Differential symmetry breaking operators from a line bundle to a vector bundle over real projective spaces

TL;DR

This paper classifies and constructs differential symmetry breaking operators between line and vector bundles over real projective spaces, analyzing their factorization, branching laws, and associated representations.

Contribution

It provides a comprehensive classification and explicit construction of symmetry breaking operators, along with their factorization identities and branching laws for generalized Verma modules.

Findings

Explicit classification of symmetry breaking operators

Factorization identities for these operators

Branching laws for associated representations

Abstract

In this paper we classify and construct differential symmetry breaking operators $\mathbb{D}$ from a line bundle over the real projective space $\mathbb{R}\mathbb{P}^n$ to a vector bundle over $\mathbb{R}\mathbb{P}^{n-1}$. We further determine the factorization identities of $\mathbb{D}$ and the branching laws of the corresponding generalized Verma modules of $\mathfrak{sl}(n+1,\mathbb{C})$. By utilizing the factorization identities, the $SL(n,\mathbb{R})$-representations realized on the image $\text{Im}(\mathbb{D})$ are also investigated.