Construction and classification of differential symmetry breaking operators for principal series representations of the pair $(SO_0(4,1), SO_0(3,1))$ for special parameters

Abstract

We construct and give a complete classification of all the differential symmetry breaking operators $\mathbb{D}_{λ, ν}^{N,m}: C^\infty(S^3, \mathcal{V}_λ^{2N+1}) \rightarrow C^\infty(S^2, \mathcal{L}_{m, ν})$, between the spaces of smooth sections of a vector bundle of rank $2N+1$ over the $3$-sphere $\mathcal{V}_λ^{2N+1} \rightarrow S^3$, and a line bundle over the $2$-sphere $\mathcal{L}_{m, ν} \rightarrow S^2$ in the special case $|m| = N$.