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Secant varieties of adjoint varieties: Orbit decomposition. (English) Zbl 0986.14034
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 defect, that is, their dimension is less than the generic dimension. This was discussed in a previous paper by the first author [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.

##### MSC:
 14N30 Adjunction problems 14M17 Homogeneous spaces and generalizations 14N15 Classical problems, Schubert calculus 14N10 Enumerative problems (combinatorial problems) in algebraic geometry 17B40 Automorphisms, derivations, other operators for Lie algebras and super algebras
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