  We prove that a strengthened form of the local Langlands conjecture is valid
throughout the principal series of any connected split reductive $p$-adic
group. The method of proof is to establish the presence of a very simple
geometric structure, in both the smooth dual and the Langlands parameters. We
prove that this geometric structure is present, in the same way, for the
general linear group, including all of its inner forms. With these results as
evidence, we give a detailed formulation of a general geometric structure
conjecture.
