A Characterization of the Unitary Highest Weight Modules by Euclidean Jordan Algebras

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(ca(J)(c)). In particular, we find an quadratic element in U(co(J)(C)). A prime ideal in U(co(J)(C)) equals the Joseph ideal if and only if it contains this quadratic element.