Projections of nilpotent orbits in a simple Lie algebra and shared orbits

TL;DR

This paper studies projections of nilpotent orbits in simple Lie algebras, establishing conditions for shared orbits, classifying pairs with certain properties, and correcting previous classifications in the literature.

Contribution

It introduces new criteria for shared nilpotent orbits via projections, classifies pairs with these properties, and identifies an omission in prior classifications.

Findings

Property (P1) implies (P2) for all orbits.

Classification of pairs (H, O) with property (P1).

Correction of previous classification in the literature.

Abstract

Let $G$ be a simple algebraic group with $\mathfrak g=Lie(G)$ and $\mathcal O\subset\mathfrak g$ a nilpotent orbit. If $H$ is a reductive subgroup of $G$ with $Lie(H)=\mathfrak h$, then $\mathfrak g=\mathfrak h\oplus\mathfrak m$, where $\mathfrak m=\mathfrak h^\perp$. We consider the natural projections $\phi: \bar{\mathcal O}\to\mathfrak h$ and $\psi:\bar{\mathcal O}\to\mathfrak m$, and two related properties of the pair $(H,\mathcal O)$: $(P_1)$: $\bar{\mathcal O}\cap\mathfrak m={0}$ and $(P_2)$: $H$ has a dense orbit in $\mathcal O$. We show that $(P_1)$ implies $(P_2)$ for all $\mathcal O$ and these properties are equivalent for $\mathcal O=\mathcal O_{min}$, the minimal nilpotent orbit. If $(P_1)$ holds, then $\phi$ is finite, and $\phi(\bar{\mathcal O})$ is the closure of a nilpotent H-orbit $\mathcal O'$. We prove that $\mathcal O$ is contained in the closure of the G-orbit $G{\cdot}\mathcal O'$ and obtain the classification of pairs $(H,\mathcal O)$ with property $(P_1)$. The orbit $\mathcal O'$ is "shared" in the sense of Brylinski and Kostant. Using our classification, we detect an omission in the list of pairs $(H,G)$ having a shared orbit that is given in "Nilpotent orbits, normality, and hamiltonian group actions", J.A.M.S., 7 (1994), 269--298. It is also proved that if $(P_1)$ holds for $(H, \mathcal O_{min})$, then both varieties $\phi(\mathcal O_{min})$ and $\psi(\mathcal O_{min})$ generate the same closed subvariety of $\mathfrak g$.