Title: Descent of equivalences and character bijections,
Abstract: Categorical equivalences between block algebras of finite groups - such as
Morita and derived equivalences - are well-known to induce character bijections
which commute with the Galois groups of field extensions. This is the
motivation for attempting to realise known Morita and derived equivalences over
non splitting fields. This article presents various result on the theme of
descent. We start with the observation that perfect isometries induced by a
virtual Morita equivalence induce isomorphisms of centers in non-split
situations, and explain connections with Navarro's generalisation of the
Alperin-McKay conjecture. We show that Rouquier's splendid Rickard complex for
blocks with cyclic defect groups descends to the non-split case. We also prove
a descent theorem for Morita equivalences with endopermutation source.