
Journal of Lie Theory 29 (2019), No. 2, 511558 Copyright Heldermann Verlag 2019 Symmetry Breaking Operators for Line Bundles over Real Projective Spaces Jan Frahm Department Mathematik, FAU ErlangenNürnberg, Cauerstr. 11, 91058 Erlangen, Germany frahm@math.fau.de Clemens Weiske Department Mathematik, FAU ErlangenNürnberg, Cauerstr. 11, 91058 Erlangen, Germany weiske@math.fau.de [Abstractpdf] \newcommand{\GL}{\rm GL} \newcommand{\R}{\mathbb{R}} \newcommand{\RP}{\mathbb{R}{\rm P}} The space of smooth sections of an equivariant line bundle over the real projective space $\RP^n$ forms a natural representation of the group $\GL(n+1,\R)$. We explicitly construct and classify all intertwining operators between such representations of $\GL(n+1,\R)$ and its subgroup $\GL(n,\R)$, intertwining for the subgroup. Intertwining operators of this form are called symmetry breaking operators, and they describe the occurrence of a representation of $\GL(n,\R)$ inside the restriction of a representation of $\GL(n+1, \R)$. In this way, our results contribute to the study of branching problems for the real reductive pair $(\GL(n+1,\mathbb{R}),\GL(n,\mathbb{R}))$.\par The analogous classification is carried out for intertwining operators between algebraic sections of line bundles, where the Lie group action of $\GL(n,\mathbb{R})$ is replaced by the action of its Lie algebra $\mathfrak{gl}(n,\mathbb{R})$, and it turns out that all intertwining operators arise as restrictions of operators between smooth sections. Keywords: Symmetry breaking operators, real projective spaces, general linear group, intertwining operators, HarishChandra modules, principal series. MSC: 22E46; 17B15, 46F12 [ Fulltextpdf (326 KB)] for subscribers only. 