Hochschild homology of reductive -adic groups
Maarten Solleveld
Radboud Universiteit Nijmegen, the Netherlands
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Abstract
Consider a reductive -adic group , its (complex-valued) Hecke algebra , and the Harish-Chandra–Schwartz algebra . We compute the Hochschild homology groups of and of , and we describe the outcomes in several ways. Our main tools are algebraic families of smooth -representations. With those we construct maps from and to modules of differential -forms on affine varieties. For , this provides a description of the cocentres of these algebras in terms of nice linear functions on the Grothendieck group of finite length (tempered) -representations. It is known from [J. Algebra 606 (2022), 371–470] that every Bernstein ideal of is closely related to a crossed product algebra of the form . Here denotes the regular functions on the variety of unramified characters of a Levi subgroup of , and is a finite group acting on . We make this relation even stronger by establishing an isomorphism between and , although we have to say that in some cases it is necessary to twist by a 2-cocycle. Similarly, we prove that the Hochschild homology of the two-sided ideal of is isomorphic to , where denotes the Lie group of unitary unramified characters of . In these pictures of and , we also show how the Bernstein centre of acts. Finally, we derive similar expressions for the (periodic) cyclic homology groups of and of and we relate that to topological K-theory.
Cite this article
Maarten Solleveld, Hochschild homology of reductive -adic groups. J. Noncommut. Geom. (2024), published online first
DOI 10.4171/JNCG/556