A characterization of unitarity of some highest weight Harish-Chandra modules

Abstract Let L ⁢ ( λ ) {L(\lambda)} be a highest weight Harish-Chandra module with highest weight λ. When the associated variety of L ⁢ ( λ ) {L(\lambda)} is not maximal, that is, not equal to the nilradical of the corresponding parabolic subalgebra, we prove that the unitarity of L ⁢ ( λ ) {L(\lambda)} can be determined by a simple condition on the value of z = ( λ + ρ , β ∨ ) {z=(\lambda+\rho,\beta^{\vee})} , where ρ is half the sum of positive roots and β is the highest root. In the proof, certain distinguished antichains of positive noncompact roots play a key role. By using these antichains, we are also able to provide a uniform formula for the Gelfand–Kirillov dimension of all highest weight Harish-Chandra modules, generalizing our previous result for the case of unitary highest weight Harish-Chandra modules.