Symmetry Breaking Differential Operators for Tensor Products of Spinorial Representations

TL;DR

This paper constructs symmetry breaking differential operators for tensor products of spinorial representations, revealing new intertwining operators that connect these representations with differential forms under conformal group actions.

Contribution

The paper introduces novel symmetry breaking differential operators for tensor products of spinorial representations, expanding the understanding of conformal symmetry and representation theory.

Findings

Explicit construction of symmetry breaking operators

Operators intertwine spinorial and form representations

Enhances understanding of conformal symmetry in spinor contexts

Abstract

Let $\mathbb S$ be a Clifford module for the complexified Clifford algebra $\mathbb{C}\ell(\mathbb R^n)$, $\mathbb S'$ its dual, $\rho$ and $\rho'$ be the corresponding representations of the spin group ${\rm Spin}(n)$. The group $G= {\rm Spin}(1,n+1)$ is a (twofold) covering of the conformal group of $\mathbb R^n$. For $\lambda, \mu\in \mathbb C$, let $\pi_{\rho, \lambda}$ (resp. $\pi_{\rho',\mu}$) be the spinorial representation of $G$ realized on a (subspace of) $C^\infty(\mathbb R^n,\mathbb S)$ (resp. $C^\infty(\mathbb R^n,\mathbb S')$). For $0\leq k\leq n$ and $m\in \mathbb N$, we construct a symmetry breaking differential operator $B_{k;\lambda,\mu}^{(m)}$ from $C^\infty(\mathbb R^n \times \mathbb R^n,\mathbb{S}\,\otimes\, \mathbb{S}')$ into $C^\infty(\mathbb R^n, \Lambda^*_k(\mathbb R^n) \otimes \mathbb{C})$ which intertwines the representations $\pi_{\rho, \lambda}\otimes \pi_{\rho',\mu} $ and $\pi_{\tau^*_k,\lambda+\mu+2m}$, where $\tau^*_k$ is the representation of ${\rm Spin}(n)$ on the space $\Lambda^*_k(\mathbb R^n) \otimes \mathbb{C}$ of complex-valued alternating $k$-forms on $\mathbb{R}^n$.