I will present joint work with Robert Boltje and Markus Linckelmann on the structure of   the Picard group  of self Morita equivalences of a block of a finite group algebra  over  a complete discrete valuation ring. In particular, I will give a description of the subgroup  of  endopermutation source  Morita  equivalences in terms of the underlying fusion system, Dade group, and source algebra.

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 ${\bf N}_G(P)$.