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

p 进等变 K 理论和 K 同调的几何 Chern 特征

• 批准号: 441787895
• 负责人: Professor Dr. Thomas Schick
• 金额: --
• 依托单位: Mathematisches Institut (MI)
• 依托单位国家: 德国
• 项目类别: Priority Programmes
• 资助国家: 德国
• 项目状态: 未结题
• 来源: https://gepris.dfg.de/gepris/projekt/441787895?language=en
• 关键词: Geometric; Chern; characters; adic; equivariant

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.

在局部紧群及其表示的研究中，p-add群，或者更一般的完全不连通群，构成了一个重要的特例，是对已有研究的李群的补充。这种完全不连通群G的表示理论的重要方面是由它们的约化C*-代数的K-理论决定的。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-同调联系联系在一起。该项目的目标是给出真G-空间X的等变K-同调的几何刻画，其中G是完全不连通的局部紧群。我们的目标是建立一个基于空间的循环模型，该模型包含额外的指数理论信息。然后，我们计划以几何的方式构造一个与可计算的等变同调的Chen特征标同构；其次，作为该Chen特征标的一个构件，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。我们将利用这个模型以几何的方式构造一个等变K-理论的Chen特征标，并最终得到二变等变K-理论和一个双变等变陈氏特征标的几何构造。将与以前的非几何构造(特别是对于紧群和离散群)进行比较，这将为应用于p-add群的表示理论以及通过使用它们的非阿基米德完备性来更深入地理解(离散)算术群及其应有的行为铺平道路。