Generalised Hecke algebras and C^*-completions

Abstract

For a Hecke pair $(G, H)$ and a finite-dimensional representation $\sigma$ of $H$ on $V_\sigma$ with finite range we consider a generalised Hecke algebra $\H_\sigma(G, H)$, which we study by embedding the given Hecke pair in a Schlichting completion $(G_\sigma, H_\sigma)$ that comes equipped with a continuous extension $\sigma$ of $H_\sigma$. There is a (non-full) projection $p_\sigma\in C_c(G_\sigma, {\cc B}(V_\sigma))$ such that $\H_\sigma(G, H)$ is isomorphic to $p_\sigma C_c(G_\sigma, {\cc B}(V_\sigma))p_\sigma$. 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 $C^*(G_\sigma, {\cc B}(V_\sigma))$. By letting $\sigma$ vary, we can compare these ideals. The main focus is on the case with $\dim\sigma=1$ and applications include $ax+b$-groups and the Heisenberg group.