Describe Bernstein decompositions via the building action

Determine an explicit, action-theoretic description of the Bernstein decompositions of the G-equivariant K-homology K_*^G(BG) and the G-equivariant chamber homology CH_*^G(BG), directly in terms of the action of the reductive p-adic group G on its Bruhat–Tits building BG, rather than through decompositions obtained from representation-theoretic isomorphisms.

Background

In Section 6.2 the paper relates topological K-theory and periodic cyclic homology for C_r^*(G) and S(G) and connects these to equivariant K-homology and chamber homology of the Bruhat–Tits building BG via the Baum–Connes assembly map. The authors establish natural isomorphisms and note that all terms not involving BG admit canonical Bernstein decompositions, and they compute the corresponding summands explicitly for algebras associated to Bernstein components.

Using the assembly map and the isomorphism identifying chamber homology with compactly supported Hochschild homology, the authors obtain natural Bernstein decompositions of K_^G(BG) and CH_^G(BG). However, they point out that a direct geometric description of these decompositions in terms of the action of G on BG is lacking, reflecting the difficulty of recovering the Bernstein decomposition from restrictions of G-representations to compact open subgroups.