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.

The author was supported by an NWO Vidi grant “A Hecke algebra approach to the local Langlands correspondence” (nr. 639.032.528).