Commutativity of invariant differential operators on vector bundles on Hermitian symmetric spaces
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866912907704401920 |
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| author | van Haastrecht, Robin Zhang, Genkai Zhao, Yufeng |
| author_facet | van Haastrecht, Robin Zhang, Genkai Zhao, Yufeng |
| contents | Let $G/K$ be a Hermitian symmetric space and $V_τ$ an irreducible representation of $K$. We study the ring $\mathcal D^G(G/K, V_τ)$ of $G$-invariant differential operators on sections of vector bundles $G\times_{(K, τ)} V_τ$ over $G/K$ defined by a finite-dimensional representation $(V_τ, τ)$ of $K$. We classify irreducible representations $(V_τ, τ)$ such that $\mathcal D^G(G/K, V_τ)$ is commutative. We construct eigenfunctions for the differential operators and study the invariance property of the eigenvalues under the Weyl group for the restricted real root system of $G$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_14864 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Commutativity of invariant differential operators on vector bundles on Hermitian symmetric spaces van Haastrecht, Robin Zhang, Genkai Zhao, Yufeng Representation Theory Let $G/K$ be a Hermitian symmetric space and $V_τ$ an irreducible representation of $K$. We study the ring $\mathcal D^G(G/K, V_τ)$ of $G$-invariant differential operators on sections of vector bundles $G\times_{(K, τ)} V_τ$ over $G/K$ defined by a finite-dimensional representation $(V_τ, τ)$ of $K$. We classify irreducible representations $(V_τ, τ)$ such that $\mathcal D^G(G/K, V_τ)$ is commutative. We construct eigenfunctions for the differential operators and study the invariance property of the eigenvalues under the Weyl group for the restricted real root system of $G$. |
| title | Commutativity of invariant differential operators on vector bundles on Hermitian symmetric spaces |
| topic | Representation Theory |
| url | https://arxiv.org/abs/2602.14864 |