Green functions and Glauberman degree-divisibility. (English) Zbl 1491.20026

Geck, Meinolf

Ann. Math. (2) 192, No. 1, 229-249 (2020).

Let \(\Gamma \) be a finite group and \(S\) be a finite solvable group acting as a group of automorphisms on \(\Gamma \) where \(S\) and \(\Gamma \) have coprime orders. G. Glauberman [Can. J. Math. 20, 1465–1488 (1968; Zbl 0167.02602)] introduced a bijection \(\theta \longmapsto \theta ^{\ast }\) between the set Irr\(_{S}(\Gamma )\) of \(S\)-invariant irreducible characters of \(\Gamma \) and the set Irr\((C_{\Gamma }(S))\) of irreducible characters of the centralizer of \(S\) in \(\Gamma \). The Glauberman degree-divisibility conjecture states that \(\theta ^{\ast }(1)\) always divides \(\theta (1)\ \)and this conjecture has been proved in special cases. In particular, B. Hartley and A. Turull [J. Reine Angew. Math. 451, 175–219 (1994; Zbl 0797.20007)] proved that the conjecture holds if \( \Gamma \) is a group of Lie type and a specific set of congruence conditions on Green functions on \(\Gamma \) are satisfied. They used this to prove the conjecture for some general classes of simple linear groups. In the present paper the author shows how Hartley and Turull’s approach [loc. cit.] via congruence conditions on Green functions can be used to prove Glauberman’s long-standing conjecture [loc. cit.] in complete generality. The proof pulls together extensive work on character sheaves by G. Lusztig [Ann. Math. (2) 131, No. 2, 355–408 (1990; Zbl 0695.20024)] and T. Shoji [Adv. Math. 111, No. 2, 244–313, 314–354 (1995; Zbl 0832.20065)].

Reviewer: John D. Dixon (Ottawa)

MSC:

20C33	Representations of finite groups of Lie type
20C15	Ordinary representations and characters
20G40	Linear algebraic groups over finite fields

Keywords:

Glauberman correspondence; finite groups of Lie type; Green functions; character sheaves

Citations:

Zbl 0167.02602; Zbl 0797.20007; Zbl 0695.20024; Zbl 0832.20065

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