Algebraic and analytic properties of invariant differential operators on a homogeneous space of complexity $1$
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arXiv
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866916936266285056 |
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| author | Fang, Hanlong Li, Xiaocheng Zhang, Yunfeng |
| author_facet | Fang, Hanlong Li, Xiaocheng Zhang, Yunfeng |
| contents | Denote by $SL_3(\mathbb R)$ the special linear group of degree 3 over the real numbers, $A$ the subgroup consisting of the diagonal matrices with positive entries. In this paper, we study the algebraic and analytic properties of the invariant differential operators on the homogeneous space $SL_3(\mathbb R)/A$. Firstly, we specify the noncommutative algebra of invariant differential operators in terms of generators and their relations. Secondly, we describe the center of this algebra and prove that all of its symmetric elements are essentially self-adjoint. Thirdly, for the first time on homogeneous spaces, we identify several essentially self-adjoint invariant differential operators which do not lie in the center of the algebra of invariant differential operators. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_00529 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Algebraic and analytic properties of invariant differential operators on a homogeneous space of complexity $1$ Fang, Hanlong Li, Xiaocheng Zhang, Yunfeng Representation Theory 43A85, 22E46, 22E30 Denote by $SL_3(\mathbb R)$ the special linear group of degree 3 over the real numbers, $A$ the subgroup consisting of the diagonal matrices with positive entries. In this paper, we study the algebraic and analytic properties of the invariant differential operators on the homogeneous space $SL_3(\mathbb R)/A$. Firstly, we specify the noncommutative algebra of invariant differential operators in terms of generators and their relations. Secondly, we describe the center of this algebra and prove that all of its symmetric elements are essentially self-adjoint. Thirdly, for the first time on homogeneous spaces, we identify several essentially self-adjoint invariant differential operators which do not lie in the center of the algebra of invariant differential operators. |
| title | Algebraic and analytic properties of invariant differential operators on a homogeneous space of complexity $1$ |
| topic | Representation Theory 43A85, 22E46, 22E30 |
| url | https://arxiv.org/abs/2301.00529 |