On Completions of Hecke Algebras

Abstract

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.