论文标题
旋转表示的张量产品的对称破坏差分运算符
Symmetry Breaking Differential Operators for Tensor Products of Spinorial Representations
论文作者
论文摘要
令$ \ mathbb s $为复杂的Clifford Algebra $ \ Mathbb {C} \ ell（\ Mathbb r^n）$，$ \ Mathbb s'$ dual，$ρ$和$ρ'$是Spin Group $ {\ RM Spin}（N）$的相应表示。组$ g = {\ rm spin}（1，n+1）$是$ \ mathbb r^n $的共形组的（双重）覆盖。对于$λ，μ\ in \ mathbb c $，令$π_{ρ，λ} $（supd。usp。$π_{ρ'，μ} $）是在$ c^of）上实现的$ g $的旋转表示形式， r^n，\ mathbb s'）$）。对于$ 0 \ leq k \ leq n $和$ m \ in \ mathbb n $，我们构建了一个对称性破坏差异操作员$ b_ {k;λ，μ}^{（m）} $来自$ c^\ infty（\ c^\ infty（\ c^\ infty） \ Mathbb {s}'）$ in $ c^\ infty（\ Mathbb r^n，λ^*_ k（\ Mathbb r^n）\ otimes \ otimes \ Mathbb {c}）$，这些$交织在一起，该$交织在一起表示$π_{ρ，λ} \λ} \ outime $π_{τ^*_ k，λ+μ+2m} $，其中$τ^*_ k $是$ {\ rm spin}（n）$的表示，$ {\ rm spin}（n）$的表示$ \ mathbb {r}^n $。
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 $\mathbb{R}^n$.
