Strongly Harmonic Forms for Representations in the Discrete Series

LetGbe a semisimple connected Lie group and letKbe a maximal compact subgroup. Assume that rankG=rank K, and letT⊂Kbe a Cartan subgroup ofG. The quotientG/Tcarries an indefiniteG-invariant hermitian form. The standard ∂ Dolbeault operator has a formal adjoint differential operator ∂*invwith respect to the invariant hermitian form. Letsdenote the complex dimension ofK/T. We form the indefinite harmonic space Hs(G/T, Lχ+2ρ)={(0, s)+Lχ+2ρ×valued forms in Ker ∂∩Ker ∂*inv}. In this paper we show that under some positivity conditions onχthe cohomology space Hs(G/T, Lχ) contains a copy of the representation in the discrete series ofGwith parameterχ.