Abelian Ideals and the Variety of Lagrangian Subalgebras

TL;DR

This paper establishes a bijection between certain group orbits on Lagrangian subalgebras and abelian ideals in a Borel subalgebra, revealing a deep connection in the structure of semisimple Lie algebras.

Contribution

It introduces a novel correspondence between orbits of a semidirect product group action and abelian ideals, linking geometric and algebraic structures in Lie theory.

Findings

Number of orbits equals 2^{rank of g}

Bijection between orbits and abelian ideals

Connection to Peterson's theorem

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 $G \ltimes \mathfrak g^{\ast}$ 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.