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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 $τ\in \widehat{SO(3)}$, let $E_τ$ be the homogeneous vector bundle over $\mathbb{R}^3$ associated with $τ$. An interesting problem consists in studying the set of bounded linear operators over the sections of $E_τ$ that are invariant under the action of $SO(3)\ltimes \mathbb{R}^3$. Such operators are in correspondence with the $End(V_τ)$-valued, bi-$τ$-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 $τ$-spherical functions. We first present a set of generators of the algebra of $SO(3)\ltimes \mathbb{R}^3$-invariant differential operators on $E_τ$. We also give an explicit form for the $τ$-spherical Fourier transform, we deduce an inversion formula and we use it to give a characterization of $End(V_τ)$-valued, bi-$τ$-equivariant, functions on $\mathbb{R}^3$.