Symmetry Breaking Differential Operators for Tensor Products of Spinorial Representations

Abstract

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