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

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.