On Completions of Hecke Algebras
| Autor: | Solleveld, M., Aubert, A.-M., Mishra, M., Roche, A., Spallone, S. |
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| Přispěvatelé: | Aubert, A.-M., Mishra, M., Roche, A., Spallone, S. |
| Rok vydání: | 2019 |
| Předmět: | |
| Zdroj: | Aubert, A.-M.; Mishra, M.; Roche, A. (ed.), Representations of Reductive p-adic Groups: International Conference, IISER, Pune, India, 2017, pp. 207-262 Progress in Mathematics ISBN: 9789811366277 Aubert, A.-M.; Mishra, M.; Roche, A. (ed.), Representations of Reductive p-adic Groups: International Conference, IISER, Pune, India, 2017, 207-262. Singapore : Springer Singapore STARTPAGE=207;ENDPAGE=262;TITLE=Aubert, A.-M.; Mishra, M.; Roche, A. (ed.), Representations of Reductive p-adic Groups: International Conference, IISER, Pune, India, 2017 |
| Popis: | Let G be a reductive p-adic group and let \(\mathcal H (G)^{\mathfrak s}\) be a Bernstein block of the Hecke algebra of G. We consider two important topological completions of \(\mathcal H (G)^{\mathfrak s}\): a direct summand \(\mathcal S (G)^{\mathfrak s}\) of the Harish-Chandra–Schwartz algebra of G and a two-sided ideal \(C_r^* (G)^{\mathfrak s}\) of the reduced \(C^*\)-algebra of G. These are useful for the study of all tempered smooth G-representations. We suppose that \(\mathcal H (G)^{\mathfrak s}\) is Morita equivalent to an affine Hecke algebra \(\mathcal H (\mathcal R,q)\) – as is known in many cases. The latter algebra also has a Schwartz completion \(\mathcal S (\mathcal R,q)\) and a \(C^*\)-completion \(C_r^* (\mathcal R,q)\), both defined in terms of the underlying root datum \(\mathcal R\) and the parameters q. We prove that, under some mild conditions, a Morita equivalence \(\mathcal H (G)^{\mathfrak s}\sim _M \mathcal H (\mathcal R,q)\) extends to Morita equivalences \(\mathcal S (G)^{\mathfrak s}\sim _M \mathcal S (\mathcal R,q)\) and \(C_r^* (G)^{\mathfrak s}\sim _M C_r^* (\mathcal R,q)\). 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. |
| Databáze: | OpenAIRE |
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