## Self-dual projective algebraic varieties associated with symmetric spaces.(English)Zbl 1093.14072

Popov, Vladimir L. (ed.), Algebraic transformation groups and algebraic varieties. Proceedings of the conference on interesting algebraic varieties arising in algebraic transformation group theory, Vienna, Austria, October 22–26, 2001. Berlin: Springer (ISBN 3-540-20838-0/hbk). Encyclopaedia of Mathematical Sciences 132. Invariant Theory and Algebraic Transformation Groups 3, 131-167 (2004).
Let $$G$$ be a complex semi-simple algebraic group, and $${\mathcal G}$$ its Lie algebra. Let $$\theta$$ be an involution of $${\mathcal G}$$. Let $${\mathcal P}$$ be the $$-1$$ eigenvectors of $$\theta$$. Let $$N({\mathcal G})$$, $$N({\mathcal P})$$ be the variety of nilpotent elements of $${\mathcal G},{\mathcal P}$$ respectively. In this paper the authors give a geometric characterization of those $$G$$-orbit closures in $$N({\mathcal G})$$, $$N({\mathcal P})$$ whose projectivizations are self-dual. This paper makes an important contribution to the variety of nilpotent elements.
For the entire collection see [Zbl 1051.14003].

### MSC:

 14M17 Homogeneous spaces and generalizations 14L30 Group actions on varieties or schemes (quotients) 14N05 Projective techniques in algebraic geometry 17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)

### Keywords:

semi-simple Lie algebra; self-dual symmetric spaces
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