K-theory and the connection index

TL;DR

This paper investigates the K-theory of unramified principal series representations of split p-adic groups, establishing a link between the rank of K_0 and the connection index, and connecting it to the Baum-Connes conjecture.

Contribution

It demonstrates that the rank of K_0 for the unramified principal series equals the connection index and relates this to the Baum-Connes conjecture and Iwahori C*-algebra generators.

Findings

Rank of K_0 equals the connection index f(G).

Explicit relation between K-theory generators and Iwahori C*-algebra.

Connection to a refined Baum-Connes conjecture.

Abstract

Let G denote a split simply connected almost simple p-adic group. The classical example is the special linear group SL(n). We study the K-theory of the unramified unitary principal series of G and prove that the rank of K_0 is the connection index f(G). We relate this result to a recent refinement of the Baum-Connes conjecture, and show explicitly how generators of K_0 contribute to the K-theory of the Iwahori C*-algebra I(G).