Geometric structure for the principal series of a split reductive p-adic group with connected centre

Let G be a split reductive p-adic group with connected centre. We show that each Bernstein block in the principal series of G admits a definite geometric structure, namely that of an extended quotient. For the Iwahori-spherical block, this extended quotient has the formT // W where T is a maximal torus in the Langlands dual group of G and W is the Weyl group of G.