Representation theoretic embedding of twisted Dirac operators

TL;DR

This paper investigates the representation-theoretic properties of twisted Dirac operators on symmetric spaces associated with non-compact semisimple Lie groups, identifying specific kernel representations in a particular example.

Contribution

It introduces a new framework for analyzing twisted Dirac operators on symmetric spaces and identifies kernel representations in a specific Lie group setting.

Findings

Identification of kernel representations for a specific Lie group triple.

Development of a representation-theoretic approach to twisted Dirac operators.

Insights into the structure of Dirac operator kernels in symmetric space contexts.

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