## Journal of Mathematics of Kyoto University

### The secant varieties of nilpotent orbits

Yasuhiro Omoda

#### Abstract

Let $\mathfrak{g}$ be a complex simple Lie algebra. We have the adjoint representation of the adjoint group $G$ on $\mathfrak{g}$. Then $G$ acts on the projective space $\mathbb{P}_{\mathfrak{g}}$. We consider the closure $X$ of the image of a nilpotent orbit in $\mathbb{P}_{\mathfrak{g}}$. The $i$-secant variety $Sec^{(i)}X$ of a projective variety $X$ is the closure of the union of projective subspaces of dimension $i$ in the ambient space $\mathbb{P}$ spanned by $i+1$ points on $X$. In particular we call the 1-secant variety the secant variety. In this paper we give explicit descriptions of the secant and the higher secant varieties of nilpotent orbits of complex classical simple Lie algebras.

#### Article information

Source
J. Math. Kyoto Univ., Volume 48, Number 1 (2008), 49-71.

Dates
First available in Project Euclid: 14 August 2009

https://projecteuclid.org/euclid.kjm/1250280975

Digital Object Identifier
doi:10.1215/kjm/1250280975

Mathematical Reviews number (MathSciNet)
MR2437891

Zentralblatt MATH identifier
1170.14037

#### Citation

Omoda, Yasuhiro. The secant varieties of nilpotent orbits. J. Math. Kyoto Univ. 48 (2008), no. 1, 49--71. doi:10.1215/kjm/1250280975. https://projecteuclid.org/euclid.kjm/1250280975