Secant varieties of adjoint varieties: Orbit decomposition
Let \(\mathfrak g\) be a complex simple Lie algebra. By the adjoint variety \(X(\mathfrak{g})\) the authors mean the unique closed orbit of the action of \(G\) (the inner automorphism group of \(\mathfrak g\)) on \({\mathfrak P}(\mathfrak g) \) induced by the adjoint representation of \(G\) on \(\mathfrak g\). Adjoint varieties are interesting because their secant varieties have \textit{defect}, that is, their dimension is less than the generic dimension. This was discussed in a previous paper by the first author [\textit{H. Kaji}, Math. Contemp. 14, 75-87 (1998; Zbl 0939.14030)]. The action of \(G\) on \(X(\mathfrak g)\) lifts to an action on the secant variety of \(X(\mathfrak g)\), and the purpose of the present paper is to give the orbit decomposition of this action for the different complex simple Lie algebras. A structure theorem is proved, which states that the secant variety decomposes as the union of a dense orbit and a finite number of projectivizations of nilpotent orbits. The nilpotent orbits have a unique maximal orbit relative to the closure ordering. A complete classification of these maximal orbits is given.
scientific article; zbMATH DE number 1448636
scientific article