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The Alperin-McKay conjecture is a longstanding open conjecture in therepresentation theory of finite groups. Sp\"ath showed that the Alperin-McKayconjecture holds if the so-called inductive Alperin-McKay (iAM) condition holdsfor all finite simple groups. In a previous paper, the author has proved thatit is enough to verify the inductive condition for quasi-isolated blocks ofgroups of Lie type. In this paper we show that the verification of theiAM-condition can be further reduced in many cases to isolated blocks. As aconsequence of this we obtain a proof of the Alperin-McKay conjecture for2-blocks of finite groups with abelian defect.

Quasi-isolated blocks and the Alperin-McKay conjecture