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.

If π is a (g, K)−module, then D induces an operatorwhere Spin is a spin module for C(s). If π is unitary, then π ⊗ Spin admits a K−invariant inner product , such that D is self adjoint with respect to this inner product. It follows that D 2 ≥ 0 on π ⊗ Spin. Using the above formula for D 2 , we find that Cas g + ρ g 2 ≤ ∆(Cas k ) + ρ k 2 on any K−type τ occurring in π ⊗ Spin. Another way of putting this is dirineq dirineqfor any τ occurring in π ⊗ Spin, where Λ is the infinitesimal character of π. This is the Dirac inequality mentioned above. These ideas are generalized by Vogan [V] and Huang-Pandžić [HP1] as follows. For an arbitrary admissible (g, K) module π, we define Dirac cohomology of π asThe main result about H D is the following theorem conjectured by Vogan. t:basicTheorem 1.2.[HP1] Assume that H D (π) is nonzero, and contains an irreducibleConversely, if π is unitary and τ = xΛ − ρ k is the highest weight of a K−type occuring in π ⊗ Spin, then this K−type is contained in H D (π).Note that for unitary π, the multiplicity of τ in H D (π) is the same as the multiplicity of τ in π ⊗ Spin.These results might suggest that difficulties should arise in passing between K−types of π and K−types of π ⊗ Spin. For unitary π, the situation is however greatly simplified by the Dirac inequality. Namely, together with (1.1), Theorem 1.2 shows that the infinitesimal characters τ + ρ k of K−types in Dirac cohomology have minimal possible norm. This means that whenever such E τ appears in the tensor product of a K−type E µ of π and a K−type E γ of Spin, it necessarily appears as the PRV component [PRV], i.e.,where µ − denotes the lowest weight of E µ .Assume now that g and k have equal rank, e.g. g 0 = sp(2n, R) or g 0 = u(p, q), and assume (as we may) that Λ is g-dominant. Then the above x must belong toThe condition that xΛ is regular and integral for K puts further restrictions on both x and Λ which will be made precise later. We will make use of the following decomposition of the K−module Spin: spindec spindec (1.4) Spin = σ∈W 1 E σρg −ρ k .

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In this paper we determine the Dirac cohomology of certain irreducible Harish-Chandra modules of a semisimple connected Lie group G with finite center: irreducible finite-dimensional modules and unitary Aq(λ) modules. We also comment on the relationship to (g, K)-cohomology.

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