Representation Theoretic Embedding Of Twisted Dirac Operators

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 boolean AND H is compact. We study the action on L/L boolean AND 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, R) x SL(2, R), Delta(SL(2, R) x SL(2, R)), SL(2, R) x SO(2)), we identify certain representations of L which lie in the kernel of D-G/H (E).