Symmetry Breaking Operators for Line Bundles over Real Projective Spaces

Jan Frahm¹ ; Clemens Weiske¹

¹Department Mathematik, FAU Erlangen-Nürnberg, Cauerstr. 11, 91058 Erlangen, Germany

Journal of Lie Theory, Tome 29 (2019) no. 2, pp. 511-558

Résumé

\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.

Détail

Classification : 22E46, 17B15, 46F12
Mots-clés : Symmetry breaking operators, real projective spaces, general linear group, intertwining operators, Harish-Chandra modules, principal series