\text{Irr}_{p'}(G)\) lying over \(\nu\), that is, satisfying \[ \langle\chi,\text{Ind}_Z^G(\nu)\rangle_G\neq 0, \] where \(\langle\,,\,\rangle_G\) is the usual scalar product on the space of class functions on \(G\).
The article under review is concerned with the relative McKay conjecture, which asserts that, for every linear character \(\nu\) of \(Z\), then one has the equality \[ |\text{Irr}_{p'}(G|\nu)|=|\text{Irr}_{p'}(\text{Nor}_G(P)|\nu)|. \] One of the motivations for considering this question is the fact that in order to prove the inductive McKay condition, one has to show in particular that the relative McKay conjecture holds for some perfect central extensions of finite simple groups.
Let \(\mathbf G\) be a connected reductive algebraic group defined over a finite field \(\mathbb F_q\), with corresponding Frobenius map \(F\colon\mathbf G\to\mathbf G\). Let \(G:=\mathbf G^F\) denote the (finite) group of \(F\)-fixed points of \(\mathbf G\). Assume that the \(q\) is power of a prime number \(p\) which is good for \(\mathbf G\), that is, \(p\) does not divide the coefficients of the highest root of the root system associated to each simple factor of \(\mathbf G\). Let \(\mathbf B\) be an \(F\)-stable Borel subgroup of \(\mathbf G\), with unipotent radical denoted by \(\mathbf U\) (which is \(F\)-stable). Note that, if \(\mathbf U\) is not trivial, then \(p\) divides the order of the finite group \(G\), and \(U:=\mathbf U^F\) is a \(p\)-Sylow subgroup of \(G\). Moreover, one has \(\text{Nor}_G(U)=B\), where \(B:=\mathbf B^F\).
If the centre of \(\mathbf G\) is connected, the McKay conjecture is true for \(G\) at the prime \(p\). It is proved in the article that the relative McKay conjecture also holds in this case.
The question is more difficult when the centre of \(\mathbf G\) is disconnected. Denote by \(A(\mathbf G):=\text{Z}(\mathbf G)/\text{Z}(\mathbf G)^0\) the group of components of the centre \(Z(\mathbf G)\) of \(\mathbf G\).
The main result of the article is the following: if the group \(H^1(F,A(\mathbf G))\) of the \(F\)-classes of \(A(\mathbf G)\) is trivial or has prime order, then, for every linear character \(\nu\) of \(Z\), one has \[ |\text{Irr}_{p'}(G|\nu)|=|\text{Irr}_{p'}(B|\nu)|. \] In particular, the relative MacKay conjecture holds for \(G\) at \(p\) for a simple simply connected group \(\mathbf G\) of type \(B_n\), \(C_n\), \(E_6\) or \(E_7\). Moreover, the paper provides an explicit computation of the number of semisimple classes of \(G\) for any simple algebraic group \(\mathbf G\).

MSC:

20C33 Representations of finite groups of Lie type
20G40 Linear algebraic groups over finite fields
20G05 Representation theory for linear algebraic groups
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