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Journal of Lie Theory 23 (2013), No. 3, 747--778
Copyright Heldermann Verlag 2013

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

Zhanqiang Bai
Dept. of Mathematics, Hong Kong University, of Science and Technology, Clear Water Bay, Kowloon, Hong Kong


\def\c{{\frak c}} \def\o{{\frak o}} \def\C{{\Bbb C}} Let $\c\o(J)$ be the conformal algebra of a simple Euclidean Jordan algebra $J$. We show that a (non-trivial) unitary highest weight $\c\o(J)$-module has the smallest positive Gelfand-Kirillov dimension if and only if a certain quadratic relation is satisfied in the universal enveloping algebra $U(\c\o(J)_\C)$. In particular, we find an quadratic element in $U(\c\o(J)_\C)$. A prime ideal in $U(\c\o(J)_\C)$ equals the Joseph ideal if and only if it contains this quadratic element.

Keywords: Euclidean Jordan algebras, unitary highest weight module, quadratic relation, Joseph Ideal.

MSC: 22E47, 17B10, 17C99

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