Periodic cyclic homology of Hecke algebras and their Schwartz completions

Periodic cyclic homology of certain nuclear algebras

Relying of properties of the inductive tensor product, we construct cyclic type homology theories for certain nuclear algebras. In this context we establish continuity theorems. We compute the periodic cyclic homology of the Schwartz algebra of p-adic GL(n) in terms of compactly supported de Rham cohomology of the tempered dual of GL(n).

A non-commutative geometry approach to the representation theory of reductive $p$-adic groups: Homology of Hecke algebras, a survey and some new results

We survey some of the known results on the relation between the homology of the {\em full} Hecke algebra of a reductive $p$-adic group $G$, and the representation theory of $G$. Let us denote by $\CIc(G)$ the full Hecke algebra of $G$ and by $\Hp_*(\CIc(G))$ its periodic cyclic homology groups. Let $\hat G$ denote the admissible dual of $G$. One of the main points of this paper is that the groups $\Hp_*(\CIc(G))$ are, on the one hand, directly related to the topology of $\hat G$ and, on the other hand, the groups $\Hp_*(\CIc(G))$ are explicitly computable in terms of $G$ (essentially, in terms…

Chern character for the Schwartz algebra of p-adic GL(n)

We construct a Chern character map from the K-theory of the reduced C^* algebra of the p-adic GL(n) with values in the periodic cyclic homology of the Schwartz algebra of this group. We prove that this map is an isomorphism after tensoring with C by comparing an explicit formula, stated in the algebraic case by Cuntz and Quillen, with the classical Chern character. This Chern character is a crucial ingredient in the proof of the Baum-Connes conjecture for the p-adic GL(n) due to Baum, Higson and Plymen.

Hecke operators in KK-theory and the K-homology of Bianchi groups

Let $Γ$ be a torsion-free arithmetic group acting on its associated global symmetric space $X$. Assume that $X$ is of non-compact type and let $Γ$ act on the geodesic boundary $\partial X$ of $X$. Via general constructions in KK-theory, we endow the K-groups of the arithmetic manifold $X/Γ$, of the reduced group C*-algebra of $Γ$ and of the boundary crossed product algebra associated to the action of $Γ$ on $\partial X$, with Hecke operators. The K-theory and K-homology groups of these C*-algebras are related by a Gysin six-term exact sequence. In the case when $Γ$ is a group of real hyperboli…

Some Fréchet algebras for which the Chern character is an isomorphism

Using similarities between topological $K$-theory and periodic cyclic homology we show that, after tensoring with $\mathbb C$, for certain Fréchet algebras the Chern character provides an isomorphism between these functors. This is applied to prove that the Hecke algebra and the Schwartz algebra of a reductive $p$-adic group have isomorphic periodic cyclic homology. The main theorem in the first version was incorrect for algebras related to noncompact manifolds. This has no effect on the results concerning p-adic groups. In the appendix we show that an analogous cohomological result does hold …

Periodic cyclic homology of reductive p-adic groups

Let G be a reductive p-adic group, H(G) its Hecke algebra and S(G) its Schwartz algebra. We will show that these algebras have the same periodic cyclic homology. This might be used to provide an alternative proof of the Baum-Connes conjecture for G, modulo torsion. As preparation for our main theorem we prove two results that have independent interest. Firstly a general comparison theorem for the periodic cyclic homology of finite type algebras and certain Fréchet completions thereof. Secondly a refined form of the Langlands classification for G, which clarifies the relation between the smooth…

Restriction to compact subgroups in the cyclic homology of reductive p-adic groups

Restriction of functions from a reductive p-adic group G to its compact subgroups defines an operator on the Hochschild and cyclic homology of the Hecke algebra of G. We study the commutation relations between this operator and others coming from representation theory: Jacquet functors, idempotents in the Bernstein centre, and characters of admissible representations.

约化 $p$-进群的 Hochschild 同调

Hochschild homology of reductive $p$-adic groups

本文研究约化 $p$-进群的 Hecke 与 Harish-Chandra–Schwartz 代数，利用光滑表示族和 Bernstein 分解证明其 Hochschild、循环同调与交叉积代数对应，并联系 K-理论。