Representation theoretic embedding of twisted Dirac operators

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 $D_{G/H}(E)$.