The index of subalgebras and strange coadjoint orbits

Abstract

For an algebraic group $Q$ with $\mathsf{Lie}Q=\mathfrak{q}$, we develop a method for estimating the index of a subalgebra $\mathfrak{h}$ in $\mathfrak{q}$ via the use of coadjoint $Q$-orbits in ${\mathfrak{q}}^{\ast }$. Let ${\mathfrak{q}}^{\xi }$ denote the stabiliser of $\xi \in {\mathfrak{q}}^{\ast }$. In the special case when ${\mathfrak{q}}^{\xi }\oplus \mathfrak{h}=\mathfrak{q}$, our estimate implies that $\mathsf{ind}\mathfrak{h}=0$. Using our theory, we also answer a question of Duflo. An orbit $Q\cdot \eta \subset {\mathfrak{q}}^{\ast }$ is said to be strange, if ${\mathfrak{q}}^{\eta }\oplus \mathfrak{h}=\mathfrak{q}$ for some $\mathfrak{h}$. In the second part of the paper, we study strange orbits for a semisimple algebra $\mathfrak{g}$. It is shown that an orbit $\mathcal{O}\subset \mathfrak{g}\simeq {\mathfrak{g}}^{\ast }$ is strange whenever the complexity of $\mathcal{O}$ is at most 1. Furthermore, if $\mathcal{S}\subset \mathfrak{g}$ is a sheet containing a strange nilpotent orbit, then all orbits in $\mathcal{S}$ are strange. We also show that strange orbits in ${\mathfrak{sl}}_n$ are not as sparse, as one might expect, and discuss some conjectures on strange orbits.

25 pp