Orbits and invariants for coisotropy representations
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909442783576064 |
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| author | Panyushev, Dmitri I. |
| author_facet | Panyushev, Dmitri I. |
| contents | For a subgroup $H$ of a reductive group $G$, let $\mathfrak m\subset \mathfrak g^*$ be the cotangent space of $eH\in G/H$. The linear action $(H:\mathfrak m)$ is the coisotropy representation. It is known that the complexity and rank of $G/H$ (denoted $c$ and $r$, respectively) are encoded in properties of $(H:\mathfrak m)$. We complement existing results on $c$, $r$, and $(H:\mathfrak m)$, especially for quasiaffine varieties $G/H$. If the algebra of invariants $k[\mathfrak m]^H$ is finitely generated, then we establish a connection between the nullcones in $\mathfrak m$ and $\mathfrak g^*$. Two other topics considered are (i) a relationship between varieties $G/H$ of complexity at most 1 and the homological dimension of the algebra of invariants $k[\mathfrak m]^H$ and (ii) the Poisson structure of $k[\mathfrak m]^H$ and Poisson-commutative subalgebras in $k[\mathfrak m]^H$ with maximal transcendence degree. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_01897 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Orbits and invariants for coisotropy representations Panyushev, Dmitri I. Representation Theory Algebraic Geometry 14L30, 14R20, 14M27, 17B20 For a subgroup $H$ of a reductive group $G$, let $\mathfrak m\subset \mathfrak g^*$ be the cotangent space of $eH\in G/H$. The linear action $(H:\mathfrak m)$ is the coisotropy representation. It is known that the complexity and rank of $G/H$ (denoted $c$ and $r$, respectively) are encoded in properties of $(H:\mathfrak m)$. We complement existing results on $c$, $r$, and $(H:\mathfrak m)$, especially for quasiaffine varieties $G/H$. If the algebra of invariants $k[\mathfrak m]^H$ is finitely generated, then we establish a connection between the nullcones in $\mathfrak m$ and $\mathfrak g^*$. Two other topics considered are (i) a relationship between varieties $G/H$ of complexity at most 1 and the homological dimension of the algebra of invariants $k[\mathfrak m]^H$ and (ii) the Poisson structure of $k[\mathfrak m]^H$ and Poisson-commutative subalgebras in $k[\mathfrak m]^H$ with maximal transcendence degree. |
| title | Orbits and invariants for coisotropy representations |
| topic | Representation Theory Algebraic Geometry 14L30, 14R20, 14M27, 17B20 |
| url | https://arxiv.org/abs/2405.01897 |