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Corpus ID: 222272257

Abelian Ideals and the Variety of Lagrangian Subalgebras

@article{Evens2020AbelianIA,
title={Abelian Ideals and the Variety of Lagrangian Subalgebras},
author={S. Evens and Y. Li},
journal={arXiv: Representation Theory},
year={2020}
}

For a semisimple algebraic group $G$ of adjoint type with Lie algebra $\mathfrak g$ over the complex numbers, we establish a bijection between the set of closed orbits of the group $G \ltimes \mathfrak g^{\ast}$ acting on the variety of Lagrangian subalgebras of $\mathfrak g \ltimes \mathfrak g^{\ast}$ and the set of abelian ideals of a fixed Borel subalgebra of $\mathfrak g$. In particular, the number of such orbits equals $2^{\text{rk} \mathfrak g}$ by Peterson's theorem on abelian ideals.