Generalised Hecke algebras and C^*-completions

Abstract

For a Hecke pair $(G, H)$ and a finite-dimensional representation $σ$ of $H$ on $V_σ$ with finite range we consider a generalised Hecke algebra $\H_σ(G, H)$, which we study by embedding the given Hecke pair in a Schlichting completion $(G_σ, H_σ)$ that comes equipped with a continuous extension $σ$ of $H_σ$. There is a (non-full) projection $p_σ\in C_c(G_σ, {\cc B}(V_σ))$ such that $\H_σ(G, H)$ is isomorphic to $p_σC_c(G_σ, {\cc B}(V_σ))p_σ$. We study the structure and properties of $C^*$-completions of the generalised Hecke algebra arising from this corner realisation, and via Morita-Fell-Rieffel equivalence we identify, in some cases explicitly, the resulting proper ideals of $C^*(G_σ, {\cc B}(V_σ))$. By letting $σ$ vary, we can compare these ideals. The main focus is on the case with $\dimσ=1$ and applications include $ax+b$-groups and the Heisenberg group.