Spherical analysis on homogeneous vector bundles of the 3-dimensional euclidean motion group

We consider $\mathbb{R}^3$ as a homogeneous manifold for the action of the motion group given by rotations and translations. For an arbitrary $\tau\in \widehat{SO(3)}$, let $E_\tau$ be the homogeneous vector bundle over $\mathbb{R}^3$ associated with $\tau$. An interesting problem consists in studying the set of bounded linear operators over the sections of $E_\tau$ that are invariant under the action of $SO(3)\ltimes \mathbb{R}^3$. Such operators are in correspondence with the $End(V_\tau)$-valued, bi-$\tau$-equivariant, integrable functions on $\mathbb{R}^3$ and they form a commutative algebra with the convolution product. We develop the spherical analysis on that algebra, explicitly computing the $\tau$-spherical functions. We first present a set of generators of the algebra of $SO(3)\ltimes \mathbb{R}^3$-invariant differential operators on $E_\tau$. We also give an explicit form for the $\tau$-spherical Fourier transform, we deduce an inversion formula and we use it to give a characterization of $End(V_\tau)$-valued, bi-$\tau$-equivariant, functions on $\mathbb{R}^3$.