Almost commuting scheme of symplectic matrices and quantum Hamiltonian reduction
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866911961066766336 |
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| author | Goyal, Pallav |
| author_facet | Goyal, Pallav |
| contents | Losev introduced the scheme $X$ of almost commuting elements (i.e., elements commuting upto a rank one element) of $\mathfrak{g}=\mathfrak{sp}(V)$ for a symplectic vector space $V$ and discussed its algebro-geometric properties. We construct a Lagrangian subscheme $X^{nil}$ of $X$ and show that it is a complete intersection of dimension $\text{dim}(\mathfrak{g})+\frac{1}{2}\text{dim}(V)$ and compute its irreducible components.
We also study the quantum Hamiltonian reduction of the algebra $\mathcal{D}(\mathfrak{g})$ of differential operators on the Lie algebra $\mathfrak{g}$ tensored with the Weyl algebra with respect to the action of the symplectic group, and show that it is isomorphic to the spherical subalgebra of a certain rational Cherednik algebra of Type $C$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2212_13436 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Almost commuting scheme of symplectic matrices and quantum Hamiltonian reduction Goyal, Pallav Representation Theory Algebraic Geometry Quantum Algebra Losev introduced the scheme $X$ of almost commuting elements (i.e., elements commuting upto a rank one element) of $\mathfrak{g}=\mathfrak{sp}(V)$ for a symplectic vector space $V$ and discussed its algebro-geometric properties. We construct a Lagrangian subscheme $X^{nil}$ of $X$ and show that it is a complete intersection of dimension $\text{dim}(\mathfrak{g})+\frac{1}{2}\text{dim}(V)$ and compute its irreducible components. We also study the quantum Hamiltonian reduction of the algebra $\mathcal{D}(\mathfrak{g})$ of differential operators on the Lie algebra $\mathfrak{g}$ tensored with the Weyl algebra with respect to the action of the symplectic group, and show that it is isomorphic to the spherical subalgebra of a certain rational Cherednik algebra of Type $C$. |
| title | Almost commuting scheme of symplectic matrices and quantum Hamiltonian reduction |
| topic | Representation Theory Algebraic Geometry Quantum Algebra |
| url | https://arxiv.org/abs/2212.13436 |