Hochschild homology of reductive $p$-adic groups
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2022
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| author | Solleveld, Maarten |
| author_facet | Solleveld, Maarten |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2203_06435 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Hochschild homology of reductive $p$-adic groups Solleveld, Maarten Representation Theory K-Theory and Homology 16E40, 20G25, 22E50, 20C08 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. |
| title | Hochschild homology of reductive $p$-adic groups |
| topic | Representation Theory K-Theory and Homology 16E40, 20G25, 22E50, 20C08 |
| url | https://arxiv.org/abs/2203.06435 |