Conjectures about p-adic groups and their noncommutative geometry

TL;DR

This paper explores conjectures connecting the representation theory and geometry of reductive p-adic groups, proposing new ideas and strategies related to the local Langlands correspondence and noncommutative geometry.

Contribution

It introduces several new conjectures about the geometric structure of Bernstein components and relates them to major conjectures like the local Langlands and Baum--Connes conjectures.

Findings

Proposes new conjectures on the geometric structure of Bernstein components.

Relates these conjectures to the local Langlands and Baum--Connes conjectures.

Suggests a reduction strategy for the local Langlands correspondence.

Abstract

Let G be any reductive p-adic group. We discuss several conjectures, some of them new, that involve the representation theory and the geometry of G. At the heart of these conjectures are statements about the geometric structure of Bernstein components for G, both at the level of the space of irreducible representations and at the level of the associated Hecke algebras. We relate this to two well-known conjectures: the local Langlands correspondence and the Baum--Connes conjecture for G. In particular, we present a strategy to reduce the local Langlands correspondence for irreducible G-representations to the local Langlands correspondence for supercuspidal representations of Levi subgroups.