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This paper completes the construction of arbitrary order conformallyinvariant differential operators in higher spin spaces. Jan Slov\'{a}k hasclassified all conformally invariant differential operators on locallyconformally flat manifolds. We complete his results in higher spin theory bygiving explicit expressions for arbitrary order conformally invariantdifferential operators, where by conformally invariant we mean equivariant withrespect to the conformal group of $\mathbb{S}^m$ acting in Euclidean space$\mathbb{R}^m$. We name these the fermionic operators when the order is odd andbosonic operators when the order is even. Our approach explicitly usesconvolution type operators to construct conformally invariant differentialoperators. These convolution type operators are examples of Knapp-Steinoperators and they can be considered as the inverses of the correspondingdifferential operators. Intertwining operators of these convolution typeoperators are provided and intertwining operators of differential operatorsfollow immediately. This reveals that our convolution type operators anddifferential operators are all conformally invariant. This also gives us aclass of conformally invariant convolution type operators in higher spinspaces. Their inverses, when they exist, are conformally invariantpseudo-differential operators. Further we use Stein Weiss gradient operatorsand representation theory for the Spin group to naturally motivate theRarita-Schwinger operators.

Construction of Arbitrary Order Conformally Invariant Operators in Higher Spin Spaces