The orbit method for the Baum-Connes Conjecture for algebraic groups over local function fields

TL;DR

This paper adapts the orbit method to verify the Baum-Connes conjecture for specific algebraic groups over local function fields, extending previous methods to positive characteristic non-archimedean fields.

Contribution

It modifies the orbit method for the Baum-Connes conjecture to handle linear algebraic groups over local function fields, verifying the conjecture for new classes of groups.

Findings

Verified Baum-Connes conjecture for certain Levi-decomposable groups

Extended orbit method to positive characteristic local fields

Confirmed conjecture for the Jacobi group

Abstract

The main purpose of this paper is to modify the orbit method for the Baum-Connes conjecture as developed by Chabert, Echterhoff and Nest in their proof of the Connes-Kasparov conjecture for almost connected groups \cite{MR2010742} in order to deal with linear algebraic groups over local function fields (i.e., non-archimedean local fields of positive characteristic). As a consequence, we verify the Baum-Connes conjecture for certain Levi-decomposable linear algebraic groups over local function fields. One of these is the Jacobi group, which is the semidirect product of the symplectic group and the Heisenberg group.