Symmetry breaking operators for line bundles over real projective spaces

The space of smooth sections of an equivariant line bundle over the real projective space $\mathbb{R}{\mathrm{P}}^n$ forms a natural representation of the group $\mathrm{G}\mathrm{L}(n+1,\mathbb{R})$. We explicitly construct and classify all intertwining operators between such representations of ${\rm GL}(n+1,$\mathbb{R}$)$anditssubgroup${\rm GL}(n,$\mathbb{R}$)$, intertwining for the subgroup. Intertwining operators of this form are called symmetry breaking operators, and they describe the occurrence of a representation of $\mathrm{G}\mathrm{L}(n,\mathbb{R})$ inside the restriction of a representation of ${\rm GL}(n+1,$\mathbb{R}$)$. In this way, our results contribute to the study of branching problems for the real reductive pair $({\rm GL}(n+1,$\mathbb{R}$),{\rm GL}(n,$\mathbb{R}$))$. The analogous classification is carried out for intertwining operators between algebraic sections of line bundles, where the Lie group action of $\mathrm{G}\mathrm{L}(n,\mathbb{R})$ is replaced by the action of its Lie algebra $\mathfrak{g}\mathfrak{l}(n,\mathbb{R})$, and it turns out that all intertwining operators arise as restrictions of operators between smooth sections.