Symmetry Breaking Differential Operators for Tensor Products of Spinorial Representations
英文摘要：
Let $\mathbb S$ be a Clifford module for the complexified Clifford algebra $\mathbb{C}\ell(\mathbb R^n)$, $\mathbb S'$ its dual, $ρ$ and $ρ'$ 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 $λ, μ\in \mathbb C$, let $π_{ρ, λ}$ (resp. $π_{ρ',μ}$) 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 $B_{k;λ,μ}^{(m)}$ from $C^\infty(\mathbb R^n \times \mathbb R^n,\mathbb{S}\,\otimes\, \mathbb{S}')$ into $C^\infty(\mathbb R^n, Λ^*_k(\mathbb R^n) \otimes \mathbb{C})$ which intertwines the representations $π_{ρ, λ}\otimes π_{ρ',μ} $ and $π_{τ^*_k,λ+μ+2m}$, where $τ^*_k$ is the representation of ${\rm Spin}(n)$ on the space $Λ^*_k(\mathbb R^n) \otimes \mathbb{C}$ of complex-valued alternating $k$-forms on $\mathbb{R}^n$.
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