约化$p$ -进群表示论的非交换几何方法：Hecke代数的同调，综述和一些新结果
A non-commutative geometry approach to the representation theory of reductive $p$-adic groups: Homology of Hecke algebras, a survey and some new results
摘要：
本文综述了关于约化$p$ -adic群$G$的{ \em全} Hecke代数的同调与$G$的表示论之间关系的一些已知结果。用$\CIc(G)$表示$G$的全Hecke代数，用$\Hp_*(\CIc(G))$表示其周期循环同调群。令$\hat G$表示$G$的容许对偶。本文的主要观点之一是，一方面，群$\Hp_*(\CIc(G))$与$\hat G$的拓扑直接相关，另一方面，群$\Hp_*(\CIc(G))$可以根据$G$（本质上是根据$G$的共轭类及其稳定子的上同调）显式地计算. $\Hp_*(\CIc(G))$和$\hat G$的拓扑之间的关系被建立为将$\Hp_*(A)$与$\Prim(A)$的拓扑联系起来的更一般原理的一部分，对于任何有限型代数$A$，$A$的本原理想谱。我们提供了几个新的例子，详细说明这一原则。在本文中，我们还证明了一些新的结果，主要是为了更好地解释和联系在一起的结果，在这里。例如，我们计算$\maO(X) \rtimes Γ$的Hochschild同调，$\maO(X) \rtimes Γ$是光滑复代数簇$X$上正则函数环与有限群$Γ$的交叉积。我们还概述了一个非常试探性的计划，使用这些结果来构建和分类的尖表示$G$。在文章的最后，我们还回顾了Hochschild和循环同调的定义。
英文摘要：
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 of the conjugacy classes of $G$ and the cohomology of their stabilizers). The relation between $\Hp_*(\CIc(G))$ and the topology of $\hat G$ is established as part of a more general principle relating $\Hp_*(A)$ to the topology of $\Prim(A)$, the primitive ideal spectrum of $A$, for any finite typee algebra $A$. We provide several new examples illustrating in detail this principle. We also prove in this paper a few new results, mostly in order to better explain and tie together the results that are presented here. For example, we compute the Hochschild homology of $\maO(X) \rtimes Γ$, the crossed product of the ring of regular functions on a smooth, complex algebraic variety $X$ by a finite group $Γ$. We also outline a very tentative program to use these results to construct and classify the cuspidal representations of $G$. At the end of the paper, we also recall the definitions of Hochschild and cyclic homology.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
