Projective Duality and Principal Nilpotent Elements of Symmetric Pairs

It is shown that projectivized irreducible components of nilpotent cones of complex symmetric spaces are projective self-dual algebraic varieties. Other properties equivalent to their projective self-duality are found. 1. Let g be a semisimple complex Lie algebra, let G be the adjoint group of g, and let θ ∈ Aut g be an element of order 2. We set k := {x ∈ g | θ(x) = x}, p := {x ∈ g | θ(x) = −x}. Then k and p 6= 0, the subalgebra k is reductive, and g = k⊕p is a Z2-grading of the Lie algebra g, cf., e.g., [OV]. Denote by G the adjoint group of g. The connected reductive algebraic subgroup K of G with the Lie algebra k is the adjoint group of k. Denoting the automorphism of G induced by θ also by θ, let Kθ be the fixed point group of θ. Then K is the identity component of Kθ. Let N (g) and N (p) be Zariski closed sets of all nilpotent elements in g and p respectively. They are cones (i.e., stable with respect to scalar multiplications and contain 0). We have N (p) = N (g) ∩ p. The cone N (g) is irreducible, [K2], but N (p), in general, is not, cf., e.g. [Se]. Consider the adjoint action of G on g. Then p and N (p) are Kθ-stable. There are only finitely many G-orbits (resp., K-orbits) in N (g) (resp., N (p)), [Dy], [K1], [KR]. Therefore N (g) (resp., every irreducible component of N (p)) contains an open G-orbit N (g)pr (resp., K-orbit). Its elements are called principal nilpotent elements of g (resp., p). All principal nilpotent elements of p constitute a single 1991 Mathematics Subject Classification. Primary 14L, 14M17, 17B70; Secondary 17B20, 14N05.