Mackey Imprimitivity and commuting tuples of homogeneous normal operators

In this semi-expository article, we investigate the relationship between the imprimitivity introduced by Mackey several decades ago and commuting $d$- tuples of homogeneous normal operators. The Hahn-Hellinger theorem gives a canonical decomposition of a $*$- algebra representation $\rho$ of $C_0(\mathbb{S})$ (where $\mathbb S$ is a locally compact Hausdorff space) into a direct sum. If there is a group $G$ acting transitively on $\mathbb{S}$ and is adapted to the $*$- representation $\rho$ via a unitary representation $U$ of the group $G$, in other words, if there is an imprimitivity, then the Hahn-Hellinger decomposition reduces to just one component, and the group representation $U$ becomes an induced representation, which is Mackey's imprimitivity theorem. We consider the case where a compact topological space $S\subset \mathbb {C}^d$ decomposes into finitely many $G$- orbits. In such cases, the imprimitivity based on $S$ admits a decomposition as a direct sum of imprimitivities based on these orbits. This decomposition leads to a correspondence with homogeneous normal tuples whose joint spectrum is precisely the closure of $G$- orbits.