A characterization of the unitary highest weight modules by Euclidean Jordan algebras (Q2856385)

The aim of the article under review is to generalize results of Meng on unitary highest weight modules of the conformal algebras of simple (finite dimensional) Euclidean Jordan algebras. To describe the results more precisely, first recall that, in [J. Lie Theory 18, No. 3, 697--715 (2008; Zbl 1160.81029)], \textit{G. Meng} showed that a (non-trivial) unitary highest weight module \(V\) of \(\mathfrak{so}(2,m+1)\) attains the smallest positive Gelfand-Kirillov dimension if and only if the action of the universal enveloping algebra \(\mathcal{U}(\mathfrak{so}(2,m+1))\) on \(V\) satisfies a certain quadratic relation. Observe that \(\mathfrak{so}(2, m+1)\) is the conformal algebra \(\mathfrak{co}(\Gamma(n))\) of the simple Euclidean Jordan algebra \(\Gamma(n)\). In this paper the author shows that such a relationship also holds for other simple Euclidean Jordan algebras, namely, \(\mathcal{H}_n(\mathbb{R})\), \(\mathcal{H}_n(\mathbb{C})\), \(\mathcal{H}_n(\mathbb{H})\), and \(\mathcal{H}_3(\mathbb{O})\).NEWLINENEWLINEOne of the main ideas of the proof is the use of classification results on the highest weights that attain the smallest Gelfand-Kirillov dimensions for \(\mathfrak{co}(J)\) for simple Euclidean Jordan algebras \(J\). See, for example, [\textit{T. J. Enright} and \textit{M. Hunziker}, Represent. Theory 8, 15--51 (2004; Zbl 1054.22009); \textit{T. J. Enright} and \textit{J. F. Willenbring}, Ann. Math. (2) 159, No. 1, 337--375 (2004; Zbl 1087.22011); \textit{A. Joseph}, Ann. Sci. Éc. Norm. Supér. (4) 25, No. 1, 1--45 (1992; Zbl 0752.17007)].