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 $\mathfrak q^*$. Let $\mathfrak q^\xi$ denote the stabiliser of $\xi\in\mathfrak q^*$. 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 $Q{\cdot}\eta\subset\mathfrak q^*$ 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 ${\mathcal O}\subset\mathfrak g\simeq\mathfrak g^*$ 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.