# On the Baum--Connes conjecture with coefficients for linear algebraic groups

## TL;DR

This paper investigates the Baum--Connes conjecture with coefficients, proving it for certain classes of linear algebraic groups over local and global fields, although the proof remains incomplete.

## Contribution

It extends the Baum--Connes conjecture to new classes of linear algebraic groups over various fields, under specific conditions.

## Key findings

- Proved the conjecture for linear algebraic groups over non-archimedean local fields
- Extended results to groups over adeles of global fields with amenable Lie groups at archimedean places
- Included all closed subgroups of the above classes

## Abstract

We prove the Baum--Connes conjecture with arbitrary coefficients for some classes of groups: (1) Linear algebraic groups over a non-archimedean local field. (2) Linear algebraic groups over the adeles of a global field k, provided that at every archimedean place of k the associated Lie group is amenable. (3) All closed subgroups of the above groups. This includes linear algebraic groups over global fields - with the same condition as in (2). Update: proof of main results incomplete, maybe not repairable.