Let co(J) be the conformal algebra of a simple Euclidean Jordan
algebra J. We show that a (non-trivial) unitary highest weight
co(J)-module has the smallest positive Gelfand-Kirillov dimension
if and only if a certain quadratic relation is satisfied in the universal
enveloping algebra U(co(J)Cβ). In particular, we find
an quadratic element in U(co(J)Cβ). A prime ideal in
U(co(J)Cβ) equals the Joseph ideal if and only if it
contains this quadratic element.Comment: 34pages, accepted by Journal of Lie Theor