The principal series of p -adic groups with disconnected center:

Let G be a split connected reductive group over a local non-Archimedean field. We classify all irreducible complex G-representations in the principal series, irrespective of the (dis)connectedness of the center of G. This leads to a local Langlands correspondence for principal series representations of G. It satisfies all expected properties, in particular it is functorial with respect to homomorphisms of reductive groups. At the same time, we show that every Bernstein component s in the principal series has the structure of an extended quotient of Bernstein's torus by Bernstein's finite group (both attached to s).