Abstract
Abstract We study the behavior of Dirac cohomology under Howe’s $$\Theta $$ Θ -correspondence in the case of complex reductive dual pairs. More precisely, if $$(G_1,G_2)$$ ( G 1 , G 2 ) is a complex reductive dual pair with $$G_1$$ G 1 and $$G_2$$ G 2 viewed as real groups, we describe those Harish-Chandra modules $$\pi _1$$ π 1 of $$G_1$$ G 1 with nonzero Dirac cohomology whose $$\Theta $$ Θ -liftings $$\Theta (\pi _1)$$ Θ ( π 1 ) still have nonzero Dirac cohomology. In this case, we compute explicitly the Dirac cohomology of $$\Theta (\pi _1)$$ Θ ( π 1 ) .
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