On completions of Hecke algebras

TL;DR

This paper proves that Morita equivalences between Hecke algebras of p-adic groups extend to their Schwartz and C*-completions, enabling computation of the topological K-theory of these groups' reduced C*-algebras.

Contribution

It establishes conditions under which Morita equivalences extend to topological completions, facilitating K-theory calculations for p-adic groups.

Findings

Morita equivalence extends to Schwartz completions.

Morita equivalence extends to C*-completions.

Conditions are verified in known cases.

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.