Symmetric weights and \(s\)-representations
In his classification of irreducible polar representations, \textit{J. Dadok} [Trans. Am. Math. Soc. 288, 125-137 (1985; Zbl 0565.22010)] assigned to an irreducible representation of a compact Lie group \(K\) with highest weight \(\lambda \) an integer \(k(\lambda)\). This can be characterized as the smallest positive integer \(l\) such that \(w\in U^l v_\lambda\) where \(w\) and \(v_\lambda\) are vectors of smallest respectively highest weight and \(U^l\) is the \(l\)-th filtration of the enveloping algebra of the Lie algebra of \(K\). Dadok proved that \(k(\lambda)\leq 4\) resp. \(k(\lambda)\leq 2\) for polar representations of real resp. complex type. Using results of \textit{M. Wang} and \textit{W. Ziller} [Symmetric spaces and strongly isotropy irreducible spaces, Math. Ann. 296, No. 2, 285-326 (1993; Zbl 0804.53075)] the authors show that a (not necessarily polar) faithful irreducible representation \(V\) of a compact connected Lie group \(K\) with symmetric highest weight \(\lambda \) is the isotropy representation of a symmetric space if \(k(\lambda)=4\) and \(V\) is of real type. If \(k(\lambda)=2\) and \(V\) is of complex type then \(V\) is the isotropy representation of a Hermitian symmetric space. The authors give a geometric interpretation of the upper bound of \(k(\lambda)\) in terms of the osculating spaces of the orbits. They show that the values \(k(\lambda)=4\) resp. \(k(\lambda)=2\) are maximal for representations of class \({\mathcal O}_2\) (such as polar representations). Generally, an irreducible orthogonal representation \(V\) is of class \({\mathcal O}_d\) if \(V\) is spanned by the derivatives \(c^{(l)}(0)\) of order \(l\leq d\) of curves \(c\) in an orbit \(Kp\subset V\) with \(p=c(0)\neq 0\).
scientific article; zbMATH DE number 1547130
scientific article