Local-global conjectures relate the representation theory of finite groups with that of their local subgroups. Their best-known representative is the McKay conjecture. The Galois-McKay conjecture claims that the bijection from the McKay conjecture can be chosen such that it is equivariant under certain Galois automorphisms.
Starting with the McKay conjecture in 2007, many local-global conjectures have been reduced to a question about simple groups. Since then, the resulting inductive conditions have been investigated and verified for many simple groups. In this talk, we introduce the Galois-McKay conjecture, the corresponding inductive condition, and ideas for the verification of the inductive condition for some groups of Lie type in defining characteristic.