An equivariant bijection of irreducible Brauer characters above the Dade-Glauberman-Nagao correspondence

The Glauberman correspondence and its generalisation, the Dade–Glauberman–Nagao (DGN) correspondence, play an important role in studying local-global counting conjectures and their reductions to (quasi-)simple groups. These reduction theorems require an additional set of compatibility conditions for the DGN correspondence. In this paper, we prove that there exists a bijection of irreducible Brauer characters above the DGN correspondence that is equivariant with Galois automorphisms and group automorphisms and preserves vertices. Our proof utilizes the framework of $\text{hH}$-triples developed by Navarro–Späth–Vallejo. The results establish a reduction theorem for the Galois Alperin weight conjecture.