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

Abstract Let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>L</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>λ</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {L(\lambda)} be a highest weight Harish-Chandra module with highest weight λ. When the associated variety of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>L</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>λ</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {L(\lambda)} is not maximal, that is, not equal to the nilradical of the corresponding parabolic subalgebra, we prove that the unitarity of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>L</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>λ</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {L(\lambda)} can be determined by a simple condition on the value of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>z</m:mi> <m:mo>=</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>λ</m:mi> <m:mo>+</m:mo> <m:mi>ρ</m:mi> </m:mrow> <m:mo>,</m:mo> <m:msup> <m:mi>β</m:mi> <m:mo>∨</m:mo> </m:msup> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {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.