Geometric Chern characters for p-adic equivariant K-theory and K-homology

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In the study of locally compact groups and their representations, the p-adic groups, or more generally totally disconnected groups, form an important special case, complementary to the much studied Lie groups. Important aspects of the representation theory of such a totally disconnected group G are governed by the K-theory of their reduced C*-algebra. The Baum-Connes conjecture (known in many cases) identifies this with the G-equivariant K-homology of the universal proper G-space.The goal of the project is a geometric description of equivariant K-homology for a proper G-space X, where G is a totally disconnected locally compact group. We aim for a cycle model based on spaces generalizing Bruhat-Tits buildings, containing additional index theoretic information. We then plan to construct in a geometric way a Chern character isomorphism to a computable equivariant homology.Secondly and as one building block for this Chern character, we plan to develop a new and particularly convenient model for the classifying space of G-equivariant K-theory for such a totally disconnected group G. We will use this to construct in a geometric way a Chern character for equivariant K-theory, and ultimately, a geometric construction of bivariant equivariant K-theory and a bivariant equivariant Chern character. A comparison with previous, non-geometric constructions (in particular for compact and discrete groups) will be carried out.This opens the way for applications in representation theory of p-adic groups and a deeper K-theoretic understanding of (discrete) arithmetic groups and their proper actions via the use of their non-Archimedean completions.