This thesis studies the problem that for an irreducible unitary representation, what kind of its K-types contributes to the Dirac cohomology. We introduce spin-norm and spin-lowest K-type, which offer the right framework to answer the problem. Based on our study of the spin-norm, reduction along a pencil, and by tracing certain bottom layer K-types, we verify that if G is on the following list: real G2, F II, E IV; complex G2, F4, E 6; and X is any irreducible unitary ( g , K) module which contains some unitarily small K-types, then only these K-types can contribute to the Dirac cohomology of X. These results also give partial support to Conjecture 7.13 of [Salamanca-Riba and Vogan, On the classification of unitary representations of reductive Lie groups, Ann. of Math. 148 (1998), 1067--1133]. Moreover, for G complex, we reveal the relation between HD( LS (Z)) and HD(Z ). This result reduces the classification of unitary representations with non-zero Dirac cohomology to the classification of the spherical ones with nonzero Dirac cohomology on the Levi level.
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