Quantum Hamiltonian Reduction for Polar Representations
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
| Veröffentlicht: |
2021
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| _version_ | 1866929296581328896 |
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| author | Bellamy, G. Levasseur, T. Nevins, T. Stafford, J. T. |
| author_facet | Bellamy, G. Levasseur, T. Nevins, T. Stafford, J. T. |
| contents | Let $G$ be a reductive complex Lie group with Lie algebra $\mathfrak{g}$ and suppose that $V$ is a polar $G$-representation. We prove the existence of a radial parts map $\mathrm{rad}: \mathcal{D}(V)^G\to A_κ$ from the $G$-invariant differential operators on $V$ to the spherical subalgebra $A_κ$ of a rational Cherednik algebra. Under mild hypotheses $\mathrm{rad}$ is shown to be surjective.
If $V$ is a symmetric space, then $\mathrm{rad}$ is always surjective, and we determine exactly when $A_κ$ is a simple ring. When $A_κ$ is simple, we also show that the kernel of $\mathrm{rad}$ is $\left(\mathcal{D}(V)τ(\mathfrak{g}\right)^G$, where $τ:\mathfrak{g}\to \mathcal{D}(V)$ is the differential of the $G$-action. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2109_11467 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Quantum Hamiltonian Reduction for Polar Representations Bellamy, G. Levasseur, T. Nevins, T. Stafford, J. T. Representation Theory 13N10, 16S32, 16S80, 20G05, 22E46 Let $G$ be a reductive complex Lie group with Lie algebra $\mathfrak{g}$ and suppose that $V$ is a polar $G$-representation. We prove the existence of a radial parts map $\mathrm{rad}: \mathcal{D}(V)^G\to A_κ$ from the $G$-invariant differential operators on $V$ to the spherical subalgebra $A_κ$ of a rational Cherednik algebra. Under mild hypotheses $\mathrm{rad}$ is shown to be surjective. If $V$ is a symmetric space, then $\mathrm{rad}$ is always surjective, and we determine exactly when $A_κ$ is a simple ring. When $A_κ$ is simple, we also show that the kernel of $\mathrm{rad}$ is $\left(\mathcal{D}(V)τ(\mathfrak{g}\right)^G$, where $τ:\mathfrak{g}\to \mathcal{D}(V)$ is the differential of the $G$-action. |
| title | Quantum Hamiltonian Reduction for Polar Representations |
| topic | Representation Theory 13N10, 16S32, 16S80, 20G05, 22E46 |
| url | https://arxiv.org/abs/2109.11467 |