Representation theoretic embedding of twisted Dirac operators

Let G G be a non-compact connected semisimple real Lie group with finite center. Suppose L"> L encoding="application/x-tex">L is closed subgroup of acting transitively on symmetric space G slash upper H"> / H encoding="application/x-tex">G/H such that L intersection ∩ encoding="application/x-tex">L\cap H compact. We study the action encoding="application/x-tex">L/L\cap Dirac operator D Subscript H Baseline left-parenthesis E right-parenthesis"> D ( E stretchy="false">) encoding="application/x-tex">D_{G/H}(E) sections an E"> encoding="application/x-tex">E -twist spin bundle over . As byproduct, in case alttext="left-parenthesis comma right-parenthesis equals S 2 double-struck R times normal Delta O , = S 2 R × O encoding="application/x-tex">(G,H,L)=(SL(2,{\mathbb R})\times SL(2,{\mathbb R}),\Delta (SL(2,{\mathbb R})),SL(2,{\mathbb SO(2)) , we identify certain representations which lie kernel