Abstract:Let $G$ be a non-compact connected semisimple real Lie group with finite center. Suppose $L$ is a non-compact connected closed subgroup of $G$ acting transitively on a symmetric space $G/H$ such that $L\cap H$ is compact. We study the action on $L/L\cap H$ of a Dirac operator $D_{G/H}(E)$ acting on sections of an $E$-twist of the spin bundle over $G/H$. As a byproduct, in the case of $(G,H,L)=(SL(2,{\mathbb R})\times SL(2,{\mathbb R}),\Delta (SL(2,{\mathbb R})\times SL(2,{\mathbb R})),SL(2,{\mathbb R})\times SO(2))$, we identify certain representations of $L$ which lie in the kernel of $D_{G/H}(E)$.

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中文翻译：

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扭曲狄拉克算子的表示理论嵌入

摘要：令$G$为一个非紧连通半单实李群，其中心有限。假设$L$ 是$G$ 的非紧连通闭子群，在对称空间$G/H$ 上传递作用，使得$L\cap H$ 是紧致的。我们研究了狄拉克算子 $D_{G/H}(E)$ 在 $L/L\cap H$ 上作用于 $G/H$ 上的自旋束的 $E$-twist 部分的作用。作为副产品，在 $(G,H,L)=(SL(2,{\mathbb R})\times SL(2,{\mathbb R}),\Delta (SL(2,{\ mathbb R})\times SL(2,{\mathbb R})),SL(2,{\mathbb R})\times SO(2))$，我们确定了 $L$ 的某些表示位于内核中$D_{G/H}(E)$。

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