Extended quotients in the principal series of reductive p-adic groups

TL;DR

This paper proves the geometric conjecture for the principal series of split connected reductive p-adic groups, linking the structure of their smooth duals to extended quotients and the local Langlands correspondence.

Contribution

It establishes the geometric conjecture for principal series in split reductive p-adic groups using Springer and Langlands parameters, highlighting the role of extended affine Weyl groups.

Findings

Confirmed the geometric conjecture for principal series

Linked extended quotients with local Langlands correspondence

Emphasized the role of two-sided cells in affine Weyl groups

Abstract

The geometric conjecture developed by the authors in [1,2,3,4] applies to the smooth dual Irr(G) of any reductive p-adic group G. It predicts a definite geometric structure - the structure of an extended quotient - for each component in the Bernstein decomposition of Irr(G). In this article, we prove the geometric conjecture for the principal series in any split connected reductive p-adic group G. The proof proceeds via Springer parameters and Langlands parameters. As a consequence of this approach, we establish strong links with the local Langlands correspondence. One important feature of our approach is the emphasis on two-sided cells in extended affine Weyl groups.