Projections of nilpotent orbits in a simple Lie algebra and shared orbits
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2024
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| _version_ | 1866914971380613120 |
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| author | Panyushev, Dmitri I. |
| author_facet | Panyushev, Dmitri I. |
| contents | 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 $ϕ: \bar{\mathcal O}\to\mathfrak h$ and $ψ:\bar{\mathcal O}\to\mathfrak m$, 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 $ϕ$ is finite, and $ϕ(\bar{\mathcal O})$ 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 $ϕ(\mathcal O_{min})$ and $ψ(\mathcal O_{min})$ generate the same closed subvariety of $\mathfrak g$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_09876 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Projections of nilpotent orbits in a simple Lie algebra and shared orbits Panyushev, Dmitri I. Representation Theory 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 $ϕ: \bar{\mathcal O}\to\mathfrak h$ and $ψ:\bar{\mathcal O}\to\mathfrak m$, 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 $ϕ$ is finite, and $ϕ(\bar{\mathcal O})$ 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 $ϕ(\mathcal O_{min})$ and $ψ(\mathcal O_{min})$ generate the same closed subvariety of $\mathfrak g$. |
| title | Projections of nilpotent orbits in a simple Lie algebra and shared orbits |
| topic | Representation Theory |
| url | https://arxiv.org/abs/2410.09876 |