K-theory of rank one reductive p-adic groups and Bernstein blocks

TL;DR

This paper establishes a colimit formula for the K-theory spectra of rank one reductive p-adic groups with regular coefficients, and extends the result to Bernstein blocks in the complex split case using Roche's types.

Contribution

It provides a new colimit formula for the K-theory of rank one p-adic groups and improves it to Bernstein blocks in the complex split case.

Findings

Colimit formula for K-theory spectra of rank one p-adic groups.

Extension of the formula to Bernstein blocks in the complex split case.

Use of Roche's types to refine the K-theory description.

Abstract

We prove a colimit formula for the K-theory spectra of reductive p-adic groups of rank one with regular coefficients in terms of the K-theory of certain compact open subgroups. Furthermore, in the complex case, we show, using the construction of types provided by Roche, that this result can be improved to obtain a formula for the K-theory spectrum of every principal series Bernstein block if the group is split.