On the Brauer–Glauberman correspondence

The purpose of the present paper is two-fold: on the one hand, to show the existence of a correspondence unifying Brauer's and Glauberman's ones (see Theorem 4.6), and, on the other hand, to give an alternative proof of Watanabe's equivalence in [Atumi Watanabe, The Glauberman character correspondence and perfect isometries for blocks of finite groups, J. Algebra 216 (1999) 548–565]. By the way, we give a short proof of the coincidence of the Clifford extensions associated with a pair of Glauberman correspondent irreducible representations (see Corollary 4.16), a question that, surprisingly enough, has only been partially solved recently (see [Morton Harris, Markus Linckelmann, On the Glauberman and Watanabe correspondences for blocks of finite p-solvable groups, Trans. Amer. Math. Soc. 354 (2002) 3435–3453] and [Shigeo Koshitani, Gerhard Michler, Glauberman correspondence of p-blocks of finite groups, J. Algebra 243 (2001) 504–517]).