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

Abstract

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