The McKay conjecture and Galois automorphisms

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A reduction theorem for the McKay conjecture (2007, background): “still, Navarro proposed [16] a version of the McKay conjecture involving Galois automorphisms, and Turull [21] has refined that to a statement involving fields of character values and Schur indices.”
A reduction theorem for the Alperin–McKay conjecture (2012, methodology): “We hope to use these methods in a forthcoming project for a reduction theorem of the Navarro conjecture from [31].”
A reduction theorem for the Alperin weight conjecture (2011, background): “in a unique block B of G having defect group D1. Again, by Lemma 3.3, we deduce that D1 is G-conjugate to D .B y Lemma 3.2 of [ 35 ], we then know that the unique block ¯ bi contained in bi is the Brauer first main correspondent of ¯”
The inductive Alperin–McKay and Blockwise Alperin Weight conditions for blocks with cyclic defect groups and odd primes (2013, unknown): “re ci ∈ Bl(NN(Di)) is the block with (ci)N = b. The characters of Irrnex(ca−1) can be identiﬁed with the characters of the block ca−1 ∈ Bl(NN(Da−1)/Da−1), that is dominated in ca−1. By Lemma (3.3) of [Nav04] the block ca−1 has defect group D/Da−1. Successively applying this procedure we obtain an Au”