Strengthening the McKay Conjecture to include local fields and local Schur indices

A central problem in the representation theory of finite groups is, given a prime p, the study of relationships between the representations of G and those of p-local subgroups of G. The simplest of these is perhaps the McKay Conjecture, which concerns the number of irreducible characters of degree prime to p. Even though this conjecture remains open, in an effort to understand it, a number of strengthenings of it have been proposed. Recently, Isaacs and Navarro, and later Navarro, have proposed interesting stronger conjectures indicating that congruences modulo p of the degree of the characters, and the invariance under certain type of fixed Galois automorphism could be taken into account as well. We propose an even further strengthening of McKayʹs Conjecture which takes into account, in addition, the p-local field of definition of the irreducible characters, as well as their p-local Schur indices.