On completions of Hecke algebras
Abstract
Let G be a reductive p-adic group and let H(G)^s be a Bernstein block of the Hecke algebra of G. We consider two important topological completions of H(G)^s: a direct summand S(G)^s of the Harish-Chandra--Schwartz algebra of G and a two-sided ideal C*_r (G)^s of the reduced C*-algebra of G. These are useful for the study of all tempered smooth G-representations. We suppose that H(G)^s is Morita equivalent to an affine Hecke algebra H(R,q) -- as is known in many cases. The latter algebra also has a Schwartz completion S(R,q) and a C*-completion C*_r (R,q), both defined in terms of the underlying root datum R and the parameters q. We prove that, under some mild conditions, a Morita equivalence between H(G)^s and H(R,q) extends to Morita equivalences between S(G)^s and S(R,q), and between C*_r (G)^s and C*_r (R,q). We also check that our conditions are fulfilled in all known cases of such Morita equivalences between Hecke algebras. This is applied to compute the topological K-theory of the reduced C*-algebra of a classical p-adic group.
Keywords
Cite
@article{arxiv.1612.01776,
title = {On completions of Hecke algebras},
author = {Maarten Solleveld},
journal= {arXiv preprint arXiv:1612.01776},
year = {2020}
}
Comments
Second, abridged version. The argument for the preservation of temperedness in the first version was incomplete. To repair this, the new paragraphs 1.2, 2.2 and 3.2 were written and arguments in the proof of Theorem 3.4 were worked out further