Let G G be a connected semisimple Lie group with finite center. Let K K be the maximal compact subgroup of G G corresponding to a fixed Cartan involution θ \theta . We prove a conjecture of Vogan which says that if the Dirac cohomology of an irreducible unitary ( g , K ) (\mathfrak {g},K) -module X X contains a K K -type with highest weight γ \gamma , then X X has infinitesimal character γ + ρ c \gamma +\rho _{c} . Here ρ c \rho _{c} is the half sum of the compact positive roots. As an application of the main result we classify irreducible unitary ( g , K ) (\mathfrak {g},K) -modules X X with non-zero Dirac cohomology, provided X X has a strongly regular infinitesimal character. We also mention a generalization to the setting of Kostant’s cubic Dirac operator.

By calculating the symmetric subgroups Aut(u 0 ) θ and their involution classes, we classify the Klein four subgroups Γ of Aut(u 0 ) for each compact simple Lie algebra u 0 up to conjugation. This leads to a new approach of classification of semisimple symmetric pairs and Z 2 × Z 2 -symmetric spaces. We also determine the fixed point subgroup Aut(u 0 ) Γ .

No abstract