Symmetry breaking operators for line bundles over real projective spaces

TL;DR

This paper explicitly constructs and classifies symmetry breaking operators for line bundle sections over real projective spaces, advancing the understanding of branching problems in the representation theory of real reductive groups.

Contribution

It provides a complete classification of intertwining operators between representations of ${ m GL}(n+1, eal)$ and ${ m GL}(n, eal)$, including both smooth and algebraic sections.

Findings

Classification of all symmetry breaking operators for line bundles over $ eal P^n$

Explicit construction of intertwining operators between these representations

All algebraic intertwining operators are restrictions of smooth operators

Abstract

The space of smooth sections of an equivariant line bundle over the real projective space $\mathbb{R}{\rm P}^n$ forms a natural representation of the group ${\rm GL}(n+1,\mathbb{R})$. We explicitly construct and classify all intertwining operators between such representations of ${\rm GL}(n+1,\mathbb{R})$ and its subgroup ${\rm GL}(n,\mathbb{R})$, intertwining for the subgroup. Intertwining operators of this form are called symmetry breaking operators, and they describe the occurrence of a representation of ${\rm GL}(n,\mathbb{R})$ 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 ${\rm GL}(n,\mathbb{R})$ 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.