  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.
