Character correspondences induced by magic representations

Abstract

Let G be a finite group, K a normal subgroup of G and H a subgroup such that G = HK, and set L = H \cap K. Suppose \theta \in Irr K and \phi \in Irr L, and \phi\ occurs in \theta_L with multiplicity n > 0. A projective representation of degree n on H/L is defined in this situation; if this representation is ordinary, it yields a bijection between Irr(G | \theta) and Irr(H | \phi). The behavior of fields of values and Schur indices under this bijection is described. A modular version of the main result is proved. We show that the theory applies if n and the order of H/L are coprime. Finally, assume that P <= G is a p-group with P \cap K = 1 and PK normal in G, that H = N_G(P), and that \theta\ and \phi\ belong to blocks of p-defect zero which are Brauer correspondents with respect to the group P. Then every block of F_p[G] or Q_p[G] lying over \theta\ is Morita-equivalent to its Brauer correspondent with respect to P. This strengthens a result of Turull [Above the Glauberman correspondence, Advances in Math. 217 (2008), 2170--2205].