Coadjoint orbits of low dimension for nilradicals of Borel subalgebras in classical types
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908469263597568 |
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| author | Ignatev, Mikhail Petukhov, Alexey |
| author_facet | Ignatev, Mikhail Petukhov, Alexey |
| contents | Let $\mathfrak g$ be a classical simple Lie algebra over an algebraically closed field $\mathbb F$ of characteristic zero or large enough, and let $\mathfrak n$ be a maximal nilpotent subalgebra of $\mathfrak g$. The main tool in representation theory of $\mathfrak n$ is the orbit method, which classifies primitive ideals in the universal enveloping algebra ${\rm U}(\mathfrak n)$ and unitary representations of the unipotent group $N=\exp(\mathfrak n)$ in terms of coadjoint orbits on the dual space $\mathfrak n^*$. In the paper, we describe explicitly coadjoint orbits of low dimension for $\mathfrak n$ as above. The answer is given in terms of subsets of positive roots. As a corollary, we provide a way to calculate the number of irreducible complex representations of dimensions $q$, $q^2$ and $q^3$ for a maximal unipotent subgroup $N(q)$ in a classical Chevalley group $G(q)$ over a finite field $\mathbb F_q$ with $q$ elements. It turned out that this number is a polynomial in $q-1$ with nonnegative integer coefficients, which agrees with Isaac's conjecture. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_20332 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Coadjoint orbits of low dimension for nilradicals of Borel subalgebras in classical types Ignatev, Mikhail Petukhov, Alexey Representation Theory Group Theory Rings and Algebras 16D60, 22E65, 22E25, 22E30, 53D17 Let $\mathfrak g$ be a classical simple Lie algebra over an algebraically closed field $\mathbb F$ of characteristic zero or large enough, and let $\mathfrak n$ be a maximal nilpotent subalgebra of $\mathfrak g$. The main tool in representation theory of $\mathfrak n$ is the orbit method, which classifies primitive ideals in the universal enveloping algebra ${\rm U}(\mathfrak n)$ and unitary representations of the unipotent group $N=\exp(\mathfrak n)$ in terms of coadjoint orbits on the dual space $\mathfrak n^*$. In the paper, we describe explicitly coadjoint orbits of low dimension for $\mathfrak n$ as above. The answer is given in terms of subsets of positive roots. As a corollary, we provide a way to calculate the number of irreducible complex representations of dimensions $q$, $q^2$ and $q^3$ for a maximal unipotent subgroup $N(q)$ in a classical Chevalley group $G(q)$ over a finite field $\mathbb F_q$ with $q$ elements. It turned out that this number is a polynomial in $q-1$ with nonnegative integer coefficients, which agrees with Isaac's conjecture. |
| title | Coadjoint orbits of low dimension for nilradicals of Borel subalgebras in classical types |
| topic | Representation Theory Group Theory Rings and Algebras 16D60, 22E65, 22E25, 22E30, 53D17 |
| url | https://arxiv.org/abs/2507.20332 |