Chain-center duality for locally compact groups

TL;DR

This paper investigates the chain-center duality property in locally compact groups, establishing conditions under which this duality holds, thus generalizing previous results for compact groups.

Contribution

It proves that various classes of locally compact groups satisfy chain-center duality, extending M"{u}ger's result from compact groups to broader categories.

Findings

Chain-center duality holds for compact-by-abelian extensions.

Connected nilpotent groups satisfy chain-center duality.

Countable discrete icc and connected semisimple groups also satisfy the duality.

Abstract

The chain group $C(G)$ of a locally compact group $G$ has one generator $g_{\rho}$ for each irreducible unitary $G$-representation $\rho$, a relation $g_{\rho}=g_{\rho'}g_{\rho"}$ whenever $\rho$ is weakly contained in $\rho'\otimes \rho"$, and $g_{\rho^*}=g_{\rho}^{-1}$ for the representation $\rho^*$ contragredient to $\rho$. $G$ satisfies chain-center duality if assigning to each $g_{\rho}$ the central character of $\rho$ is an isomorphism of $C(G)$ onto the dual $\widehat{Z(G)}$ of the center of $G$. We prove that $G$ satisfies chain-center duality if it is (a) a compact-by-abelian extension, (b) connected nilpotent, (c) countable discrete icc or (d) connected semisimple; this generalizes M. M\"{u}ger's result compact groups satisfy chain-center duality.