Journal of Lie Theory 29 (2019), No. 2, 511--558
Copyright Heldermann Verlag 2019
Symmetry Breaking Operators for Line Bundles over Real Projective Spaces
Jan Frahm
Department Mathematik, FAU Erlangen-Nürnberg, Cauerstr. 11, 91058 Erlangen, Germany
frahm@math.fau.de
Clemens Weiske
Department Mathematik, FAU Erlangen-Nürnberg, Cauerstr. 11, 91058 Erlangen, Germany
weiske@math.fau.de
[Abstract-pdf]
\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, Harish-Chandra modules, principal series.
MSC: 22E46; 17B15, 46F12
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