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

在局部紧群及其表示的研究中，p进群，或者更一般的全不连通群，形成了一个重要的特例，补充了大量研究的李群。这样一个完全不连通群G的表示论的重要方面是由它们的约化C*-代数的K-理论所支配的。Baum-Connes猜想（在许多情况下已知）将其与泛真G-空间的G-等变K-同调等同起来。该项目的目标是对真G-空间X的等变K-同调进行几何描述，其中G是完全不连通的局部紧群。我们的目标是一个周期模型的基础上推广Bruhat-Tits建筑物的空间，包含额外的指标理论信息。 然后，我们计划用几何方法构造一个可计算等变同调的Chern特征标同构。其次，作为这个Chern特征标的一个构造块，我们计划为这样一个完全不连通群G的G-等变K-理论的分类空间建立一个新的特别方便的模型。我们将利用这一点以几何的方式构造等变K-理论的一个Chern特征标，并最终构造双变等变K-理论和双变等变Chern特征标。与以前的非几何结构（特别是紧致和离散群）进行比较，这为p-adic群的表示论中的应用开辟了道路，并通过使用它们的非阿基米德完备化对（离散）算术群及其适当作用有了更深的K理论理解。

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.