Symmetry breaking differential operators for tensor products of spinorial representations.

Abstract

Let $\mathbb S$ be a Clifford module for the complexified Clifford algebra $C\ell(\mathbb R^n)$, $\mathbb S'$ its dual, $\rho$ and $\rho'$ be the corresponding representations of the spin group $Spin(\mathbb R^n)$. The group $G=Spin(\mathbb R^{1,n+1})$ is the (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 $G$ on $ \mathbb S$-valued $\lambda$-densities (resp. $\mathbb S'$-valued $\mu$-densities) on $\mathbb R^n$. 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 $C^\infty(\mathbb R^n, \Lambda^*_k(\mathbb R^n))$ 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 $Spin(\mathbb R^n)$ on $\Lambda^*_k(\mathbb R^n)$.