Sylow normalizers and Galois action on characters

The Navarro conjecture states that the actions of a particular subgroup of Galois automorphisms on the two sets of characters involved in the McKay conjecture should be permutation isomorphic. This conjecture predicted that the local condition that a Sylow $p$-subgroup $P$ of a finite group $G$ is self-normalizing can be characterized in terms of the character theory of $G$; a prediction that has been recently verified for all primes. In the same spirit, we restrict our attention to the character theory of the principal $p$-block and analyze the relation with the structure of ${\bf N}_G(P)$.

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