Geometric Chern characters in p-adic equivariant K-theory

Totally disconnected groups play an important role for arithmetic questions, in particular groups like $Sl_n(\mathbb{Q}_p)$. They have nice actions on their Bruhat-Tits building (with the special property that the orbits are discrete). Among the important invariants of such actions are the equivariant K-theory and equivariant K-homology, closely related also to the representation theory of these groups. For example, the equivariant K-homology of the Bruhat-Tits building features as the left hand side of the Baum-Connes conjecture for $Sl_n(\mathbb{Q}_p)$.
Chern characters are an important tool for the computation of these equivariant K-theory groups. We present a new and very geometric construction of an equivariant Chern character, taking values in a suitably defined Alexander-Spanier cohomology. It takes values in an equivariant version of Alexander-Spanier cohomology. As a main result we prove that this Chern character is an isomorphism after tensor product with the complex numbers.