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

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 $|m| = N$.

💡 Research Summary

The paper addresses the construction and complete classification of differential symmetry breaking operators (DSBOs) for principal series representations of the real reductive pair ((G,G’)=(SO_0(4,1),SO_0(3,1))). The authors focus on the concrete geometric setting where the representation spaces are realized as smooth sections of a rank‑(2N+1) vector bundle (\mathcal V^{2N+1}\lambda\to S^3) and a line bundle (\mathcal L{m,\nu}\to S^2). The main goal is to determine, for given complex parameters (\lambda,\nu), a natural number (N) and an integer (m), the space \