AbstractsMathematics

This thesis mainly contains two parts. In the first part, let (G,K) be a Hermitian symmetric pair. We give a formula on the Gelfand-Kirillov dimension of unitary highest weight (g,K)-modules. By using this formula, we give a characterization for the highest weights of unitary highest weight (g,K)-modules which have the smallest positive Gelfand-Kirillov dimension (called minimal GK-dimension). In the second part, 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 minimal GK-dimension if and only if a certain quadratic relation (Q1) is satisfied in the universal enveloping algebra U(co(J)C). In particular, we find a quadratic element Q’1 in U(co(J)C). And the annihilator ideal of an irreducible highest weight co(J)-module equals the Joseph ideal if and only if it contains this quadratic element Q’1.