Representation theoretic embedding of twisted Dirac operators

Abstract

Let G G be a non-compact connected semisimple real Lie group with finite center. Suppose L L is a non-compact connected closed subgroup of G G acting transitively on a symmetric space G / H G/H such that L ∩ H L\cap H is compact. We study the action on L / L ∩ H L/L\cap H of a Dirac operator D G / H ( E ) D_{G/H}(E) acting on sections of an E E -twist of the spin bundle over G / H G/H . As a byproduct, in the case of ( G , H , L ) = ( S L ( 2 , R ) × S L ( 2 , R ) , Δ ( S L ( 2 , R ) × S L ( 2 , R ) ) , S L ( 2 , R ) × S O ( 2 ) ) (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 L which lie in the kernel of D G / H ( E ) D_{G/H}(E) .