  Given a Lie group $G$, a compact subgroup $K$ and a representation
$\tau\in\hat K$, we assume that the algebra of $\text{End}(V_\tau)$-valued,
bi-$\tau$-equivariant, integrable functions on $G$ is commutative. We present
the basic facts of the related spherical analysis, putting particular emphasis
on the r\^ole of the algebra of $G$-invariant differential operators on the
homogeneous bundle $E_\tau$ over $G/K$. In particular, we observe that, under
the above assumptions, $(G,K)$ is a Gelfand pair and show that the Gelfand
spectrum for the triple $(G,K,\tau)$ admits homeomorphic embeddings in $\mathbb
C^n$.
  In the second part, we develop in greater detail the spherical analysis for
$G=K\ltimes H$ with $H$ nilpotent. In particular, for $H=\mathbb R^n$ and
$K\subset SO(n)$ and for the Heisenberg group $H_n$ and $K\subset U(n)$, we
characterize the representations $\tau \in \hat K$ giving a commutative
algebra. \end{abstract}
