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| Format: | Preprint |
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2007
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| Online Access: | https://arxiv.org/abs/0707.0570 |
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| _version_ | 1866909484149899264 |
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| author | Neretin, Yuri |
| author_facet | Neretin, Yuri |
| contents | We obtain explicit formulas for the spinor representation $ρ$ of the real orthosymplectic supergroup $\mathrm{OSp}(2p|2q,\mathbb{R})$ by integral 'Gauss--Berezin' operators. Next, we extend $ρ$ to a complex domain and get a representation of a larger semigroup, which is a counterpart of Olshanski subsemigroups in semisimple Lie groups. Further, we show that $ρ$ can be extended to an operator-valued function on a certain domain in the Lagrangian super-Grassmannian (graphs of elements of the supergroup $\mathrm{OSp}(2p|2q,\mathbb{C})$ are Lagrangian super-subspaces) and show that this function is a 'representation' in the following sense: we consider Lagrangian subspaces as linear relations and composition of two Lagrangian relations in general position corresponds to a product of Gauss--Berezin operators |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_0707_0570 |
| institution | arXiv |
| publishDate | 2007 |
| record_format | arxiv |
| spellingShingle | Gauss--Berezin integral operators and spinors over supergroups $\mathrm{OSp}(2p|2q)$, and Lagrangian super-Grasmannians Neretin, Yuri Representation Theory Mathematical Physics 17B10, 58A50, 30H20, 22E46 We obtain explicit formulas for the spinor representation $ρ$ of the real orthosymplectic supergroup $\mathrm{OSp}(2p|2q,\mathbb{R})$ by integral 'Gauss--Berezin' operators. Next, we extend $ρ$ to a complex domain and get a representation of a larger semigroup, which is a counterpart of Olshanski subsemigroups in semisimple Lie groups. Further, we show that $ρ$ can be extended to an operator-valued function on a certain domain in the Lagrangian super-Grassmannian (graphs of elements of the supergroup $\mathrm{OSp}(2p|2q,\mathbb{C})$ are Lagrangian super-subspaces) and show that this function is a 'representation' in the following sense: we consider Lagrangian subspaces as linear relations and composition of two Lagrangian relations in general position corresponds to a product of Gauss--Berezin operators |
| title | Gauss--Berezin integral operators and spinors over supergroups $\mathrm{OSp}(2p|2q)$, and Lagrangian super-Grasmannians |
| topic | Representation Theory Mathematical Physics 17B10, 58A50, 30H20, 22E46 |
| url | https://arxiv.org/abs/0707.0570 |