Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 17 (2021), 049, 23 pages      arXiv:2012.09625      https://doi.org/10.3842/SIGMA.2021.049

Symmetry Breaking Differential Operators for Tensor Products of Spinorial Representations

Jean-Louis Clerc and Khalid Koufany
Université de Lorraine, CNRS, IECL, F-54000 Nancy, France

Received January 12, 2021, in final form May 06, 2021; Published online May 13, 2021

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 $\mathbb{R}^n$.

Key words: Clifford algebra; spinors; tensor product; conformal analysis; symmetry breaking differential operators.

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