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

TL;DR

This paper reveals that each Bernstein block in the principal series of a split reductive p-adic group with connected centre can be described as an extended quotient, providing a geometric perspective on their structure.

Contribution

It establishes that Bernstein blocks in the principal series have a geometric structure as extended quotients, specifically identifying the Iwahori-spherical block with a form involving the dual group.

Findings

Bernstein blocks admit a geometric structure as extended quotients.

The Iwahori-spherical block is isomorphic to $T//W$ involving the dual group.

Provides a new geometric interpretation for the principal series.

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 $\mathcal{G}$.