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]).