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We prove a simple formula that calculates the associated variety of a highest weight Harish-Chandra module directly from its highest weight. We also give a formula for the Gelfand--Kirillov dimension of highest weight Harish-Chandra module which is uniform across Cartan types and is valid for arbitrary infinitesimal character., Comment: 34 pages

It is shown by Barchini, Kable, and Zierau that conformally invariant systems of differential operators yield explicit homomorphisms between certain generalized Verma modules. In this paper we determine whether or not the homomorphisms arising from such systems of first and second order differential operators associated to maximal parabolic subalgebras of quasi-Heisenberg type are standard.