Projections of Nilpotent Orbits in a Simple Lie Algebra and Shared Orbits

Let G be a simple algebraic group and \(\mathcal {O}\subset {\mathfrak g}={\mathrm {Lie\,}}G\) a nilpotent orbit. If H is a reductive subgroup of G with \({\mathfrak h}={\mathrm {Lie\,}}H\), then \({\mathfrak g}={\mathfrak h}\oplus {\mathfrak m}\), where \({\mathfrak m}={\mathfrak h}^\perp \). We consider the natural projections \(\varvec{\varphi }: \overline{\mathcal {O}}\rightarrow \mathfrak {h}\) and \(\varvec{\psi }: \overline{\mathcal {O}}\rightarrow \mathfrak {m}\) and two related properties of \((H, \mathcal {O})\):

$$ (\mathcal {P}_1): \overline{\mathcal {O}}\cap {\mathfrak m}=\{0\}; \qquad (\mathcal {P}_2): H \text { has a dense orbit in } \mathcal {O}. $$

It is shown that either of these properties implies that H is semisimple. We prove that \((\mathcal {P}_1)\) implies \((\mathcal {P}_2)\) for all \(\mathcal {O}\) and the converse holds for \(\mathcal {O}_\textsf{min}\), the minimal nilpotent orbit. If \((\mathcal {P}_1)\) holds, then \(\varvec{\varphi }\) is finite and \([\varvec{\varphi }(e),\varvec{\psi }(e)]=0\) for all \(e\in \mathcal {O}\). Then \(\overline{\varvec{\varphi }(\mathcal {O})}\) is the closure of a nilpotent H-orbit \(\mathcal {O}'\). The orbit \(\mathcal {O}'\) is “shared” in the sense of Brylinski–Kostant (J. Am. Math. Soc. 7(2), 269–298 1994). We obtain a classification of all pairs \((H,\mathcal {O})\) with property \((\mathcal {P}_1)\) and discuss various relations between \(\mathcal {O}\) and \(\mathcal {O}'\). In particular, we detect an omission in the list of pairs of simple groups (H, G) having a shared orbit that was given by Brylinski and Kostant. It is also proved that \((\mathcal {P}_1)\) for \((H,\mathcal {O}_\textsf{min})\) implies that \(\overline{G{\cdot }\varvec{\varphi }(\mathcal {O}_\textsf{min})}=\overline{G{\cdot }\varvec{\psi }(\mathcal {O}_\textsf{min})}\).