The index of subalgebras and strange coadjoint orbits
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866913168487350272 |
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| author | Panyushev, Dmitri I. |
| author_facet | Panyushev, Dmitri I. |
| contents | 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 $\mathfrak q^ξ$ denote the stabiliser of $ξ\in\mathfrak q^*$. In the special case when $\mathfrak q^ξ\oplus\mathfrak h=\mathfrak q$, our estimate implies that $\mathsf{ind\,}\mathfrak h=0$. Using our theory, we also answer a question of Duflo. An orbit $Q{\cdot}η\subset\mathfrak q^*$ is said to be strange, if $\mathfrak q^η\oplus\mathfrak h=\mathfrak q$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_28796 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The index of subalgebras and strange coadjoint orbits Panyushev, Dmitri I. Representation Theory 17B08, 17B20, 14L30, 17B63 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 $\mathfrak q^ξ$ denote the stabiliser of $ξ\in\mathfrak q^*$. In the special case when $\mathfrak q^ξ\oplus\mathfrak h=\mathfrak q$, our estimate implies that $\mathsf{ind\,}\mathfrak h=0$. Using our theory, we also answer a question of Duflo. An orbit $Q{\cdot}η\subset\mathfrak q^*$ is said to be strange, if $\mathfrak q^η\oplus\mathfrak h=\mathfrak q$ 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. |
| title | The index of subalgebras and strange coadjoint orbits |
| topic | Representation Theory 17B08, 17B20, 14L30, 17B63 |
| url | https://arxiv.org/abs/2605.28796 |