Self–Dual Algebraic Varieties and Nilpotent Orbits

We give a construction of nonsmooth self-dual projective algebraic varieties. They appear as certain projectivized orbit closures for some linear actions of reductive algebraic groups. Applying this construction to adjoint representations, we obtain geometric characterization of distinguished nilpotent elements of semisimple Lie algebras [BC1], [BC2] (i.e., nilpotent elements whose centralizer contains no nonzero semisimple elements) as nilpotent elements whose projectivized orbit closures are self-dual projective algebraic varieties. In particular we show that the projectivized nilpotent cone of every semisimple Lie algebra is a self-dual projective algebraic variety. We also apply this construction to isotropy representations of symmetric spaces and introduce the notion of (−1)-distinguished nilpotent element, the counterpart of the notion of distinguished element. The projectivized orbit closures of (−1)-distinguished elements are self-dual projective algebraic varieties as well. Under different guises dual varieties of projective algebraic varieties have been considered in various branches of mathematics for over a hundred years. In fact, the dual variety is the generalization to algebraic geometry of the Legendre transform in classical mechanics, and the biduality theorem essentially ∗ Supported in part by INTAS, grant INTAS–OPEN–97–1570, and by The Erwin Schrödinger International Institute for Mathematical Physics (Austria). 1991 Mathematics Subject Classification. 14Mxx, 14L30.