论文标题
hochschild还原$ p $ - 亚种集团
Hochschild homology of reductive $p$-adic groups
论文作者
论文摘要
考虑一个还原的$ p $ - 亚种$ g $，其（复杂值）Hecke代数$ h（g）$和Harish-Chandra--Schwartz代数$ S（G）$。我们计算$ h（g）$和$ s（g）$的Hochschild同源性小组，我们以多种方式描述了结果。 我们的主要工具是代数为$ g $的代数家庭。我们将构建从$ hh_n（h（g））$和$ hh_n（s（g））$构建的地图到仿射品种的微分$ n $ forms的模块。对于$ n = 0 $，这提供了对这些代数的cocentres的描述，该代数在有限长度的Grothendieck组上的好线性功能（pervered）$ g $ - 表述。 从较早的工作中知道，每个伯恩斯坦理想的$ h（g）$ h（g）$ $与$ o（t）\ rtimes w $的交叉产品代数密切相关。这里$ o（t）$表示Levi子组$ l $ $ g $的多样性$ t $ t $的常规功能，而$ w $是在$ t $上作用的有限群体。我们通过在$ hh_*（h（g）^s）$和$ hh_*（o（t）\ rtimes w）$之间建立同构的同构来使这种关系变得更加牢固，尽管我们必须说，在某些情况下，有必要由2个cocycle扭动$ c [w] $。 同样，我们证明了$ s（g）$的双面理想$ s（g）^s $的Hochschild同源性是$ HH_**（c^\ infty（t_u）\ rtimes w）$的同构，其中$ t_u $表示$ l $的单位不合格字符。在这些$ hh_*（h（g））$和$ hh_*（s（g））$的图片中，我们还展示了$ h（g）$ ACTS的伯恩斯坦中心。 最后，我们为$ h（g）$和$ s（g）$的（周期性）周期性同源组提供了类似的表达方式，我们将其与拓扑K理论联系起来。
Consider a reductive $p$-adic group $G$, its (complex-valued) Hecke algebra $H(G)$ and the Harish-Chandra--Schwartz algebra $S(G)$. We compute the Hochschild homology groups of $H(G)$ and of $S(G)$, and we describe the outcomes in several ways. Our main tools are algebraic families of smooth $G$-representations. With those we construct maps from $HH_n (H(G))$ and $HH_n (S(G))$ to modules of differential $n$-forms on affine varieties. For $n = 0$ this provides a description of the cocentres of these algebras in terms of nice linear functions on the Grothendieck group of finite length (tempered) $G$-representations. It is known from earlier work that every Bernstein ideal $H(G)^s$ of $H(G)$ is closely related to a crossed product algebra of the from $O(T) \rtimes W$. Here $O(T)$ denotes the regular functions on the variety $T$ of unramified characters of a Levi subgroup $L$ of $G$, and $W$ is a finite group acting on $T$. We make this relation even stronger by establishing an isomorphism between $HH_* (H(G)^s)$ and $HH_* (O(T) \rtimes W)$, although we have to say that in some cases it is necessary to twist $C[W]$ by a 2-cocycle. Similarly we prove that the Hochschild homology of the two-sided ideal $S(G)^s$ of $S(G)$ is isomorphic to $HH_* (C^\infty (T_u) \rtimes W)$, where $T_u$ denotes the Lie group of unitary unramified characters of $L$. In these pictures of $HH_* (H(G))$ and $HH_* (S(G))$ we also show how the Bernstein centre of $H(G)$ acts. Finally, we derive similar expressions for the (periodic) cyclic homology groups of $H(G)$ and of $S(G)$ and we relate that to topological K-theory.
