On distinguished orbits of reductive representations
Let \(G\) be a real reductive Lie group and let \(\tau :G\to\text{GL}(V)\) be a real reductive representation of \(G\) with (restricted) moment map \(m_{\mathfrak{g}} :V\setminus\{0\}\to\mathfrak{g}\). The author introduces the notion of nice space of a real reductive representation to study the problem of how to determine if a \(G\)-orbit is distinguished (i.e. it contains a critical point of the norm squared of \(m_{\mathfrak{g}}\)). He gives an elementary proof of the well-known convexity theorem of Atiyah-Guillemin-Sternberg in the present context and uses it to give an easy-to-check sufficient condition for a \(G\)-orbit of an element in a nice space to be distinguished. In the case where \(G\) is algebraic and \(\tau\) is a rational representation, the above condition is also necessary (making heavy use of recent results of Michael Jablonski), obtaining a generalization of Nikolayevsky's nice basis criterion. The author also provides useful characterizations of nice spaces in terms of the weights of \(\tau\). Finally, some applications to ternary forms are presented.
scientific article; zbMATH DE number 6340912
scientific article