Abelian Ideals and the Variety of Lagrangian Subalgebras

Abstract

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 $G⋉{\mathfrak{g}}^{\ast }$ acting on the variety of Lagrangian subalgebras of $\mathfrak{g}⋉{\mathfrak{g}}^{\ast }$ 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.

References in corpus (1)

• Abelian ideals in a Borel subalgebra of a complex simple Lie algebra