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

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 $\mathfrak g$.